diff --git a/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/2301.01945v1.pdf.txt b/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/2301.01945v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7180f446075b08f2a884d856288067f9a9f01e3b --- /dev/null +++ b/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/2301.01945v1.pdf.txt @@ -0,0 +1,3160 @@ +Optimizing density-functional simulations for two-dimensional metals +Kameyab Raza Abidi and Pekka Koskinen∗ +NanoScience Center, Department of Physics, University of Jyv¨askyl¨a, 40014 Jyv¨askyl¨a, Finland +(Dated: January 6, 2023) +Unlike covalent two-dimensional (2D) materials like graphene, 2D metals have non-layered struc- +tures due to their non-directional, metallic bonding. While experiments on 2D metals are still scarce +and challenging, density-functional theory (DFT) provides an ideal approach to predict their basic +properties and assist in their design. However, DFT methods have been rarely benchmarked against +metallic bonding at low dimensions. Therefore, to identify optimal DFT attributes for a desired +accuracy, we systematically benchmark exchange-correlation functionals from LDA to hybrids and +basis sets from plane waves to local basis with different pseudopotentials. With 1D chain, 2D hon- +eycomb, 2D square, 2D hexagonal, and 3D bulk metallic systems, we compare the DFT attributes +using bond lengths, cohesive energies, elastic constants, densities of states, and computational costs. +Although today most DFT studies on 2D metals use plane waves, our comparisons reveal that local +basis with often-used PBE exchange-correlation is well sufficient for most purposes, while plane +waves and hybrid functionals bring limited improvement compared to the greatly increased compu- +tational cost. These results ease the demands for generating DFT data for better interaction with +experiments and for data-driven discoveries of 2D metals incorporating machine learning algorithms. +I. +INTRODUCTION +The discovery of graphene nearly two decades ago +sparked an entire new research field of two-dimensional +(2D) materials [1]. The 2D materials pedigree has ex- +panded ever since, thanks to unique properties and vi- +sions for novel applications [2–5]. +Most 2D materials +are covalently bound and have layered structures eas- +ily exfoliable from three-dimensional (3D) bulk matter +[6, 7]. However, in contrast to directional covalent bond- +ing, non-directional metallic bonding prefers large coor- +dination numbers, which renders low-dimensional metal +structures energetically unfavourable. Despite this pref- +erence for large coordination, in 2014 atomically thin sta- +ble iron patches were discovered in graphene pores [8]. +This discovery has been followed by rapid progress in re- +search on 2D metals and alloys, making 2D metals a full +member the 2D materials family [9–14]. +The wavering stability of 2D metals makes experi- +ments challenging, whereby research relies heavily on +computations. A reasonable description of metallic bond- +ing requires electronic structure simulations, which has +made the density-functional theory (DFT) [15, 16] the +workhorse method for modeling 2D metals [17–31]. Most +DFT studies have chosen plane wave (PW) basis sets [32] +and the non-empirical Perdew-Burke-Ernzerhof (PBE) +exchange-correlation functional [33]. +These choices for +DFT attributes are plausible in the context of delocal- +ized electrons in periodic systems that are still lacking +experimental data. However, DFT attributes have not +been systematically benchmarked for metallic bonding +at low dimensions. It is not certain whether these stan- +dard choices are efficient and accurate enough or they if +simply waste computational resources. +∗ pekka.j.koskinen@jyu.fi +The DFT attributes consist of few central choices. The +first choice is the flavor of exchange-correlation (xc) func- +tional, the level of which is of central importance for con- +sistent results. A functional performing well in some sys- +tems may perform poorly in others. Here we make use +of several xc-functionals to obtain a systematic picture +of their performance in low-dimensional metallic bond- +ing [34]. The second choice is the type of basis function. +Plane waves are suitable for periodic systems, whose elec- +trons fill out the entire simulation cell. Unfortunately, +the non-periodic directions of low-dimensional systems +require large vacuum regions that make PW simulations +inefficient compared to modeling bulk. Thus, an addi- +tional choice in PW simulations is an optimum size of the +vacuum. In this respect, PW and grid-based DFT share +the same challenges [35, 36]. Another alternative for ba- +sis is linear combination of atomic orbitals (LCAO), and +controlling its size provides a powerful handle to trade +between accuracy and efficiency [37]. +The choice of basis type has implications beyond mere +accuracy. For example, PW is not suitable for studying +electron transport using nonequilibrium Green’s function +method in nanoscaled devices [38]. In addition, with the +coming of data science and machine learning in materials +science, lots of consistent DFT data is required for ma- +chine learning -enabled 2D metals studies [39–43]. This +efficiency demand calls for a critical examination of the +necessity of PW method to model metallic bonding in +low dimensions. +Third choice for periodic systems is the number of k- +points along periodic directions for the desired accuracy. +Fourth choice is the level of Fermi-broadening of elec- +tronic states, which is partly a physical choice but mostly +a necessity for rapid convergence of the self-consistent it- +eration of the electron density. In practice, there are a +plethora of other choices to make for numerical stability +and speedup, but they are often chosen as default val- +ues that have been previously fine-tuned for each DFT +arXiv:2301.01945v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +2 +FIG. 1. Schematics of the systems with different dimensional- +ities and coordination numbers C: 1D chain (C = 2), 2D hon- +eycomb (C = 3), 2D square (C = 4), 2D hexagonal (C = 6), +and 3D bulk (C = 12). The quadrilaterals show the simula- +tion cells. +code. In this article, we consider the above-mentioned +choices of DFT attributes regarding xc-functionals, basis +sets, vacuum, k-point sampling, and Fermi-broadening, +and juxtapose their performance against various prop- +erties of selected low-dimensional metal systems. +The +selected systems include a one-dimensional chain (coor- +dination number C = 2), three two-dimensional lattices +(C = 3, 4, and 6), and a 3D bulk (C = 12) (Figure 1). +These systems enable comparative analysis of the perfor- +mance of DFT attributes in various dimensions. Being +low-dimensional systems, these structures are prone to +various symmetry-breaking deformations, such as out-of- +plane buckling in 2D or Peierls distortions in 1D [26, 44]. +However, in order to enable unambiguous comparison of +the effect of dimensionality and coordination and avoid +making unfounded conclusions based on incomplete set +of deformations, we retain our focus on these ideal, non- +deformed systems. +We also compare the performance +and speed of DFT to the density-functional tight-binding +(DFTB) method, which is the next-in-line approximation +to DFT [45]. One of our main conclusions is that, for +general purposes, DFT-LCAO can be chosen over the de- +fault DFT-PW without compromising accuracy, a choice +which enables simulating transport and helps generating +DFT data more effortlessly. +Our treatise will advance +DFT modeling of 2D metals and help boosting the inter- +action with experiments. +II. +COMPUTATIONAL METHODS +The basic idea DFT is to use the variational principle +to generate exact ground state energy and density for the +systems of interest [15]. The ground state energy E is a +functional of the electron density (n), +E[n] = T[n] + Eext[n] + EH[n] + Exc[n] , +(1) +where T[n] is the Kohn-Sham kinetic energy for the fic- +titious non-interacting electron system, Eext[n] is the ex- +TABLE I. Exchange-correlation functionals used in this work. +Functional and its family +Refs. +Local Density Approximation (LDA) +[15, 55] +Generalized Gradient Approximation (GGA) +[56] +RPBE +[57] +PW91 +[58, 59] +PBE +[33] +Hybrid Functionals +[60] +B3LYP +[61] +PBE0 +[62] +HSE03 (screening ω = 0.15 Bohr−1) +[63] +HSE06 (screening ω = 0.11 Bohr−1) +[64] +ternal potential energy, EH[n] is the Hartree energy, and +Exc[n] is the exchange-correlation energy. The xc term +attempts to capture the complex features of many-body +quantum mechanics, and a variety of approximate xc +functionals have been developed for different purposes +[34]. +As a result, the quality of xc functional mostly +determines the quality of the results. +Here, using the +QuantumATK (S-2021.06) DFT implementation [46], we +explore the set of eight xc functionals ranging from local +density approximation to hybrid functionals (Table I). +We used two types of basis sets, plane waves and +LCAOs. The wave-function energy cutoff for plane waves +was 800 eV. Cutoff needed no separate analysis for low- +dimensional metals, because it depends only on element +and pseudopotential [47]. +For LCAOs, we used three +variants: LCAO-M(edium), LCAO-H(igh), and LCAO- +U(ltra). These variants derive from the numerical basis +sets of the FHI-aims package [48], but are further opti- +mized for computational speed of the LCAO calculator. +For example, for Ag the radial functions for Medium ba- +sis are 3s/2p/1d (14), for High 4s/3p/5d/1f (35), and +for Ultra 4s/3p/5d/2f/1g (51), with brackets displaying +the total number of orbitals per atom [37, 48]. Local ba- +sis sets were used in conjunction with norm-conserving +PseudoDojo pseudopotentials [49]. +Further, the total energy convergence criteria for self- +consistent electron density was ≤ 10−7eV. System ge- +ometries were optimized to forces below 1 meV�A +−1 and +stresses below 0.3 meV�A +−3 using the LBFGS [50] algo- +rithm. +The k-points were sampled by the Monkhorst- +Pack method [51]. All calculations were spin-polarized +and the initial guess for lattice parameters were adopted +from the Atlas of 2D metals [20]. +To complement the results with various DFT at- +tributes with wider context, we analyzed the systems +with Ag also with DFTB method at the level of self- +consistent charge [45, 52]. The Ag parametrizations were +taken from earlier studies [53, 54]. + +(a) 1D Chain +(b) 2D Honeycomb (hc) +(c) 2D Square (sq) +OODD +(d) 2D Hexagonal (hex) +(e) 3D Bulk +X3 +III. +RESULTS AND DISCUSSION +A. +Convergence Analysis +We made various systematic convergence analyses for +the group of coinage metals Cu, Ag, and Au [17–19]. +Computational and experimental studies have shown +that the free-standing monolayer patches of these met- +als are stabilized by graphene pores [13, 22, 24, 31]. The +analyses were done using PBE xc-functional [33], projec- +tor augmented waves (PAW) for core electrons [65], and +plane waves for valence electrons. +a. +k-point convergence: +The k-point convergence +was studied using the 2D systems with a converged vac- +uum of 15 �A in the non-periodic direction (as confirmed +below). +The total energy is practically converged at +30 × 30 × 1 k-point sampling, and we define the energy +tolerance using this value, +∆E = ENk×Nk×1 − E30×30×1 . +(2) +Apart from rapid convergence at very few k-points, the +convergence is exponential. Chosen relative energy tol- +erance can therefore be approximated by +log δ = A1 + B1L , +(3) +where δ =| ∆E | /E3D is an (approximate) relative en- +ergy tolerance, the ratio between energy tolerance to the +3D cohesive energy E3D [66]. The length L = acNk, the +product of simulation box length and the number of k- +points in corresponding direction, is the maximum period +of the Bloch wave function. Using L as the convergence +parameter helps identifying the required k-point sam- +pling for variable simulation cell sizes in later research. +The k-point convergence is not monotonic; more k- +points does not necessarily mean better accuracy (Fig- +ure 2). However, for different system symmetries and cell +shapes and sizes, the ansatz (3) works satisfactorily. Lin- +ear regression analysis to the data gives the parameters +A1 = −1.29 and B1 = −0.036 �A +−1 (Figure 2). Inverting +Eq. (3), we can obtain an optimal number of k-points for +given simulation cell size ac and desired accuracy δ as +Nk(δ) = ceil +�L(δ) +ac +� +, +(4) +where ceil(x) = ⌈x⌉ maps x to the least integer greater +than or equal to x. For instance, with relative accuracy +δ = 10−3 one obtains the Nk = ⌈47 ˚A/ac⌉, suggesting +Γ-point calculations for 4.7-nm-sized simulation cells. In +subsequent analyses, we use Nk = 13, suggesting ∼ δ = +10−2.5...−3 relative tolerance. +b. +Vacuum convergence: +Using plane waves requires +periodicity in all directions, regardless of system dimen- +sions. Low-dimensional systems need therefore a large +vacuum region in the non-periodic direction to avoid spu- +rious interactions with periodic images of the system. +Larger vacuum means more volume and computational +FIG. 2. The k-point convergence of total energy for 2D sys- +tems made of coinage metals. δ is the relative energy toler- +ance and L is the maximum period of the Bloch function [cf. +Eq.(4)]. The linear fit refers to Eq. (3). +cost, implying a need to minimize the vacuum without +affecting the energy. For a complete picture, we inves- +tigate vacuum convergence not only in 2D systems and +but also in 1D chains and free atoms. +We normalize atoms’ dimensions by their van der +Waals radii RvdW and consider the normalized vacuum +Lnorm = Lvac/RvdW , where Lvac is the vacuum along the +non-periodic direction (i.e., the separation between peri- +odic images.) The total energy is practically converged +at 8-˚A vacuum, and we define the energy tolerance as +∆E = E(Lvac) − E(8 �A) and relative energy tolerance +again as δ = ∆E/E3D. The tolerance converges roughly +exponentially, log δ = A2 + B2Lnorm (Figure 3). Conse- +quently, the vacuum for a desired relative energy accu- +racy for a given element can be estimated from +Lvac(δ) = RvdW +(log δ − A2) +B2 +, +(5) +where the parameters A2 = 2.38 and B2 = −1.65 were +obtained by linear regression. For instance, the relative +tolerance δ = 10−3 requires Lvac = 3.3 × RvdW . +In +subsequent analysis, if not said otherwise, we will use +Lvac = 10 �A, which for Ag means δ = 10−4.2, in rough +alignment with k-point convergence. +Still, such a single estimate is indicative at best. The +vacuum convergence follows roughly the coordination +number, free atom converging the slowest, hexagonal sys- +tem the fastest (Figure 3). +This suggests that for a +given element the vacuum should be set by the lowest- +coordinated atom—or by the free atom to be on the +safe side. After all, a modest 16 % increase in vacuum +(Lnorm = 2.5 → 3.0) may increase the relative accuracy +by an order of magnitude. Thus, a single fit as above +is not the best guideline and the vacuum convergence is +best considered by case basis, especially in the presence + +Au +100 +Au. +Cu. +Ag. +SO +SC +Auhc +10-1 +Linear fit +10-2 +10-3 +10-5 +10-6 +10-7 +10-8 +20 +40 +60 +80 +100 +120 +0 +140 +L (A)4 +FIG. 3. Vacuum convergence of the total energy for 1D and +2D systems made of coinage metals. δ is the relative energy +tolerance and Lnorm is vacuum normalized in terms of van der +Waals radii. Free atom vacuum convergences are added for +comparison. +of possible charge transfer. +B. +Effect of Fermi broadening +In principle the Fermi-broadening is a physical param- +eter intimately linked to the electronic temperature T; +in practice it is frequently used as a technical parameter +to accelerate the self-consistency convergence. The tech- +nical attitude towards broadening is evident in available +methods other than the Fermi-function. Computational +literature shows a plethora of different values for Fermi- +broadening, but its effect is rarely discussed in detail. For +insulators and semiconductors the broadening is inconse- +quential, but for metals it matters. In this section, we +want to investigate its effect on the energetics systemat- +ically, for sheer completeness and future reference. +Ideally, broadening should be chosen to enable rapid +convergence without conflicting too much with other con- +vergence parameters. We investigated the effect of broad- +ening by increasing the electronic temperature T from +10−5 K to 1000 K and looked at the energy difference +∆E(T) = E(T) − E(10−5 K). +(6) +The temperature 10−5 K was the smallest that enabled +robust convergence for all systems. Vacuum was 15 ˚A +for all systems. As a result, 1D systems were most sensi- +tive to the broadening, 3D bulk systems were least sen- +sitive (Figure 4). This result is plausible, because the +density of states is the smallest for 1D systems. In 2D +and 3D systems there are more k-points, density of states +at Fermi-level is greater, and state occupations average +over a larger set of states, consequently diminishing the +influence of broadening. The 2D systems show energy +FIG. 4. The effect of electronic temperature on the cohesion +energy of coinage metals in different dimensions. +variation around ∼ 10 meV upon increasing temperature +to 1000 K, corresponding to 86 meV energy broaden- +ing (Figure 4). For the remainder of the calculations in +this article, we used the electronic temperature of 580 K +(�=0.05 eV). +C. +Performance of exchange-correlation functionals +We investigated the performance of xc functionals by +first fixing certain attributes. +To eliminate uncertain- +ties from an insufficient description of valence electrons, +we used the most complete PW basis set and the PAW +potential to describe the core electrons. +We used the +converged number of k-points and size of vacuum from +previous analysis, as well as the recently adopted 0.05 eV +broadening. With these choices, we may concentrate on +the performance of xc-functionals without worrying too +much about artifacts from other sources. +We also investigate xc functionals by using only Ag +systems. By belonging to the same group, the coinage +metals follow similar trends and it is reasonable to expect +other metals to follow the trends of Ag. Still, we do not +claim Ag displays completely universal trends, for there +are elements that have complex many-body effects even +beyond the capabilities of DFT. +In the following, we compare the xc-functional perfor- +mance against bond lengths, cohesive energies, and elas- +tic moduli of all 1D, 2D, and 3D systems. The electronic +structure is compared in terms of later-introduced char- +acteristic figures related to the density of states at the +Fermi-level. +a. +Cohesive Energies: +The cohesive energy was de- +fined as +Ecoh = Efree − E/N , +(7) + +100 +10 +Ag1D +Ag +AuFree +AuD +Auhex +ny +AU +CuiD +CuFree +hex +10-4 +Linear fit +1.5 +2.0 +2.5 +3.0 +3.5 + norm0 +.5 +(Aa) +△E +-10 +-15 +3D +2D +1D +-20 +400 +0 +200 +600 +800 +1000 +Electronic temperature +(K5 +FIG. 5. The cohesive energies of optimized 1D, 2D (hc, sq, +and hex), and 3D systems of Ag with different xc-functionals. +where E is the energy of the system with N atoms and +Efree is the energy of free atom calculated by placing it +inside a 15-�A cube. +All functionals display similar trends, cohesive energy +increasing monotonically from 1D to 3D bulk (Figure 5). +Yet the quantitative differences are visible. LDA displays +its well-known tendency to overestimate cohesive ener- +gies. The 3D bulk cohesion shoots over the experimental +value by 23 % [66]. GGA functionals work significantly +better, where PW91 and PBE are now off by approx- +imately ≈ 13 − 14 %. +In contrast, RPBE shows con- +siderable underbinding and even less accurate cohesion +than LDA. Among hybrid functionals, the performance +of screened exchange HSE03 and HSE06 is better than +PBE0, which still suffers from the spurious Coulomb in- +teraction. B3LYP describes cohesion poorly and is out- +performed by practically all other functionals, and should +be avoided while modeling 2D metals—a conclusion not +surprising in the light of previous observations [67]. In +addition, convergence of free atom with B3LYP was dif- +ficult and required loosening the convergence criterion to +≤ 10−6 eV (loosening had an insignificant effect on the +cohesion of Figure 5). As a rule, GGA and hybrid func- +tionals outperform LDA, but a hybrid functionals do not +necessarily outperform GGA. PW91 and PBE appear as +still as fair choices for robust energetics for general pur- +poses. +b. +Dimensionality-dependence of energetics: +In 2D +metal modeling, the coordination of single metal atoms +can range from C ∼ 1 to C ∼ 6 and occasionally beyond. +The computational method should therefore capture cor- +rectly the relative energetics of atoms at different coor- +dination numbers. In other words, the cohesion should +increase with the coordination number with an appropri- +ate dependence. +Our ansatz for the C-dependence for +FIG. 6. Trends of low-dimensional energetics with different +xc-functionals. The fitted scaling exponent γ is plotted for +different xc-functionals; smaller γ means that energy depends +less linearly on the coordination number [see Eq.(7)]. +the cohesion Ecoh is +Ecoh(C) = E3D +coh × (C/12)γ , +(8) +where E3D +coh is the 3D bulk cohesion and γ is an expo- +nent that quantifies the coordination- or dimensionality- +dependence of the cohesion energy. The ansatz has the +correct asymptotic limits [Ecoh(0) = 0 and Ecoh(12) = +E3D +coh] and suffices for our purposes in this article. (We +tested also more refined ansatzes, but the conclusions +remained the same.) The exponent γ was obtained by +fitting the Eq. (8) for energies from each functional. +As the result, LDA and all GGA and HSE function- +als show roughly the same γ, the same dimensionality- +dependence in energetics (Figure 6). Especially the de- +pendencies in different GGAs are nearly identical. Only +the dependencies in B3LYP and PBE0 are clear out- +liers, PBE0 showing more linear dependence on C (γ +closer to one) and B3LYP showing more non-linear de- +pendence on C (γ further away from one). Interestingly, +although LDA badly overestimates the absolute cohe- +sion energies, the dimensionality-dependence lies some- +where in between GGAs and HSE functionals. In conclu- +sion, GGA-PBE appears to capture the dimensionality- +dependence of energetics comparably well and be still a +serious competitor to the far more costly HSE function- +als. +c. +Bond Lengths: +The bond lengths were obtained +directly from the optimized lattice constants (Figure 7). +In accordance with overbinding, LDA functional shows +small bond lengths. In 3D, the functionals PW91, PBE, +PBE0, HSE03, and HSE06 are underbinding and show +1 − 2 % too large bond lengths. +PBE0 shows short- +est bonds among hybrid functionals, and B3LYP shows +longest bonds among all functionals. Nearly all function- +als show monotonic increase of bond length with coor- + +4.0 +Ag3D +Ag. +Ag +0 +hex +3.5 +3.0 - +(eV) +2.5 +Cohesive energy ( +2.0 +1.5 +1.0- +0.5- +0.0 +LDA +RPBE +PW91 +PBE +B3LYP +PBEO +HSE03 +HSE06 +Exchange-correlation functional0.46 +0.44 - +0.42 - +0.40 - +0.38 - +0.36 - +0.34 - +0.32 - +0.30 +LDA +RPBE +PW91 +PBE +B3LYP +PBEO +HSE03HSE06 +Exchange-correlation functional6 +dination number. Only LDA functional is an exception: +it has a slightly smaller bond length for 2D hexagonal +lattice than for 1D chain. +d. +Elastic constants (theory recap): +Due to colorful +practices in the notations of low-dimensional elasticity, +and to avoid any confusion, we wish to define explicitly +the elastic constants presented in this article. +Within the linear elastic regime the stresses {σi} and +strains {εi} (i = 1 . . . 6) satisfy the generalized Hooke’s +law +σi = +6 +� +j=1 +Cijεj , +(9) +where Cij are elastic constants and expressed as a 6 × 6 +matrix and ε1 = εxx, ε2 = εyy, ε3 = εzz, ε4 = 2εyz, ε5 = +2εxz, ε6 = 2εxy, when following the Voigt notation. We +adapted the formalism of Refs. [68–72] to evaluate the +elastic constants for 1D, 2D and 3D systems. +In 3D, the strain tensor is +ϵ3D = +� +� +ε1 +ε6/2 ε5/2 +ε6/2 +ε2 +ε4/2 +ε5/2 ε4/2 +ε3 +� +� . +(10) +The elastic constants are obtained by applying selected +strains {εi} to the equilibrium simulation cell and by +calculating the partial derivatives +Cij = ∂2∆U +∂εi∂εj +. +(11) +Here ∆U(εi) = U(εi)−U(0) is the elastic energy density +per unit volume, where U(εi) is the energy density at +strain εi. For a system with cubic symmetry, the energy +FIG. 7. Optimized bond lengths of 1D, 2D (hc, sq, and hex), +and 3D systems of Ag with different xc-functionals +density is +∆U(εi) =1 +2 +� +C11ε2 +1 + C11ε2 +2 + C11ε2 +3 + C12ε1ε2 + C12ε1ε3 ++C12ε2ε1 + C12ε2ε3 + C12ε3ε1 + C12ε3ε2 ++C44ε2 +4 + C44ε2 +5 + C44ε2 +6 +� +. +(12) +For 2D systems, the strain tensor is +ϵ2D = +� +ε1 +ε6/2 +ε6/2 +ε2 +� +. +(13) +Again, the elastic constants are obtained by applying se- +lected strains {εi} to the equilibrium simulation cell and +by calculating the partial derivatives +Cij = ∂2∆U +∂εi∂εj +(14) +Here ∆U(εi) = U(εi) − U(0) is the energy density per +unit area, where U(εi) is the energy density at strain εi. +For a system with square symmetry, the energy density +is +∆U(εi) =1 +2(C11ε2 +1 + C22ε2 +2 + 2C12ε1ε2 + 2C16ε1ε6 ++2C26ε2ε6 + C66ε2 +6) +(15) +and all three elastic constants C11, C12 and C66 are in- +dependent. However, for a hexagonal system, only con- +stants C11 and C12 are independent and C66 = (C11 − +C12)/2. +Finally, for 1D systems, the strain-tensor matrix is sim- +ply ϵ1D = (ε1). Yet again, the elastic constant is obtained +by applying the strain ε1 to the equilibrium simulation +cell and by taking the partial derivative +C1 = ∂2∆U +∂2ε1 +. +(16) +Here ∆U(εi) = U(εi) − U(0) is the energy density per +unit length, where U(εi) is the energy density at strain +εi. In other words, +∆U(ε1) = 1 +2C11ε2 +1 . +(17) +Table II summarizes the formulae for the elastic con- +stants and their relations. Note that the elastic constants +in different dimensions have also different units: they are +GPa for 3D, GPa nm for 2D, and GPa nm2 for 1D (GPa +nm3−D or eV/˚AD in short, where D is the dimensional- +ity). +e. +Elastic +constants +(results): +Functionals +show +similar trends for bulk moduli, but there are quantita- +tive differences (Figure 8a). +We remind that because +the elastic moduli in different dimensions have different +units, the trend with respect to the coordination num- +ber can be compared only between different 2D lattices. +LDA overestimates the bulk moduli systematically, for +3D bulk by almost 40 %. Only for 1D chain the modulus + +Aghc +Ag3D +3.0 - +bso +2.90 - +Bond length (A) +2.80 +2.70 +2.60 +2.50 +LDA +RPBE +PW91 +PBE +B3LYP +HSE03 +HSE06 +PBEO +Exchange-correlation functional7 +FIG. 8. Elastic properties of low-dimensional systems of Ag +with different xc-functionals. Bulk moduli (a) and Young’s +moduli (b) are shown for all systems, shear moduli (c) and +Poisson’s ratio (d) are shown only for 3D and stable 2D sys- +tems. Units for moduli are GPa nm3−D, where D is the sys- +tem dimensionality. +TABLE II. Formulae for Bulk Modulus (K), Shear-modulus +(G), Young’s modulus (Y), and Poisson’s ratio (µ) for the +systems in Fig. 1. +System +K +G +Y +µ +1D +C11 +- +K +- +2Dhex/hc +C11+C12 +2 +C11−C12 +2 +4KG +K+G +K−G +K+G +2Dsq +C11+C12 +2 +C66 +C2 +11−C2 +12 +C11 +C11 +C12 +3D +C11+2C12 +3 +3C44+C11−C12 +5 +9KG +3K+G +3K−2G +2(3K+G) +is in line with HSE06. +Among GGAs, the bulk mod- +uli of PW91 and PBE are nearly the same. The hybrid +functionals have fairly similar performance, with B3LYP +again showing a striking exception, especially related to +1D modulus. These observations in bulk moduli apply +also to Young’s moduli (Figure 8b). Only GGAs show +somewhat larger stiffness and the trends in 2D moduli +for B3LYP and PBE0 are different. +The shear modulus and Poisson’s ratio are defined only +for 2D and 3D systems (Figures 8c and d). Moreover, +shear modulus is not reported for the 2D square lattice +due to instability against shear deformations. In addi- +tion, some deformations with PBE0 and B3LYP resulted +in consistent numerical errors, forcing us to omit shear +and Young’s modulus as well Poisson ratio for these func- +tionals. +In summary, the most consistent behavior in +elastic moduli is displayed by HSE and GGA function- +als. LDA, B3LYP and PBE0 functionals suffer from both +numerical challenges and deviant trends at least in some +elastic properties. +f. +Electronic structure (density of states): +To com- +plement pure energetic and geometric properties, we now +extend our investigations to electronic structure proper- +ties. Electronic structure is a complex topic with many +features. To reduce complexity and extract trends, we in- +vestigate the electronic structure simply in terms of the +density of states DOS(ϵ) and its projections DOSl(ϵ) to +s (l = 0), p (l = 1), and d (l = 2) angular momen- +tum states. In addition, we focus only on energies at the +vicinity of the Fermi-level ϵ = ϵF . +Consequently, we define the quantities +Nl = +� ∞ +−∞ +DOSl (ϵ) g (ϵ) dϵ +(18) +that give the number of l-type orbitals surrounding the +Fermi-level. The DOS is also normalized by the number +of atoms in the simulation cell. The envelope function +g(ϵ) has a Gaussian form +g (ϵ) = exp +� +−1 +2 +�ϵ − ϵf +σ +�2� +(19) + +160 - +I AgiD +Agsg +1 Ag3D +Aghc +Aghex +(a) +140 - +120 +80 - +60 - +20 - +115 +b +100 +80 - +Young's modulus +40 - +20 - +C) +40 +30 - +Shear modulus +20 - +-01 +d) +os'O +ratio +0.45 +s +0.30 - +HSE03HSE06 +Exchange-correlation functional8 +FIG. 9. Effect of xc functional on the electronic structure of +low-dimensional metals made of Ag. Heatmap visualizes the +number of s-type states (Ns), p-type states (Np), d-type states +(Nd), and the total number of states (Nt) within a ∼ 1 eV +energy window around the Fermi-level [see Eq.(18)]. +and we used σ = 1 eV energy window around ϵF . +In general, the s-orbital contribution decreases with in- +creasing coordination number for all xc functionals (Fig- +ure 9). +In 1D the main contribution comes from s- +orbitals, followed by p- and d-orbitals for all functionals. +In 2D this order is rearranged to p > s > d. In 3D this +same trend is retained by all hybrid functionals. +The +LDA, PW91, and PBE have very similar orbital contri- +bution ordering. For all xc functionals, the p contribu- +tion is the largest for honeycomb, smallest for 1D, and +smallest for hexagonal among 2D systems. The ordering +of Np with respect to different coordination number is +the same for GGAs, PBE0, and B3LYP. For HSE03 and +HSE06 all Nl are very similar. The d-orbital contribu- +tions follow trend similar to s-orbitals. The value of Nd +is the highest for LDA and the lowest for PBE0 for all +systems; the most visible difference is the generally low +Nd of all hybrid functionals, especially in 1D. +Regarding the total DOS, all GGAs produce nearly +identical Nt, apart from 3D bulk in RPBE. The total +DOS from hybrids differs somewhat from the LDA and +GGA functionals. HSE functionals show similar Nt for +C = 6 and 12 systems, but differ in other systems. Over- +all, trends in the total densities are inconsistent for LDA +and PBE0 functionals, but somewhat consistent among +GGA as well as B3LYP and HSE functionals. +g. +Conclusions on xc functionals: +To summarize, +PW91 and PBE perform similarly for forces, energies, +and densities of states, while RPBE shows underbinding, +smaller bond lengths, and smaller elastic constants. LDA +is inferior to GGA practically in all respects. Among hy- +brid functionals, the performances of HSE03 and HSE06 +aligned in all respects. B3LYP failed to improve GGA in +terms of accuracy in the lattice constants and cohesive +energies, even if its electronic structures resembled those +of HSE functionals. Cohesion energy displayed congruent +dimensionality-dependencies, apart from visibly differing +dependencies by B3LYP and PBE0 functionals. +Before reaching ultimate conclusions, however, we have +to consider the computational cost (Table III). As ex- +pected by the nonlocal character of the hybrid function- +als, already minimal-cell systems require 2 − 3 orders +of magnitude more computational time for hybrids than +for LDA and GGA, and for larger systems the differ- +ence would increase even further. Considering the low +computational cost, GGA functionals perform extremely +well compared to hybrid functionals, compared even to +the most robust HSE family. To conclude, unless the low- +dimensional metals are studied for very specific purposes, +the standard PBE indeed remains the preferred weapon +of choice for low-dimensional metals modeling. +TABLE III. Computational cost of different xc-functionals: +Time in seconds to calculate the energy of minimal-cell sys- +tems using 24 cores. The cell has one atom for all systems +except for 2D honeycomb. +LDA RPBE PW91 PBE B3LYP PBE0 HSE03 HSE06 +1D +39 +39 +44 +43 +476 +1360 +491 +1897 +hc +49 +59 +62 +58 +16786 20937 +18662 +15006 +sq +18 +24 +23 +22 +1469 +1739 +1535 +1493 +hex +16 +19 +20 +17 +1454 +1800 +1698 +1675 +3D +14 +18 +19 +17 +88553 41352 +38802 +38704 +D. +Performance of different basis sets +In this section, we choose PBE xc functional and repeat +the systematics of the previous section while this time +varying the basis set. The converged plane wave basis +gives the best results that provide the reference assessing +the performance of the three LCAO basis sets Medium, +High, and Ultra introduced in Section II. +To obtain a broader context, we compared the DFT- +LCAO with DFTB method, which uses a minimal local +basis and contains approximations speeding up the cal- +culations. Here we used the parameters available for Ag +developed earlier [53, 54]. However, parametrization can +be done in different ways, and one should not consider +these results as unique and absolute representation of +DFTB. +a. +Cohesive Energies: +The LCAO-U and LCAO-H +produce cohesive energies very close to those of PW (Fig- +ure 10). LCAO-M overbinds slightly in comparison, but +the accuracy for 2D systems is still 3 − 4 % compared to +PW. The dependence of cohesion on coordination num- +ber is reproduced with all basis sets, and differences are +difficult to see on absolute scale. DFTB follows similar +behavior, but shows significant overbinding, especially +for 3D bulk. + +1.35 +Ag3D +Aghex +Agsq +1.20 +Aghc +Agid +1.05 +Ag3D. +Aghex - +0.90 +Aghc - +0.75 +Agid. +Ag3D +Aghex +0.60 +Agsq +Aghc +0.45 +AgiD +Ag3D +0.30 +Aghex - +Agsq- +0.15 +Aghc - +Agid: +0.00 +LDA +RPBE +PW91 +PBE +B3LYP +PBEO +HSE03 +HSE06 +Exchange-correlation functional9 +FIG. 10. Cohesive energies of optimized 1D, 2D (hc, sq, and +hex), and 3D systems made of Ag with different basis sets. +Bars on the left show DFTB results with minimal basis for +comparison. +b. +Dimensionality-dependence +of +energetics: +As +with xc functionals, we investigate how basis set affects +the dependence of energetics on coordination number. +Again this dependence is analyzed via the scaling +exponent γ in Eq. (8) fitted to the cohesive energies as +a function of C. +Compared to PW, the dependence on C becomes sys- +tematically more linear as we move from Ultra to High +and ultimately to Medium basis (Figure 11). However, +still the Medium basis reproduces γ to within 5 % accu- +racy compared to PW basis. Even DFTB compares well +in the overall coordination-dependence, although there +are visible problems in capturing the DFT trends for +2D systems (the green bars for DFTB in Figure 10). +However, to state the main point, the choice of basis in- +fluences dimensionality-dependence of energetics far less +than xc functional: note that Figs. 6 and 11 have the +same scale in γ. +c. +Bond Lengths: +The LCAO-U and LCAO-H bond +lengths are very similar, accurate to within 0.77 % com- +pared to PW (Figure 12). All LCAO variants overesti- +mate all bonds, LCAO-M having the lowest performance +with 1.6 % too long bonds. DFTB no longer captures the +DFT trends in coordination-dependence. The 1D chain +bond length is larger than honeycomb and the 2D bonds +vary wildly, even if the C-ordering still remains correct. +d. +Elastic constants and moduli: +For 1D and 2D sys- +tems, elastic moduli have minor dependence on basis set +(Figure 13). The largest deviation from PW occurs for +3D bulk, for all LCAO variants. +This deviation likely +stems from the better space-filling character of PW ba- +sis. Moreover, although performing well in cohesion and +bond lengths, LCAO-M performs poorly in all elastic +properties. LCAO-U is close to PW in all respects, and +FIG. 11. Trends of low-dimensional energetics with different +basis sets. The fitted scaling exponent γ is plotted for different +basis sets; smaller γ means that energy depends less linearly +on the coordination number [see Eq.(7)]. The vertical scale is +the same as in Fig. 6. +FIG. 12. Bond lengths of optimized 1D, 2D (hc, sq, and hex), +and 3D systems made of Ag with different basis sets. +LCAO-M captures all the same trends, even if with some +quantitative differences. These results suggest that, ex- +cept perhaps for LCAO-M, LCAO basis can be reliable +for studying mechanical properties of low-dimensional +metallic systems. +The LCAO variant -dependency of +elastic properties is even smaller than the changes upon +switching from GGA to hybrid functional (compare Figs. +8 and 13). +In comparison, DFTB shows both trend differences +and large absolute differences compared to DFT-LCAO +(Figure 13). For example, the 1D elastic modulus is over- +estimated by a factor of ∼ 5. +Even the trend within +2D systems was not reproduced. It appears that the Ag +parametrization should be revised for more reliable me- + +Ag3D +Ag. +4.0 +1 +bss +Shex +3.5 +3.0 - +2.5 +2.0. +1.5 +1.0. +0.5 +0.0 +DFTB +LCAO-M +LCAO-H +LCAO-U +PW +Basis set0.46 +0.44 - +0.42 - +0.40 - +0.38 +0.36 - +0.34 - +0.32 +0.30 +DFTB +LCAO-M +LCAO-H +LCAO-U +PW +Basis setAgh +Ag +Shex +S3D +Shc +bss +3.0 +2.90 +Bond length (A) +2.80 +2.70 +2.60 +2.50 +DFTB +LCAO-M +LCAO-H +LCAO-U +PW +Basis set10 +chanical properties of low-dimensional Ag systems. +e. +Electronic structure (density of states): +Also the +electronic structure from LCAO is compared here against +PW results, +using the indicator numbers given by +Eq. (18). +For 2D structures PW gives orbital contri- +butions in order p > s > d (Figure 14). +For LCAO +this trend shuffles to s > d > p, that is, the p contri- +bution diminishes for all LCAO variants. +For 1D sys- +tem the orbital ordering for PW and LCAO basis re- +mains the same. However, still all basis sets—including +minimal-basis DFTB—show consistent C-dependence in +orbital contributions around the Fermi-level. LCAO-H +and LCAO-U results align better, while LCAO-M re- +sults are different in some respects. In summary, the C- +dependence of the total DOS in 2D metals is reproduced +by LCAO to a fair degree, but the orbital contributions +are different. +f. +Conclusions on basis sets: +To conclude, LCAO +basis competes extremely well with PW for studying +energetic and geometric properties of low-dimensional +metal systems. Even elastic moduli are reproduced rea- +sonably well by LCAO-H and LCAO-U basis, compared +to converged PW basis. The performance of LCAO-M +basis was notably modest, regarding elastic properties +and also the details of electronic structure. +The or- +bital breakup of the electronic structures at the vicin- +ity of Fermi-level for PW and LCAO variants differed +markedly. +Regarding DFTB, the Ag parametrizations clearly re- +quire revisiting. The cohesive energies are too large, bond +lengths are both large and small, and elastic moduli are +close to arbitrary. Still many of the qualitative trends +regarding C-dependence were reproduced reliably. +However, before again reaching ultimate conclusions, +we have to consider the computational cost with differ- +ent basis (Table IV). The cost was investigated by simula- +tion cells with 32−64 atoms and a couple of dozen cores. +The comparison is thus by no means unique or absolute, +but it does give a rough inkling of the computational de- +mands. As expected, DFTB outspeeds DFT by one to +three orders of magnitude. Within DFT, switching from +LCAO-M to LCAO-U results in cost increases from a fac- +tor of two (1D) up to a factor of ∼ 15 (3D). Especially +for low-dimensional systems LCAOs are faster than PW, +nearly by two orders of magnitude. +For 3D bulk PW +is very competitive against LCAO due to lacking vac- +uum region; here LCAO-U is even slower than PW. In +conclusion, unless very high accuracy is of central impor- +tance, LCAO has demonstrated a fair accuracy in most +properties and should be prioritized over PW due to its +superior efficiency. Even LCAO-M basis can be consid- +ered for simulations where the improved speed wins over +lost accuracy. +FIG. 13. Elastic properties of low-dimensional systems of Ag +with different basis sets. Bulk moduli (a) and Young’s moduli +(b) are shown for all systems, shear moduli (c) and Poisson’s +ratio (d) are shown only for 3D and stable 2D systems. Units +for moduli are GPa nm3−D, where D is the system dimen- +sionality. + +Agid +Ag3D +Aghex +Aghc +Agsq +(a) +120 - +100 + Bulk modulus +08 +60 - +40 - +20- +0 +(b) +140 - +120 - +80 +Young's r +F 09 +40 - +20 +0 +(c) +50 - +40 - +modulus +Shear r +1 +20 +0 +(d) +Fos'O +0.35 - +LCAO-M +LCAO-H +LCAO-U +PW +DFTB +Basis set11 +FIG. 14. +Effect of basis set on the electronic structure of +low-dimensional metals made of Ag. Heatmap visualizes the +number of s-type states (Ns), p-type states (Np), d-type states +(Nd), and total number of states (Nt) within a ∼ 1 eV energy +window around the Fermi-level [see Eq.(19)]. +TABLE IV. Computational cost of different basis sets: Time +in seconds to calculate the energy of systems using 24 cores. +The parenthesis contain the number of atoms in the supercell. +Systems +DFTB +LCAO-M +LCAO-H +LCAO-U +PW +1D (32) +10 +175 +265 +310 +11890 +2D hc (64) +20 +215 +355 +610 +13120 +2D sq (64) +18 +190 +300 +500 +12370 +2D hex(64) +17 +130 +290 +655 +6885 +3D (64) +19 +145 +855 +2220 +2050 +E. +Combined scanning of xc functionals and basis +sets +Above we investigated xc functionals (with PW basis) +and basis sets (with PBE functional) separately. How- +ever, the performance of xc functionals and basis sets +can be coupled. +We therefore complement our analy- +sis by combined scanning of different xc functionals with +different basis sets. The bond lengths, cohesive energies, +elastic constants, and orbital contributions to DOS ob- +tained at different basis set-xc functional -combinations +are shown in Tables V, VI, and VII in the Appendix. +For LDA, the choice of basis set did not affect the co- +hesion dependence on C (Table V). Changing the basis +set from PW to LCAO increases the cohesive energy for +C ≥ 4 and decreases it for C = 1 and 3. +Decreasing +the LCAO size also decreases the cohesion, as expected +in the light of variational principle. Bond lengths with +PW, LCAO-U and LCAO-H basis are nearly equal. With +LCAO-M bonds are longer for all systems. The elastic +properties are nearly basis-independent, with the notable +exception of LCAO-M (Table VI). Most sensitive to the +choice of basis is the electronic structure; all LCAO vari- +ants show the same trend, which however differs signifi- +cantly from PW (Table VII). +For GGAs, the performance remains robust upon re- +ducing the size of the basis set. In fact, the observations +in Subsection III D with PBE are representative for other +GGAs as well. Switching PW to LCAO-U or LCAO-H +changes bond lengths and cohesive energies less than 1 %; +less robust LCAO-M decreases cohesive energies by 4 % +and increases bond lengths by ≈ 1.5 % (Table V). Basis +set sensitivity is the smallest for PW91 and the largest +for RPBE. Elastic constants follow the accuracy trends +similar to those of energetics and geometric properties. +PBE shows some basis set sensitivity, especially for the +bulk moduli of 2D systems (Table VI). +For hybrid functionals, the matters are less systematic. +Using LCAO-M in conjunction with unscreened B3LYP +and PBE0 functionals results in significant overbinding; +bond lengths are underestimated by more than 10 % (Ta- +ble V). With LCAO-H and LCAO-U basis sets the same +xc functionals underestimate bonds only by ≈ 2 %, while +increase cohesive energies by ≤ 24 %. B3LYP and PBE0 +are thus extremely sensitive to the quality of LCAO ba- +sis. Moreover, B3LYP and PBE0 are unable to produce +elastic moduli due to persistent numerical errors. In con- +trast, the screened HSE functionals produced robust ge- +ometries, energetics and elastic properties upon changing +the size of the LCAO basis. The robustness was even +better than with PW91 and PBE, although admittedly +at a considerable computational cost. The orbital con- +tributions to DOS with PW and LCAO basis were dif- +ferent; the same effect was observed for PBE functional +(Figure 14). Among different LCAO variants, LCAO-H +and LCAO-U show similar orbital contributions for all +systems. In addition to energetic and geometric prop- +erties, the peculiarities of B3LYP and PBE0 functionals +are observable also in electronic properties (Table VII). +In general, hybrid functionals in conjunction with LCAO- +H and LCAO-U basis requires prohibitive computational +resources even for single atom. +F. +The effect of DFT implementation +In addition to DFT attributes, it is important also +to be able to rely on the DFT implementation itself. +For completeness, therefore, we briefly discuss the mag- +nitude of differences related to the numerical imple- +mentation of DFT. We calculated the cohesive ener- +gies, bond lengths, and elastic moduli also with the +GPAW code, using plane wave basis with the same +800 eV energy cutoff and default parameters [36]. The +QuantumATK/GPAW cohesive energies were 1.1671 eV / +1.1661 eV (1D), 1.5062 eV / 1.5054 eV (2D hc), 1.8293 eV +/ 1.8286 eV (2D sq), 2.0583 eV / 2.0570 eV (2D hex), +2.5326 eV / 2.5323 eV (3D), bond lenghts 2.6480 ˚A/ +2.6501 ˚A (1D), 2.6700 ˚A/ 2.6682 ˚A (2D hc), 2.6998 ˚A/ +2.700567 ˚A +(2D +sq), +2.7877 ˚A/ +2.7894 ˚A +(2D +hex), + +1.80 +Ag3D +Aghex +1.60 +Aghc +Agid +1.40 +Ag3D +Aghex +1.20 +ABsq +Aghc +1.00 +Agid +Ag3D +Aghex +0.80 +Agsq +Aghc +0.60 +Agid +Ag3D +0.40 +Aghex +0.20 +Aghc +Agid +0.00 +LCAO-U +PW +DFTB +LCAO-M +LCAO-H +Basis set12 +2.9301 ˚A/ 2.9305 ˚A (3D), and bulk moduli 18.32 GPa nm2 +/18.73 GPa nm2 (1D), 17.20 GPa nm / 17.21 GPa nm (2D +hc), 31.46 GPa nm / 31.26 GPa nm (2D sq), 38.07 GPa nm +/ 37.79 GPa nm (2D hex), 92.03 GPa / 90.37 GPa (3D). +Thus, default parameters without tuning give code- +related differences in cohesive energies ≲ 1.3 meV, in +bond lengths ≲ 0.002 ˚A, and in bulk moduli ≲ 1 % (2D +systems) or ≃ 2% (1D and 3D systems). Although the +comparison used the PBE functional and plane waves, it +is reasonable to suspect the level of differences to remain +similar also for other functionals and basis sets. Over- +all, code-related differences remain considerably smaller +than the differences originating from physical attributes. +IV. +SUMMARY AND CONCLUSION +In summary, we investigated the performance of vari- +ous DFT attributes in the modeling of low-dimensional +elemental metals. +For future reference, the number of +k-points, the size of the vacuum region, and the magni- +tude of Fermi-broadening were given tolerance-dependent +rules of thumb. Such rules help choosing combinations +of attributes that result in commensurate accuracies. +The most robust against the choice of basis set +was HSE06, followed by HSE03, PBE, PW91, RPBE +and LDA. The B3LYP produced inaccurate cohesions +and bond lengths—with the highest computational cost. +Only the electronic structure in B3LYP was in line with +other hybrid functionals. +The energetics, geometries, and elastic properties with +PW, LCAO-U, and LCAO-H basis sets were in over- +all good agreement. +The greatest disparities between +PW and LCAO methods resided in the orbital contribu- +tions to the DOS, although in the total DOS they were +moderated. +On a general level, LCAO-U and LCAO- +H performed similarly at different xc functionals; there- +fore, for general purposes, LCAO-H should be preferred +over LCAO-U due to superior efficiency (Table IV). The +LCAO-M basis worked varyingly well in many respects, +except when used in conjunction with B3LYP and PBE0 +functionals. +To conclude, in the research of metallic bonding at +low dimensions, the best value for a given cost is proba- +bly given by semi-local PW91 and PBE xc functionals in +conjunction with moderately-sized LCAO-U or LCAO- +H basis sets. +These results are encouraging for doing +large-scale, high-throughput DFT simulations to gener- +ate data for machine learning algorithms. In comparison, +DFTB is a very speedy method and is capable of simu- +lations unaccessible by DFT [73–75], but the quality of +parametrization needs to be ensured first. We hope that +our results and gentle recommendations help lifting 2D +metal research to new heights, expedite better interac- +tion with experiments, and feed machine learning algo- +rithms with quality data to drive further discoveries in +low-dimensional metals. +ACKNOWLEDGMENTS +We acknowledge the Finnish Grid and Cloud Infras- +tructure (FGCI) for computational resources. +[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D.-e. Jiang, +Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. +Firsov, Science 306, 666 (2004). +[2] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. +Khotkevich, S. V. Morozov, and A. K. Geim, Proceedings +of the National Academy of Sciences 102, 10451 (2005). +[3] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, +and M. S. Strano, Nature nanotechnology 7, 699 (2012). +[4] V. Shanmugam, R. A. Mensah, K. Babu, S. Gawusu, +A. Chanda, Y. Tu, R. E. Neisiany, M. F¨orsth, G. Sas, and +O. Das, Particle & Particle Systems Characterization 39, +2200031 (2022). +[5] X. Zhang, A. Chen, L. Chen, and Z. Zhou, Advanced +Energy Materials 12, 2003841 (2022). +[6] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, +and A. Kis, Nature Reviews Materials 2, 1 (2017). +[7] D. Geng and H. Y. Yang, Advanced Materials 30, +1800865 (2018). +[8] J. Zhao, Q. Deng, A. Bachmatiuk, G. Sandeep, A. Popov, +J. Eckert, and M. H. R¨ummeli, Science 343, 1228 (2014). +[9] Y. Ma, B. Li, and S. Yang, Mater. Chem. Front. 2, 456 +(2018). +[10] Y. Chen, Z. Fan, Z. Zhang, W. Niu, C. Li, N. Yang, +B. Chen, and H. Zhang, Chemical Reviews 118, 6409 +(2018). +[11] T. Wang, M. Park, Q. Yu, J. Zhang, and Y. Yang, Ma- +terials Today Advances 8, 100092 (2020). +[12] H. Q. Ta, R. G. Mendes, Y. Liu, X. Yang, J. Luo, +A. Bachmatiuk, T. Gemming, M. Zeng, L. Fu, L. Liu, and +M. H. R¨ummeli, Advanced Science 8, 2100619 (2021). +[13] G. Zagler, M. Reticcioli, C. Mangler, D. Scheinecker, +C. Franchini, and J. Kotakoski, 2D Materials 7, 045017 +(2020). +[14] X. Wang, C. Chen, B. Jiang, H. Duan, and K. Du, Acta +Materialia 229, 117844 (2022). +[15] P. Hohenberg and W. Kohn, Physical Review 136, B864 +(1964). +[16] W. Kohn and L. J. Sham, Physical Review 140, A1133 +(1965). +[17] L.-M. Yang, T. Frauenheim, and E. Ganz, Physical +Chemistry Chemical Physics 17, 19695 (2015). +[18] L.-M. Yang, M. Dornfeld, T. Frauenheim, and E. Ganz, +Physical Chemistry Chemical Physics 17, 26036 (2015). +[19] L.-M. Yang, T. Frauenheim, and E. Ganz, Journal of +Nanomaterials 2016, 8429510 (2016). +[20] J. Nevalaita and P. Koskinen, Physical Review B 97, +035411 (2018). +[21] J. Nevalaita and P. Koskinen, Physical Review B 98, +115433 (2018). +[22] J. Nevalaita and P. Koskinen, Nanoscale 11, 22019 + +13 +(2019). +[23] J. Nevalaita and P. Koskinen, AIP Advances 10, 065327 +(2020). +[24] S. Ono, Physical Review B 102, 165424 (2020). +[25] Y. Ren, L. Hu, Y. Shao, Y. Hu, L. Huang, and X. Shi, +Journal of Materials Chemistry C 9, 4554 (2021). +[26] S. Ono, Physical Review Materials 5, 104004 (2021). +[27] B. Anam and N. Gaston, Journal of Physics: Condensed +Matter 33, 125901 (2021). +[28] P. Kapoor, M. Sharma, and P. Ahluwalia, Physica +E: Low-dimensional Systems and Nanostructures 131, +114745 (2021). +[29] A. Kutana, T. Altalhi, Q. Ruan, J.-J. Zhang, E. S. Penev, +and B. I. Yakobson, Nanoscale Adv. 4, 1408 (2022). +[30] A. A. Sangolkar and R. Pawar, Physica status solidi (b) +259, 2100489 (2022). +[31] A. A. Sangolkar, S. Jha, and R. Pawar, Advanced Theory +and Simulations 5, 2200057 (2022). +[32] G. Kresse and J. Furthm¨uller, Physical Review B 54, +11169 (1996). +[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re- +view Letter 77, 3865 (1996). +[34] S. Lehtola, C. Steigemann, M. J. Oliveira, and M. A. +Marques, SoftwareX 7, 1 (2018). +[35] E. L. Briggs, D. J. Sullivan, and J. Bernholc, Physical +Review B 54, 14362 (1996). +[36] J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, +M. Du�lak, +L. Ferrighi, +J. Gavnholt, +C. Glinsvad, +V. +Haikola, +H. +A. +Hansen, +H. +H. +Kristoffersen, +M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljung- +berg, +O. Lopez-Acevedo, +P. G. Moses, +J. Ojanen, +T. Olsen, V. Petzold, N. A. Romero, J. Stausholm- +Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter, +B. Hammer, H. H¨akkinen, G. K. H. Madsen, R. M. Niem- +inen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz, +K. S. Thygesen, and K. W. Jacobsen, Journal of Physics: +Condensed Matter 22, 253202 (2010). +[37] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Jun- +quera, P. Ordej´on, and D. S´anchez-Portal, Journal of +Physics: Condensed Matter 14, 2745 (2002). +[38] M. Brandbyge, J.-L. Mozos, P. Ordej´on, J. Taylor, and +K. Stokbro, Physical Review B 65, 165401 (2002). +[39] J. Schmidt, M. R. G. Marques, S. Botti, and M. A. L. +Marques, npj Comput Mater 5 (2019). +[40] B. Mortazavi, +I. S. Novikov, +E. V. Podryabinkin, +S. Roche, T. Rabczuk, A. V. Shapeev, and X. Zhuang, +Applied Materials Today 20, 100685 (2020). +[41] J. Cai, X. Chu, K. Xu, H. Li, and J. Wei, Nanoscale Adv. +2, 3115 (2020). +[42] G. R. Schleder, C. M. Acosta, and A. Fazzio, ACS Ap- +plied Materials & Interfaces 12, 20149 (2020). +[43] B. Ryu, L. Wang, H. Pu, M. K. Y. Chan, and J. Chen, +Chem. Soc. Rev. 51, 1899 (2022). +[44] E. Canadell, M.-L. Doublet, and C. Iung, Orbital Ap- +proach to the Electronic Structure of Solids (Oxford Uni- +versity Press, 2012). +[45] M. Elstner, +D. Porezag, +G. Jungnickel, +J. Elsner, +M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Phys- +ical Review B 58, 7260 (1998). +[46] S. Smidstrup, T. Markussen, P. Vancraeyveld, J. Wellen- +dorff, J. Schneider, T. Gunst, B. Verstichel, D. Stradi, +P. A. Khomyakov, U. G. Vej-Hansen, M.-E. Lee, S. T. +Chill, F. Rasmussen, G. Penazzi, F. Corsetti, A. Ojan- +per¨a, K. Jensen, M. L. N. Palsgaard, U. Martinez, +A. Blom, M. Brandbyge, and K. Stokbro, Journal of +Physics: Condensed Matter 32, 015901 (2019). +[47] R. K. Harris, P. Hodgkinson, C. J. Pickard, J. R. Yates, +and V. Zorin, Magnetic Resonance in Chemistry 45, S174 +(2007). +[48] V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, +X. Ren, K. Reuter, and M. Scheffler, Computer Physics +Communications 180, 2175 (2009). +[49] M. van Setten, M. Giantomassi, E. Bousquet, M. Ver- +straete, D. Hamann, X. Gonze, and G.-M. Rignanese, +Computer Physics Communications 226, 39 (2018). +[50] D. C. Liu and J. Nocedal, Mathematical Programming +45, 503 (1989). +[51] H. J. Monkhorst and J. D. Pack, Physical Review B 13, +5188 (1976). +[52] P. Koskinen and V. M¨akinen, Computational Materials +Science 47, 237 (2009). +[53] B. Sz˝ucs, Z. Hajnal, T. Frauenheim, C. Gonz´alez, J. Or- +tega, R. P´erez, and F. Flores, Applied Surface Science +212-213, 861 (2003). +[54] B. Sz˝ucs, Z. Hajnal, R. Scholz, S. Sanna, and T. Frauen- +heim, Applied Surface Science 234, 173 (2004). +[55] J. P. Perdew and Y. Wang, Physical Review B 45, 13244 +(1992). +[56] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re- +view Letter 77, 3865 (1996). +[57] B. Hammer, L. B. Hansen, and J. K. Nørskov, Physical +Review B 59, 7413 (1999). +[58] J. P. Perdew, P. Ziesche, and H. Eschrig, Electronic struc- +ture of solids’ 91 (1991). +[59] K. Burke, J. P. Perdew, and Y. Wang, Derivation of +a generalized gradient approximation: The pw91 den- +sity functional, in Electronic density functional theory +(Springer, 1998) pp. 81–111. +[60] Y. Matsushita, K. Nakamura, and A. Oshiyama, Physical +Review B 84, 075205 (2011). +[61] A. D. Becke, The Journal of Chemical Physics 98, 5648 +(1993). +[62] J. P. Perdew, M. Ernzerhof, and K. Burke, The Journal +of Chemical Physics 105, 9982 (1996). +[63] R. R. Pela, M. Marques, and L. K. Teles, Journal of +Physics: Condensed Matter 27, 505502 (2015). +[64] J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal +of Chemical Physics 118, 8207 (2003). +[65] P. E. Bl¨ochl, Physical Review B 50, 17953 (1994). +[66] C. Kittel, Introduction to solid state physics, 8th ed. (Wi- +ley New York, 2005). +[67] J. Paier, M. Marsman, and G. Kresse, The Journal of +Chemical Physics 127, 024103 (2007). +[68] S. Zhang and R. Zhang, Computer Physics Communica- +tions 220, 403 (2017). +[69] V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, +Computer Physics Communications 267, 108033 (2021). +[70] N. W. Tschoegl, Australian Journal of Physics 11, 154 +(1958). +[71] M. Ma´zdziarz, 2D Materials 6, 048001 (2019). +[72] M. Jamal, S. Jalali Asadabadi, I. Ahmad, and H. Rahna- +maye Aliabad, Computational Materials Science 95, 592 +(2014). +[73] P. Koskinen and T. Korhonen, Nanoscale 7, 10140 +(2015). +[74] P. Koskinen, H. H¨akkinen, B. Huber, B. von Issendorff, +and M. Moseler, Physical Review Letter 98, 015701 +(2007). + +14 +[75] P. +Koskinen, +H. +H¨akkinen, +G. +Seifert, +S. +Sanna, +T. Frauenheim, and M. Moseler, New Journal of Physics +8, 9 (2006). + +15 +APPENDIX +TABLE V. Bond lengths d(�A) and Cohesive energies Ecoh(eV) for each lattice type corresponding to different DFT-attributes. +1D +Honeycomb +Square +Hexagonal +3D +DFT-Methods +d +Ecoh +d +Ecoh +d +Ecoh +d +Ecoh +d +Ecoh +DFTB +2.572 +1.691 +2.562 +2.450 +2.636 +2.804 +2.819 +2.967 +3.008 +3.891 +LDA-LCAO-M +2.584 +1.513 +2.591 +2.012 +2.623 +2.475 +2.712 +2.761 +2.840 +3.547 +LDA-LCAO-H +2.553 +1.563 +2.562 +2.105 +2.606 +2.563 +2.693 +2.858 +2.827 +3.660 +LDA-LCAO-U +2.542 +1.587 +2.553 +2.126 +2.598 +2.590 +2.685 +2.887 +2.826 +3.672 +LDA-PW +2.542 +1.591 +2.542 +2.138 +2.595 +2.586 +2.682 +2.881 +2.828 +3.638 +RPBE-LCAO-M +2.732 +0.959 +2.760 +1.198 +2.764 +1.474 +2.853 +1.677 +2.982 +2.065 +RPBE-LCAO-H +2.710 +0.989 +2.731 +1.251 +2.745 +1.531 +2.831 +1.738 +2.965 +2.130 +RPBE-LCAO-U +2.691 +1.001 +2.723 +1.262 +2.736 +1.547 +2.824 +1.756 +2.963 +2.143 +RPBE-PW +2.689 +0.992 +2.709 +1.248 +2.734 +1.523 +2.822 +1.732 +2.962 +2.100 +PW91-LCAO-M +2.679 +1.145 +2.700 +1.470 +2.717 +1.806 +2.807 +2.026 +2.941 +2.529 +PW91-LCAO-H +2.655 +1.171 +2.670 +1.522 +2.703 +1.858 +2.790 +2.083 +2.932 +2.586 +PW91-LCAO-U +2.642 +1.186 +2.668 +1.536 +2.696 +1.876 +2.785 +2.103 +2.932 +2.598 +PW91-PW +2.639 +1.185 +2.659 +1.534 +2.693 +1.862 +2.783 +2.089 +2.928 +2.560 +PBE-LCAO-M +2.690 +1.126 +2.710 +1.441 +2.724 +1.771 +2.814 +1.994 +2.945 +2.501 +PBE-LCAO-H +2.668 +1.155 +2.685 +1.497 +2.710 +1.826 +2.797 +2.053 +2.932 +2.558 +PBE-LCAO-U +2.651 +1.170 +2.677 +1.510 +2.702 +1.844 +2.790 +2.073 +2.932 +2.571 +PBE-PW +2.648 +1.167 +2.670 +1.506 +2.700 +1.829 +2.788 +2.058 +2.930 +2.533 +B3LYP-LCAO-M +2.373 +3.734 +2.410 +5.164 +2.457 +6.029 +2.558 +6.586 +2.725 +8.340 +B3LYP-LCAO-H +2.655 +1.067 +2.691 +1.426 +2.714 +1.772 +2.812 +1.977 +- +- +B3LYP-LCAO-U +2.642 +1.100 +2.679 +1.461 +2.705 +1.816 +2.803 +2.025 +- +- +B3LYP-PW +2.681 +0.944 +2.715 +1.211 +2.737 +1.470 +2.830 +1.659 +2.986 +1.963 +PBE0-LCAO-M +2.322 +4.877 +2.358 +6.807 +2.409 +7.978 +2.512 +8.657 +- +- +PBE0-LCAO-H +2.635 +1.092 +2.654 +1.523 +2.679 +1.970 +2.773 +2.219 +- +- +PBE0-LCAO-U +2.626 +1.128 +2.642 +1.567 +2.670 +2.023 +2.764 +2.277 +- +- +PBE0-PW +2.649 +0.963 +2.671 +1.288 +2.690 +1.640 +2.779 +1.879 +2.910 +2.444 +HSE03-LCAO-M +2.694 +1.030 +2.715 +1.351 +2.729 +1.696 +2.825 +1.919 +2.725 +2.436 +HSE03-LCAO-H +2.668 +1.044 +2.693 +1.385 +2.714 +1.728 +2.807 +1.949 +- +- +HSE03-LCAO-U +2.663 +1.058 +2.687 +1.396 +2.710 +1.744 +2.801 +1.966 +- +- +HSE03-PW +2.651 +1.049 +2.664 +1.392 +2.697 +1.742 +2.787 +1.971 +2.925 +2.484 +HSE06-LCAO-M +2.697 +1.061 +2.716 +1.358 +2.733 +1.707 +2.829 +1.932 +2.954 +2.431 +HSE06-LCAO-H +2.676 +1.075 +2.693 +1.391 +2.719 +1.738 +2.812 +1.961 +- +- +HSE06-LCAO-U +2.666 +1.088 +2.686 +1.402 +2.709 +1.753 +2.803 +1.978 +- +- +HSE06-PW +2.650 +1.075 +2.664 +1.396 +2.695 +1.750 +2.786 +1.982 +2.923 +2.479 +TABLE VI. Elastic constants for 1D (GPa nm2), 2D (GPa nm), and 3D (GPa) calculated by using different DFT-attributes. +1D +Honeycomb +Square +Hexagonal +3D +DFT-Methods +C11 +C11 +C12 +C66 +C11 +C12 +C66 +C11 +C12 +C66 +C11 +C12 +C66 +DFTB +88.2 +163.6 +63.8 +49.9 +57.8 +9.7 +-3.9 +42.7 +22.6 +10.1 +110.7 +102.4 +19.9 +LDA-LCAO-M +24.5 +34.3 +14.9 +9.7 +80.5 +9.0 +-5.9 +77.9 +30.7 +23.6 +163.6 +133.0 +53.4 +LDA-LCAO-H +25.2 +33.7 +17.0 +8.3 +79.5 +10.7 +-7.5 +79.1 +28.4 +25.3 +165.4 +131.0 +56.3 +LDA-LCAO-U +25.8 +34.2 +17.4 +8.4 +80.6 +12.1 +-7.6 +85.3 +27.1 +29.1 +164.3 +131.4 +54.7 +LDA-PW +24.7 +34.0 +18.3 +7.9 +79.4 +12.0 +-8.8 +79.2 +31.4 +23.9 +165.4 +131.1 +58.7 +RPBE-LCAO-M +15.5 +19.0 +8.7 +5.1 +48.6 +4.6 +-2.9 +43.5 +13.8 +14.9 +103.4 +88.0 +35.9 +RPBE-LCAO-H +15.3 +19.4 +9.0 +5.2 +47.8 +5.6 +-2.3 +48.6 +16.7 +16.0 +103.6 +83.9 +32.7 +RPBE-LCAO-U +15.1 +19.6 +8.4 +5.6 +47.9 +6.5 +-2.7 +44.6 +21.8 +11.4 +103.4 +82.9 +31.3 +RPBE-PW +16.0 +20.5 +9.4 +5.5 +48.2 +6.4 +-3.4 +49.0 +17.1 +16.0 +92.7 +72.0 +25.4 +PW91-LCAO-M +18.8 +24.1 +10.7 +6.7 +57.3 +6.0 +-3.0 +56.2 +-9.0 +32.6 +133.3 +81.2 +16.7 +PW91-LCAO-H +18.6 +24.5 +11.7 +6.6 +56.7 +7.4 +-3.4 +56.3 +21.8 +17.2 +116.4 +69.2 +19.7 +PW91-LCAO-U +19.1 +24.2 +11.0 +6.6 +56.1 +8.1 +-3.6 +56.8 +21.1 +17.8 +117.7 +68.9 +19.0 +PW91-PW +19.1 +24.7 +11.5 +6.6 +57.5 +8.2 +-4.1 +56.8 +21.3 +17.7 +109.8 +85.9 +29.7 +PBE-LCAO-M +16.9 +25.5 +8.63 +8.4 +56.2 +5.5 +-3.1 +55.2 +20.1 +17.5 +114.1 +67.6 +34.2 +PBE-LCAO-H +17.6 +23.0 +10.7 +6.2 +54.8 +6.8 +-3.6 +56.2 +18.8 +18.7 +113.2 +68.3 +20.5 +PBE-LCAO-U +18.6 +23.2 +10.7 +6.2 +55.3 +7.7 +-3.7 +55.8 +20.7 +17.5 +115.2 +68.0 +20.0 + +16 +TABLE VI. (Continued) +1D +Honeycomb +Square +Hexagonal +3D +DFT-Methods +C11 +C11 +C12 +C66 +C11 +C12 +C66 +C11 +C12 +C66 +C11 +C12 +C66 +PBE-PW +18.3 +23.4 +11.0 +6.2 +55.3 +7.7 +-4.3 +55.8 +20.4 +17.7 +107.7 +84.2 +31.0 +B3LYP-LCAO-M +65.3 +97.4 +45.6 +25.9 +160.2 +52.2 +-31.9 +168.8 +85.3 +41.8 +- +- +- +B3LYP-LCAO-H +19.4 +23.3 +10.5 +6.4 +44.9 +17.0 +-3.6 +51.3 +19.2 +16.0 +- +- +- +B3LYP-LCAO-U +20.5 +24.2 +10.7 +6.7 +46.0 +18.2 +-3.4 +52.8 +20.6 +16.1 +- +- +- +B3LYP-PW +35.9 +20.8 +9.6 +5.6 +38.9 +15.5 +-1.8 +47.2 +17.6 +14.8 +- +- +- +PBE0-LCAO-M +80.4 +115.1 +58.9 +28.1 +205.4 +60.0 +-90.1 +203.9 +103.2 +50.3 +- +- +- +PBE0-LCAO-H +20.2 +25.1 +11.1 +7.0 +48.4 +23.8 +-8.0 +58.3 +20.2 +19.1 +- +- +- +PBE0-LCAO-U +20.8 +26.3 +14.8 +5.8 +45.2 +25.2 +-8.3 +59.9 +21.9 +19.0 +- +- +- +PBE0-PW +17.6 +22.5 +11.0 +5.7 +40.0 +22.2 +-5.3 +55.1 +19.6 +17.6 +138.4 +73.0 +- +HSE03-LCAO-M +17.7 +23.1 +10.2 +6.5 +53.9 +8.3 +-3.1 +54.0 +19.6 +17.2 +96.4 +84.5 +27.9 +HSE03-LCAO-H +18.0 +23.4 +10.6 +6.4 +50.9 +10.2 +-3.3 +53.2 +20.5 +16.3 +- +- +- +HSE03-LCAO-U +17.4 +22.7 +10.9 +5.9 +50.3 +10.0 +-3.5 +53.8 +19.4 +17.2 +- +- +- +HSE03-PW +20.9 +22.3 +11.5 +5.4 +52.4 +9.1 +-4.5 +53.8 +21.2 +16.3 +96.2 +83.0 +13.7 +HSE06-LCAO-M +17.4 +23.1 +10.3 +6.4 +52.1 +8.6 +-3.1 +52.8 +19.3 +16.8 +113.5 +95.6 +36.5 +HSE06-LCAO-H +18.6 +23.2 +10.6 +6.3 +49.9 +11.0 +-3.2 +51.9 +19.5 +16.2 +- +- +- +HSE06-LCAO-U +17.1 +24.0 +9.9 +7.0 +49.1 +11.3 +-3.5 +53.3 +18.9 +17.2 +- +- +- +HSE06-PW +26.5 +21.9 +11.5 +5.2 +50.5 +11.1 +-5.3 +54.4 +20.3 +17.0 +94.2 +87.2 +14.4 +TABLE VII. Estimation of contribution of s, p, and d orbitals to the density of states by implementing different DFT-attributes +1D +Honeycomb +Square +Hexagonal +3D +DFT-Methods +Ns +Np +Nd +Ns +Np +Nd +Ns +Np +Nd +Ns +Np +Nd +Ns +Np +Nd +DFTB +1.01 +0.19 +0.15 +0.52 +0.39 +0.12 +0.36 +0.45 +0.12 +0.34 +0.42 +0.13 +0.19 +0.46 +0.15 +LDA-LCAO-M +0.97 +0.08 +0.53 +0.56 +0.14 +0.18 +0.39 +0.18 +0.20 +0.33 +0.16 +0.22 +0.20 +0.26 +0.14 +LDA-LCAO-H +1.00 +0.04 +0.79 +0.57 +0.13 +0.26 +0.41 +0.18 +0.26 +0.34 +0.16 +0.27 +0.21 +0.22 +0.18 +LDA-LCAO-U +0.99 +0.04 +0.81 +0.56 +0.13 +0.26 +0.40 +0.18 +0.27 +0.33 +0.16 +0.26 +0.21 +0.22 +0.18 +LDA-PW +0.64 +0.39 +0.31 +0.17 +0.54 +0.08 +0.10 +0.50 +0.11 +0.08 +0.44 +0.09 +0.01 +0.41 +0.02 +RPBE-LCAO-M +1.04 +0.08 +0.37 +0.67 +0.13 +0.13 +0.45 +0.17 +0.13 +0.40 +0.17 +0.18 +0.26 +0.26 +0.12 +RPBE-LCAO-H +1.06 +0.04 +0.53 +0.67 +0.12 +0.18 +0.47 +0.17 +0.17 +0.41 +0.15 +0.22 +0.26 +0.22 +0.18 +RPBE-LCAO-U +1.05 +0.04 +0.54 +0.67 +0.12 +0.18 +0.47 +0.17 +0.18 +0.40 +0.15 +0.22 +0.26 +0.22 +0.18 +RPBE-PW +0.75 +0.33 +0.23 +0.28 +0.52 +0.07 +0.16 +0.49 +0.08 +0.14 +0.43 +0.08 +0.03 +0.49 +0.02 +PW91-LCAO-M +1.01 +0.08 +0.28 +0.63 +0.13 +0.11 +0.43 +0.18 +0.11 +0.38 +0.17 +0.15 +0.24 +0.26 +0.11 +PW91-LCAO-H +1.04 +0.04 +0.58 +0.63 +0.12 +0.20 +0.45 +0.17 +0.19 +0.39 +0.16 +0.23 +0.25 +0.23 +0.17 +PW91-LCAO-U +1.03 +0.04 +0.59 +0.63 +0.12 +0.20 +0.45 +0.17 +0.19 +0.38 +0.16 +0.22 +0.25 +0.22 +0.17 +PW91-PW +0.73 +0.33 +0.23 +0.25 +0.53 +0.07 +0.14 +0.50 +0.08 +0.12 +0.44 +0.08 +0.02 +0.48 +0.02 +PBE-LCAO-M +1.02 +0.08 +0.31 +0.64 +0.13 +0.12 +0.44 +0.17 +0.12 +0.38 +0.17 +0.16 +0.24 +0.26 +0.12 +PBE-LCAO-H +1.04 +0.04 +0.59 +0.65 +0.12 +0.20 +0.46 +0.17 +0.19 +0.39 +0.15 +0.23 +0.25 +0.22 +0.17 +PBE-LCAO-U +1.04 +0.04 +0.60 +0.64 +0.12 +0.20 +0.45 +0.17 +0.19 +0.39 +0.15 +0.23 +0.25 +0.22 +0.18 +PBE-PW +0.73 +0.33 +0.23 +0.25 +0.53 +0.07 +0.14 +0.50 +0.08 +0.12 +0.44 +0.08 +0.02 +0.49 +0.02 +B3LYP-LCAO-M +0.62 +0.10 +0.01 +0.41 +0.14 +0.00 +0.29 +0.18 +0.00 +0.27 +0.15 +0.00 +0.19 +0.22 +0.01 +B3LYP-LCAO-H +0.81 +0.03 +0.02 +0.55 +0.10 +0.01 +0.41 +0.15 +-0.01 +0.37 +0.13 +0.00 +- +- +- +B3LYP-LCAO-U +0.78 +0.03 +0.02 +0.55 +0.10 +0.01 +0.41 +0.15 +-0.01 +0.37 +0.14 +0.00 +- +- +- +B3LYP-PW +0.55 +0.26 +0.02 +0.21 +0.43 +0.01 +0.13 +0.40 +0.02 +0.12 +0.35 +0.02 +0.04 +0.39 +0.01 +PBE0-LCAO-M +0.50 +0.10 +0.01 +0.37 +0.14 +0.00 +0.25 +0.19 +0.00 +0.24 +0.16 +0.00 +- +- +- +PBE0-LCAO-H +0.71 +0.03 +0.02 +0.51 +0.10 +0.01 +0.38 +0.15 +-0.01 +0.34 +0.13 +0.00 +- +- +- +PBE0-LCAO-U +0.70 +0.03 +0.02 +0.50 +0.10 +0.01 +0.37 +0.15 +-0.01 +0.34 +0.13 +0.00 +- +- +- +PBE0-PW +0.47 +0.26 +0.01 +0.18 +0.42 +0.01 +0.10 +0.39 +0.02 +0.10 +0.34 +0.01 +0.01 +0.38 +0.01 +HSE03-LCAO-M +0.92 +0.07 +0.03 +0.58 +0.12 +0.03 +0.39 +0.16 +0.02 +0.35 +0.15 +0.04 +0.23 +0.25 +0.07 +HSE03-LCAO-H +0.94 +0.04 +0.05 +0.57 +0.12 +0.05 +0.40 +0.16 +0.04 +0.35 +0.14 +0.06 +- +- +- +HSE03-LCAO-U +0.93 +0.04 +0.05 +0.57 +0.12 +0.05 +0.40 +0.16 +0.04 +0.35 +0.14 +0.06 +- +- +- +HSE03-PW +0.67 +0.30 +0.03 +0.22 +0.49 +0.01 +0.13 +0.45 +0.02 +0.11 +0.40 +0.02 +0.02 +0.46 +0.01 +HSE06-LCAO-M +0.88 +0.07 +0.03 +0.55 +0.11 +0.03 +0.38 +0.15 +0.02 +0.34 +0.14 +0.04 +0.22 +0.24 +0.06 +HSE06-LCAO-H +0.90 +0.03 +0.04 +0.55 +0.11 +0.04 +0.39 +0.15 +0.04 +0.34 +0.13 +0.05 +- +- +- +HSE06-LCAO-U +0.89 +0.03 +0.04 +0.55 +0.11 +0.05 +0.39 +0.15 +0.04 +0.34 +0.13 +0.05 +- +- +- +HSE06-PW +0.63 +0.29 +0.02 +0.21 +0.47 +0.01 +0.12 +0.43 +0.02 +0.11 +0.38 +0.02 +0.02 +0.45 +0.01 + diff --git a/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/load_file.txt b/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b899b3582e2ce82271f1c8c8d680553e94b0957a --- /dev/null +++ b/-NAzT4oBgHgl3EQf_P6F/content/tmp_files/load_file.txt @@ -0,0 +1,2283 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf,len=2282 +page_content='Optimizing density-functional simulations for two-dimensional metals Kameyab Raza Abidi and Pekka Koskinen∗ NanoScience Center, Department of Physics, University of Jyv¨askyl¨a, 40014 Jyv¨askyl¨a, Finland (Dated: January 6, 2023) Unlike covalent two-dimensional (2D) materials like graphene, 2D metals have non-layered struc- tures due to their non-directional, metallic bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' While experiments on 2D metals are still scarce and challenging, density-functional theory (DFT) provides an ideal approach to predict their basic properties and assist in their design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, DFT methods have been rarely benchmarked against metallic bonding at low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Therefore, to identify optimal DFT attributes for a desired accuracy, we systematically benchmark exchange-correlation functionals from LDA to hybrids and basis sets from plane waves to local basis with different pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' With 1D chain, 2D hon- eycomb, 2D square, 2D hexagonal, and 3D bulk metallic systems, we compare the DFT attributes using bond lengths, cohesive energies, elastic constants, densities of states, and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Although today most DFT studies on 2D metals use plane waves, our comparisons reveal that local basis with often-used PBE exchange-correlation is well sufficient for most purposes, while plane waves and hybrid functionals bring limited improvement compared to the greatly increased compu- tational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These results ease the demands for generating DFT data for better interaction with experiments and for data-driven discoveries of 2D metals incorporating machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' INTRODUCTION The discovery of graphene nearly two decades ago sparked an entire new research field of two-dimensional (2D) materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The 2D materials pedigree has ex- panded ever since, thanks to unique properties and vi- sions for novel applications [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Most 2D materials are covalently bound and have layered structures eas- ily exfoliable from three-dimensional (3D) bulk matter [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, in contrast to directional covalent bond- ing, non-directional metallic bonding prefers large coor- dination numbers, which renders low-dimensional metal structures energetically unfavourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Despite this pref- erence for large coordination, in 2014 atomically thin sta- ble iron patches were discovered in graphene pores [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' This discovery has been followed by rapid progress in re- search on 2D metals and alloys, making 2D metals a full member the 2D materials family [9–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The wavering stability of 2D metals makes experi- ments challenging, whereby research relies heavily on computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A reasonable description of metallic bond- ing requires electronic structure simulations, which has made the density-functional theory (DFT) [15, 16] the workhorse method for modeling 2D metals [17–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Most DFT studies have chosen plane wave (PW) basis sets [32] and the non-empirical Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These choices for DFT attributes are plausible in the context of delocal- ized electrons in periodic systems that are still lacking experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, DFT attributes have not been systematically benchmarked for metallic bonding at low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' It is not certain whether these stan- dard choices are efficient and accurate enough or they if simply waste computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' ∗ pekka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='koskinen@jyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='fi The DFT attributes consist of few central choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The first choice is the flavor of exchange-correlation (xc) func- tional, the level of which is of central importance for con- sistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A functional performing well in some sys- tems may perform poorly in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Here we make use of several xc-functionals to obtain a systematic picture of their performance in low-dimensional metallic bond- ing [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The second choice is the type of basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Plane waves are suitable for periodic systems, whose elec- trons fill out the entire simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Unfortunately, the non-periodic directions of low-dimensional systems require large vacuum regions that make PW simulations inefficient compared to modeling bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Thus, an addi- tional choice in PW simulations is an optimum size of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In this respect, PW and grid-based DFT share the same challenges [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Another alternative for ba- sis is linear combination of atomic orbitals (LCAO), and controlling its size provides a powerful handle to trade between accuracy and efficiency [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The choice of basis type has implications beyond mere accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For example, PW is not suitable for studying electron transport using nonequilibrium Green’s function method in nanoscaled devices [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In addition, with the coming of data science and machine learning in materials science, lots of consistent DFT data is required for ma- chine learning -enabled 2D metals studies [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' This efficiency demand calls for a critical examination of the necessity of PW method to model metallic bonding in low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Third choice for periodic systems is the number of k- points along periodic directions for the desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Fourth choice is the level of Fermi-broadening of elec- tronic states, which is partly a physical choice but mostly a necessity for rapid convergence of the self-consistent it- eration of the electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In practice, there are a plethora of other choices to make for numerical stability and speedup, but they are often chosen as default val- ues that have been previously fine-tuned for each DFT arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01945v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schematics of the systems with different dimensional- ities and coordination numbers C: 1D chain (C = 2), 2D hon- eycomb (C = 3), 2D square (C = 4), 2D hexagonal (C = 6), and 3D bulk (C = 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The quadrilaterals show the simula- tion cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In this article, we consider the above-mentioned choices of DFT attributes regarding xc-functionals, basis sets, vacuum, k-point sampling, and Fermi-broadening, and juxtapose their performance against various prop- erties of selected low-dimensional metal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The selected systems include a one-dimensional chain (coor- dination number C = 2), three two-dimensional lattices (C = 3, 4, and 6), and a 3D bulk (C = 12) (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These systems enable comparative analysis of the perfor- mance of DFT attributes in various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Being low-dimensional systems, these structures are prone to various symmetry-breaking deformations, such as out-of- plane buckling in 2D or Peierls distortions in 1D [26, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, in order to enable unambiguous comparison of the effect of dimensionality and coordination and avoid making unfounded conclusions based on incomplete set of deformations, we retain our focus on these ideal, non- deformed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We also compare the performance and speed of DFT to the density-functional tight-binding (DFTB) method, which is the next-in-line approximation to DFT [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' One of our main conclusions is that, for general purposes, DFT-LCAO can be chosen over the de- fault DFT-PW without compromising accuracy, a choice which enables simulating transport and helps generating DFT data more effortlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Our treatise will advance DFT modeling of 2D metals and help boosting the inter- action with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' COMPUTATIONAL METHODS The basic idea DFT is to use the variational principle to generate exact ground state energy and density for the systems of interest [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The ground state energy E is a functional of the electron density (n), E[n] = T[n] + Eext[n] + EH[n] + Exc[n] , (1) where T[n] is the Kohn-Sham kinetic energy for the fic- titious non-interacting electron system, Eext[n] is the ex- TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Exchange-correlation functionals used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Functional and its family Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Local Density Approximation (LDA) [15, 55] Generalized Gradient Approximation (GGA) [56] RPBE [57] PW91 [58, 59] PBE [33] Hybrid Functionals [60] B3LYP [61] PBE0 [62] HSE03 (screening ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 Bohr−1) [63] HSE06 (screening ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 Bohr−1) [64] ternal potential energy, EH[n] is the Hartree energy, and Exc[n] is the exchange-correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The xc term attempts to capture the complex features of many-body quantum mechanics, and a variety of approximate xc functionals have been developed for different purposes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As a result, the quality of xc functional mostly determines the quality of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Here, using the QuantumATK (S-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='06) DFT implementation [46], we explore the set of eight xc functionals ranging from local density approximation to hybrid functionals (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We used two types of basis sets, plane waves and LCAOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The wave-function energy cutoff for plane waves was 800 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cutoff needed no separate analysis for low- dimensional metals, because it depends only on element and pseudopotential [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For LCAOs, we used three variants: LCAO-M(edium), LCAO-H(igh), and LCAO- U(ltra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These variants derive from the numerical basis sets of the FHI-aims package [48], but are further opti- mized for computational speed of the LCAO calculator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For example, for Ag the radial functions for Medium ba- sis are 3s/2p/1d (14), for High 4s/3p/5d/1f (35), and for Ultra 4s/3p/5d/2f/1g (51), with brackets displaying the total number of orbitals per atom [37, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Local ba- sis sets were used in conjunction with norm-conserving PseudoDojo pseudopotentials [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Further, the total energy convergence criteria for self- consistent electron density was ≤ 10−7eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' System ge- ometries were optimized to forces below 1 meV�A −1 and stresses below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 meV�A −3 using the LBFGS [50] algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The k-points were sampled by the Monkhorst- Pack method [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' All calculations were spin-polarized and the initial guess for lattice parameters were adopted from the Atlas of 2D metals [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To complement the results with various DFT at- tributes with wider context, we analyzed the systems with Ag also with DFTB method at the level of self- consistent charge [45, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The Ag parametrizations were taken from earlier studies [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (a) 1D Chain (b) 2D Honeycomb (hc) (c) 2D Square (sq) OODD (d) 2D Hexagonal (hex) (e) 3D Bulk X3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Convergence Analysis We made various systematic convergence analyses for the group of coinage metals Cu, Ag, and Au [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Computational and experimental studies have shown that the free-standing monolayer patches of these met- als are stabilized by graphene pores [13, 22, 24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The analyses were done using PBE xc-functional [33], projec- tor augmented waves (PAW) for core electrons [65], and plane waves for valence electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' k-point convergence: The k-point convergence was studied using the 2D systems with a converged vac- uum of 15 �A in the non-periodic direction (as confirmed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The total energy is practically converged at 30 × 30 × 1 k-point sampling, and we define the energy tolerance using this value, ∆E = ENk×Nk×1 − E30×30×1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (2) Apart from rapid convergence at very few k-points, the convergence is exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chosen relative energy tol- erance can therefore be approximated by log δ = A1 + B1L , (3) where δ =| ∆E | /E3D is an (approximate) relative en- ergy tolerance, the ratio between energy tolerance to the 3D cohesive energy E3D [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The length L = acNk, the product of simulation box length and the number of k- points in corresponding direction, is the maximum period of the Bloch wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Using L as the convergence parameter helps identifying the required k-point sam- pling for variable simulation cell sizes in later research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The k-point convergence is not monotonic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' more k- points does not necessarily mean better accuracy (Fig- ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, for different system symmetries and cell shapes and sizes, the ansatz (3) works satisfactorily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Lin- ear regression analysis to the data gives the parameters A1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='29 and B1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='036 �A −1 (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Inverting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (3), we can obtain an optimal number of k-points for given simulation cell size ac and desired accuracy δ as Nk(δ) = ceil �L(δ) ac � , (4) where ceil(x) = ⌈x⌉ maps x to the least integer greater than or equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For instance, with relative accuracy δ = 10−3 one obtains the Nk = ⌈47 ˚A/ac⌉, suggesting Γ-point calculations for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7-nm-sized simulation cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In subsequent analyses, we use Nk = 13, suggesting ∼ δ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='−3 relative tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vacuum convergence: Using plane waves requires periodicity in all directions, regardless of system dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Low-dimensional systems need therefore a large vacuum region in the non-periodic direction to avoid spu- rious interactions with periodic images of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Larger vacuum means more volume and computational FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The k-point convergence of total energy for 2D sys- tems made of coinage metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' δ is the relative energy toler- ance and L is the maximum period of the Bloch function [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='(4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The linear fit refers to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' cost, implying a need to minimize the vacuum without affecting the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For a complete picture, we inves- tigate vacuum convergence not only in 2D systems and but also in 1D chains and free atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We normalize atoms’ dimensions by their van der Waals radii RvdW and consider the normalized vacuum Lnorm = Lvac/RvdW , where Lvac is the vacuum along the non-periodic direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=', the separation between peri- odic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=') The total energy is practically converged at 8-˚A vacuum, and we define the energy tolerance as ∆E = E(Lvac) − E(8 �A) and relative energy tolerance again as δ = ∆E/E3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The tolerance converges roughly exponentially, log δ = A2 + B2Lnorm (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Conse- quently, the vacuum for a desired relative energy accu- racy for a given element can be estimated from Lvac(δ) = RvdW (log δ − A2) B2 , (5) where the parameters A2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 and B2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='65 were obtained by linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For instance, the relative tolerance δ = 10−3 requires Lvac = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 × RvdW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In subsequent analysis, if not said otherwise, we will use Lvac = 10 �A, which for Ag means δ = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2, in rough alignment with k-point convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Still, such a single estimate is indicative at best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The vacuum convergence follows roughly the coordination number, free atom converging the slowest, hexagonal sys- tem the fastest (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' This suggests that for a given element the vacuum should be set by the lowest- coordinated atom—or by the free atom to be on the safe side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' After all, a modest 16 % increase in vacuum (Lnorm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 → 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0) may increase the relative accuracy by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Thus, a single fit as above is not the best guideline and the vacuum convergence is best considered by case basis, especially in the presence Au 100 Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' SO SC Auhc 10-1 Linear fit 10-2 10-3 10-5 10-6 10-7 10-8 20 40 60 80 100 120 0 140 L (A)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vacuum convergence of the total energy for 1D and 2D systems made of coinage metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' δ is the relative energy tolerance and Lnorm is vacuum normalized in terms of van der Waals radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Free atom vacuum convergences are added for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' of possible charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Effect of Fermi broadening In principle the Fermi-broadening is a physical param- eter intimately linked to the electronic temperature T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' in practice it is frequently used as a technical parameter to accelerate the self-consistency convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The tech- nical attitude towards broadening is evident in available methods other than the Fermi-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Computational literature shows a plethora of different values for Fermi- broadening, but its effect is rarely discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For insulators and semiconductors the broadening is inconse- quential, but for metals it matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In this section, we want to investigate its effect on the energetics systemat- ically, for sheer completeness and future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ideally, broadening should be chosen to enable rapid convergence without conflicting too much with other con- vergence parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We investigated the effect of broad- ening by increasing the electronic temperature T from 10−5 K to 1000 K and looked at the energy difference ∆E(T) = E(T) − E(10−5 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (6) The temperature 10−5 K was the smallest that enabled robust convergence for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vacuum was 15 ˚A for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As a result, 1D systems were most sensi- tive to the broadening, 3D bulk systems were least sen- sitive (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' This result is plausible, because the density of states is the smallest for 1D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 2D and 3D systems there are more k-points, density of states at Fermi-level is greater, and state occupations average over a larger set of states, consequently diminishing the influence of broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The 2D systems show energy FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The effect of electronic temperature on the cohesion energy of coinage metals in different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' variation around ∼ 10 meV upon increasing temperature to 1000 K, corresponding to 86 meV energy broaden- ing (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For the remainder of the calculations in this article, we used the electronic temperature of 580 K (�=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Performance of exchange-correlation functionals We investigated the performance of xc functionals by first fixing certain attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To eliminate uncertain- ties from an insufficient description of valence electrons, we used the most complete PW basis set and the PAW potential to describe the core electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We used the converged number of k-points and size of vacuum from previous analysis, as well as the recently adopted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 eV broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' With these choices, we may concentrate on the performance of xc-functionals without worrying too much about artifacts from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We also investigate xc functionals by using only Ag systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' By belonging to the same group, the coinage metals follow similar trends and it is reasonable to expect other metals to follow the trends of Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Still, we do not claim Ag displays completely universal trends, for there are elements that have complex many-body effects even beyond the capabilities of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In the following, we compare the xc-functional perfor- mance against bond lengths, cohesive energies, and elas- tic moduli of all 1D, 2D, and 3D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The electronic structure is compared in terms of later-introduced char- acteristic figures related to the density of states at the Fermi-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cohesive Energies: The cohesive energy was de- fined as Ecoh = Efree − E/N , (7) 100 10 Ag1D Ag AuFree AuD Auhex ny AU CuiD CuFree hex 10-4 Linear fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 norm0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 (Aa) △E 10 15 3D 2D 1D 20 400 0 200 600 800 1000 Electronic temperature (K5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The cohesive energies of optimized 1D, 2D (hc, sq, and hex), and 3D systems of Ag with different xc-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' where E is the energy of the system with N atoms and Efree is the energy of free atom calculated by placing it inside a 15-�A cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' All functionals display similar trends, cohesive energy increasing monotonically from 1D to 3D bulk (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yet the quantitative differences are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LDA displays its well-known tendency to overestimate cohesive ener- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The 3D bulk cohesion shoots over the experimental value by 23 % [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' GGA functionals work significantly better, where PW91 and PBE are now off by approx- imately ≈ 13 − 14 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In contrast, RPBE shows con- siderable underbinding and even less accurate cohesion than LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Among hybrid functionals, the performance of screened exchange HSE03 and HSE06 is better than PBE0, which still suffers from the spurious Coulomb in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' B3LYP describes cohesion poorly and is out- performed by practically all other functionals, and should be avoided while modeling 2D metals—a conclusion not surprising in the light of previous observations [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In addition, convergence of free atom with B3LYP was dif- ficult and required loosening the convergence criterion to ≤ 10−6 eV (loosening had an insignificant effect on the cohesion of Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As a rule, GGA and hybrid func- tionals outperform LDA, but a hybrid functionals do not necessarily outperform GGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' PW91 and PBE appear as still as fair choices for robust energetics for general pur- poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Dimensionality-dependence of energetics: In 2D metal modeling, the coordination of single metal atoms can range from C ∼ 1 to C ∼ 6 and occasionally beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The computational method should therefore capture cor- rectly the relative energetics of atoms at different coor- dination numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In other words, the cohesion should increase with the coordination number with an appropri- ate dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Our ansatz for the C-dependence for FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Trends of low-dimensional energetics with different xc-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The fitted scaling exponent γ is plotted for different xc-functionals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' smaller γ means that energy depends less linearly on the coordination number [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' the cohesion Ecoh is Ecoh(C) = E3D coh × (C/12)γ , (8) where E3D coh is the 3D bulk cohesion and γ is an expo- nent that quantifies the coordination- or dimensionality- dependence of the cohesion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The ansatz has the correct asymptotic limits [Ecoh(0) = 0 and Ecoh(12) = E3D coh] and suffices for our purposes in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (We tested also more refined ansatzes, but the conclusions remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=') The exponent γ was obtained by fitting the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (8) for energies from each functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As the result, LDA and all GGA and HSE function- als show roughly the same γ, the same dimensionality- dependence in energetics (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Especially the de- pendencies in different GGAs are nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Only the dependencies in B3LYP and PBE0 are clear out- liers, PBE0 showing more linear dependence on C (γ closer to one) and B3LYP showing more non-linear de- pendence on C (γ further away from one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Interestingly, although LDA badly overestimates the absolute cohe- sion energies, the dimensionality-dependence lies some- where in between GGAs and HSE functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In conclu- sion, GGA-PBE appears to capture the dimensionality- dependence of energetics comparably well and be still a serious competitor to the far more costly HSE function- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bond Lengths: The bond lengths were obtained directly from the optimized lattice constants (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In accordance with overbinding, LDA functional shows small bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 3D, the functionals PW91, PBE, PBE0, HSE03, and HSE06 are underbinding and show 1 − 2 % too large bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' PBE0 shows short- est bonds among hybrid functionals, and B3LYP shows longest bonds among all functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nearly all function- als show monotonic increase of bond length with coor- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 Ag3D Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ag 0 hex 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 - (eV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 Cohesive energy ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 LDA RPBE PW91 PBE B3LYP PBEO HSE03 HSE06 Exchange-correlation functional0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='42 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='36 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='32 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='30 LDA RPBE PW91 PBE B3LYP PBEO HSE03HSE06 Exchange-correlation functional6 dination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Only LDA functional is an exception: it has a slightly smaller bond length for 2D hexagonal lattice than for 1D chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic constants (theory recap): Due to colorful practices in the notations of low-dimensional elasticity, and to avoid any confusion, we wish to define explicitly the elastic constants presented in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Within the linear elastic regime the stresses {σi} and strains {εi} (i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 6) satisfy the generalized Hooke’s law σi = 6 � j=1 Cijεj , (9) where Cij are elastic constants and expressed as a 6 × 6 matrix and ε1 = εxx, ε2 = εyy, ε3 = εzz, ε4 = 2εyz, ε5 = 2εxz, ε6 = 2εxy, when following the Voigt notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We adapted the formalism of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [68–72] to evaluate the elastic constants for 1D, 2D and 3D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 3D, the strain tensor is ϵ3D = � � ε1 ε6/2 ε5/2 ε6/2 ε2 ε4/2 ε5/2 ε4/2 ε3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (10) The elastic constants are obtained by applying selected strains {εi} to the equilibrium simulation cell and by calculating the partial derivatives Cij = ∂2∆U ∂εi∂εj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (11) Here ∆U(εi) = U(εi)−U(0) is the elastic energy density per unit volume, where U(εi) is the energy density at strain εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For a system with cubic symmetry, the energy FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Optimized bond lengths of 1D, 2D (hc, sq, and hex), and 3D systems of Ag with different xc-functionals density is ∆U(εi) =1 2 � C11ε2 1 + C11ε2 2 + C11ε2 3 + C12ε1ε2 + C12ε1ε3 +C12ε2ε1 + C12ε2ε3 + C12ε3ε1 + C12ε3ε2 +C44ε2 4 + C44ε2 5 + C44ε2 6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (12) For 2D systems, the strain tensor is ϵ2D = � ε1 ε6/2 ε6/2 ε2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (13) Again, the elastic constants are obtained by applying se- lected strains {εi} to the equilibrium simulation cell and by calculating the partial derivatives Cij = ∂2∆U ∂εi∂εj (14) Here ∆U(εi) = U(εi) − U(0) is the energy density per unit area, where U(εi) is the energy density at strain εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For a system with square symmetry, the energy density is ∆U(εi) =1 2(C11ε2 1 + C22ε2 2 + 2C12ε1ε2 + 2C16ε1ε6 +2C26ε2ε6 + C66ε2 6) (15) and all three elastic constants C11, C12 and C66 are in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, for a hexagonal system, only con- stants C11 and C12 are independent and C66 = (C11 − C12)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Finally, for 1D systems, the strain-tensor matrix is sim- ply ϵ1D = (ε1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yet again, the elastic constant is obtained by applying the strain ε1 to the equilibrium simulation cell and by taking the partial derivative C1 = ∂2∆U ∂2ε1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (16) Here ∆U(εi) = U(εi) − U(0) is the energy density per unit length, where U(εi) is the energy density at strain εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In other words, ∆U(ε1) = 1 2C11ε2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (17) Table II summarizes the formulae for the elastic con- stants and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Note that the elastic constants in different dimensions have also different units: they are GPa for 3D, GPa nm for 2D, and GPa nm2 for 1D (GPa nm3−D or eV/˚AD in short, where D is the dimensional- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic constants (results): Functionals show similar trends for bulk moduli, but there are quantita- tive differences (Figure 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We remind that because the elastic moduli in different dimensions have different units, the trend with respect to the coordination num- ber can be compared only between different 2D lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LDA overestimates the bulk moduli systematically, for 3D bulk by almost 40 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Only for 1D chain the modulus Aghc Ag3D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 - bso 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='90 - Bond length (A) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 LDA RPBE PW91 PBE B3LYP HSE03 HSE06 PBEO Exchange-correlation functional7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic properties of low-dimensional systems of Ag with different xc-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bulk moduli (a) and Young’s moduli (b) are shown for all systems, shear moduli (c) and Poisson’s ratio (d) are shown only for 3D and stable 2D sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Units for moduli are GPa nm3−D, where D is the sys- tem dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Formulae for Bulk Modulus (K), Shear-modulus (G), Young’s modulus (Y), and Poisson’s ratio (µ) for the systems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' System K G Y µ 1D C11 K 2Dhex/hc C11+C12 2 C11−C12 2 4KG K+G K−G K+G 2Dsq C11+C12 2 C66 C2 11−C2 12 C11 C11 C12 3D C11+2C12 3 3C44+C11−C12 5 9KG 3K+G 3K−2G 2(3K+G) is in line with HSE06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Among GGAs, the bulk mod- uli of PW91 and PBE are nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The hybrid functionals have fairly similar performance, with B3LYP again showing a striking exception, especially related to 1D modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These observations in bulk moduli apply also to Young’s moduli (Figure 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Only GGAs show somewhat larger stiffness and the trends in 2D moduli for B3LYP and PBE0 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The shear modulus and Poisson’s ratio are defined only for 2D and 3D systems (Figures 8c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moreover, shear modulus is not reported for the 2D square lattice due to instability against shear deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In addi- tion, some deformations with PBE0 and B3LYP resulted in consistent numerical errors, forcing us to omit shear and Young’s modulus as well Poisson ratio for these func- tionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In summary, the most consistent behavior in elastic moduli is displayed by HSE and GGA function- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LDA, B3LYP and PBE0 functionals suffer from both numerical challenges and deviant trends at least in some elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Electronic structure (density of states): To com- plement pure energetic and geometric properties, we now extend our investigations to electronic structure proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Electronic structure is a complex topic with many features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To reduce complexity and extract trends, we in- vestigate the electronic structure simply in terms of the density of states DOS(ϵ) and its projections DOSl(ϵ) to s (l = 0), p (l = 1), and d (l = 2) angular momen- tum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In addition, we focus only on energies at the vicinity of the Fermi-level ϵ = ϵF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Consequently, we define the quantities Nl = � ∞ −∞ DOSl (ϵ) g (ϵ) dϵ (18) that give the number of l-type orbitals surrounding the Fermi-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The DOS is also normalized by the number of atoms in the simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=" The envelope function g(ϵ) has a Gaussian form g (ϵ) = exp � −1 2 �ϵ − ϵf σ �2� (19) 160 - I AgiD Agsg 1 Ag3D Aghc Aghex (a) 140 - 120 80 - 60 - 20 - 115 b 100 80 - Young's modulus 40 - 20 - C) 40 30 - Shear modulus 20 - 01 d) os'O ratio 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='30 - HSE03HSE06 Exchange-correlation functional8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Effect of xc functional on the electronic structure of low-dimensional metals made of Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Heatmap visualizes the number of s-type states (Ns), p-type states (Np), d-type states (Nd), and the total number of states (Nt) within a ∼ 1 eV energy window around the Fermi-level [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (18)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' and we used σ = 1 eV energy window around ϵF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In general, the s-orbital contribution decreases with in- creasing coordination number for all xc functionals (Fig- ure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 1D the main contribution comes from s- orbitals, followed by p- and d-orbitals for all functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 2D this order is rearranged to p > s > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In 3D this same trend is retained by all hybrid functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The LDA, PW91, and PBE have very similar orbital contri- bution ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For all xc functionals, the p contribu- tion is the largest for honeycomb, smallest for 1D, and smallest for hexagonal among 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The ordering of Np with respect to different coordination number is the same for GGAs, PBE0, and B3LYP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For HSE03 and HSE06 all Nl are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The d-orbital contribu- tions follow trend similar to s-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The value of Nd is the highest for LDA and the lowest for PBE0 for all systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' the most visible difference is the generally low Nd of all hybrid functionals, especially in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Regarding the total DOS, all GGAs produce nearly identical Nt, apart from 3D bulk in RPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The total DOS from hybrids differs somewhat from the LDA and GGA functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' HSE functionals show similar Nt for C = 6 and 12 systems, but differ in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Over- all, trends in the total densities are inconsistent for LDA and PBE0 functionals, but somewhat consistent among GGA as well as B3LYP and HSE functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Conclusions on xc functionals: To summarize, PW91 and PBE perform similarly for forces, energies, and densities of states, while RPBE shows underbinding, smaller bond lengths, and smaller elastic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LDA is inferior to GGA practically in all respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Among hy- brid functionals, the performances of HSE03 and HSE06 aligned in all respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' B3LYP failed to improve GGA in terms of accuracy in the lattice constants and cohesive energies, even if its electronic structures resembled those of HSE functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cohesion energy displayed congruent dimensionality-dependencies, apart from visibly differing dependencies by B3LYP and PBE0 functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Before reaching ultimate conclusions, however, we have to consider the computational cost (Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As ex- pected by the nonlocal character of the hybrid function- als, already minimal-cell systems require 2 − 3 orders of magnitude more computational time for hybrids than for LDA and GGA, and for larger systems the differ- ence would increase even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Considering the low computational cost, GGA functionals perform extremely well compared to hybrid functionals, compared even to the most robust HSE family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To conclude, unless the low- dimensional metals are studied for very specific purposes, the standard PBE indeed remains the preferred weapon of choice for low-dimensional metals modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Computational cost of different xc-functionals: Time in seconds to calculate the energy of minimal-cell sys- tems using 24 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The cell has one atom for all systems except for 2D honeycomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LDA RPBE PW91 PBE B3LYP PBE0 HSE03 HSE06 1D 39 39 44 43 476 1360 491 1897 hc 49 59 62 58 16786 20937 18662 15006 sq 18 24 23 22 1469 1739 1535 1493 hex 16 19 20 17 1454 1800 1698 1675 3D 14 18 19 17 88553 41352 38802 38704 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Performance of different basis sets In this section, we choose PBE xc functional and repeat the systematics of the previous section while this time varying the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The converged plane wave basis gives the best results that provide the reference assessing the performance of the three LCAO basis sets Medium, High, and Ultra introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To obtain a broader context, we compared the DFT- LCAO with DFTB method, which uses a minimal local basis and contains approximations speeding up the cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Here we used the parameters available for Ag developed earlier [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, parametrization can be done in different ways, and one should not consider these results as unique and absolute representation of DFTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cohesive Energies: The LCAO-U and LCAO-H produce cohesive energies very close to those of PW (Fig- ure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LCAO-M overbinds slightly in comparison, but the accuracy for 2D systems is still 3 − 4 % compared to PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The dependence of cohesion on coordination num- ber is reproduced with all basis sets, and differences are difficult to see on absolute scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' DFTB follows similar behavior, but shows significant overbinding, especially for 3D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 Ag3D Aghex Agsq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 Aghc Agid 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 Ag3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Aghex - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='90 Aghc - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='75 Agid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ag3D Aghex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 Agsq Aghc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 AgiD Ag3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='30 Aghex - Agsq- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 Aghc - Agid: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 LDA RPBE PW91 PBE B3LYP PBEO HSE03 HSE06 Exchange-correlation functional9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cohesive energies of optimized 1D, 2D (hc, sq, and hex), and 3D systems made of Ag with different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bars on the left show DFTB results with minimal basis for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Dimensionality-dependence of energetics: As with xc functionals, we investigate how basis set affects the dependence of energetics on coordination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Again this dependence is analyzed via the scaling exponent γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (8) fitted to the cohesive energies as a function of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Compared to PW, the dependence on C becomes sys- tematically more linear as we move from Ultra to High and ultimately to Medium basis (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, still the Medium basis reproduces γ to within 5 % accu- racy compared to PW basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Even DFTB compares well in the overall coordination-dependence, although there are visible problems in capturing the DFT trends for 2D systems (the green bars for DFTB in Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, to state the main point, the choice of basis in- fluences dimensionality-dependence of energetics far less than xc functional: note that Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 6 and 11 have the same scale in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bond Lengths: The LCAO-U and LCAO-H bond lengths are very similar, accurate to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='77 % com- pared to PW (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' All LCAO variants overesti- mate all bonds, LCAO-M having the lowest performance with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 % too long bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' DFTB no longer captures the DFT trends in coordination-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The 1D chain bond length is larger than honeycomb and the 2D bonds vary wildly, even if the C-ordering still remains correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic constants and moduli: For 1D and 2D sys- tems, elastic moduli have minor dependence on basis set (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The largest deviation from PW occurs for 3D bulk, for all LCAO variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' This deviation likely stems from the better space-filling character of PW ba- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moreover, although performing well in cohesion and bond lengths, LCAO-M performs poorly in all elastic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LCAO-U is close to PW in all respects, and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Trends of low-dimensional energetics with different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The fitted scaling exponent γ is plotted for different basis sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' smaller γ means that energy depends less linearly on the coordination number [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='(7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The vertical scale is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bond lengths of optimized 1D, 2D (hc, sq, and hex), and 3D systems made of Ag with different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LCAO-M captures all the same trends, even if with some quantitative differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These results suggest that, ex- cept perhaps for LCAO-M, LCAO basis can be reliable for studying mechanical properties of low-dimensional metallic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The LCAO variant -dependency of elastic properties is even smaller than the changes upon switching from GGA to hybrid functional (compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 8 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In comparison, DFTB shows both trend differences and large absolute differences compared to DFT-LCAO (Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For example, the 1D elastic modulus is over- estimated by a factor of ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Even the trend within 2D systems was not reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' It appears that the Ag parametrization should be revised for more reliable me- Ag3D Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 1 bss Shex 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 DFTB LCAO-M LCAO-H LCAO-U PW Basis set0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='42 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='36 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='30 DFTB LCAO-M LCAO-H LCAO-U PW Basis setAgh Ag Shex S3D Shc bss 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='90 Bond length (A) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 DFTB LCAO-M LCAO-H LCAO-U PW Basis set10 chanical properties of low-dimensional Ag systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Electronic structure (density of states): Also the electronic structure from LCAO is compared here against PW results, using the indicator numbers given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For 2D structures PW gives orbital contri- butions in order p > s > d (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For LCAO this trend shuffles to s > d > p, that is, the p contri- bution diminishes for all LCAO variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For 1D sys- tem the orbital ordering for PW and LCAO basis re- mains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, still all basis sets—including minimal-basis DFTB—show consistent C-dependence in orbital contributions around the Fermi-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' LCAO-H and LCAO-U results align better, while LCAO-M re- sults are different in some respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In summary, the C- dependence of the total DOS in 2D metals is reproduced by LCAO to a fair degree, but the orbital contributions are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Conclusions on basis sets: To conclude, LCAO basis competes extremely well with PW for studying energetic and geometric properties of low-dimensional metal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Even elastic moduli are reproduced rea- sonably well by LCAO-H and LCAO-U basis, compared to converged PW basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The performance of LCAO-M basis was notably modest, regarding elastic properties and also the details of electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The or- bital breakup of the electronic structures at the vicin- ity of Fermi-level for PW and LCAO variants differed markedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Regarding DFTB, the Ag parametrizations clearly re- quire revisiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The cohesive energies are too large, bond lengths are both large and small, and elastic moduli are close to arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Still many of the qualitative trends regarding C-dependence were reproduced reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' However, before again reaching ultimate conclusions, we have to consider the computational cost with differ- ent basis (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The cost was investigated by simula- tion cells with 32−64 atoms and a couple of dozen cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The comparison is thus by no means unique or absolute, but it does give a rough inkling of the computational de- mands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' As expected, DFTB outspeeds DFT by one to three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Within DFT, switching from LCAO-M to LCAO-U results in cost increases from a fac- tor of two (1D) up to a factor of ∼ 15 (3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Especially for low-dimensional systems LCAOs are faster than PW, nearly by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For 3D bulk PW is very competitive against LCAO due to lacking vac- uum region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' here LCAO-U is even slower than PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In conclusion, unless very high accuracy is of central impor- tance, LCAO has demonstrated a fair accuracy in most properties and should be prioritized over PW due to its superior efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Even LCAO-M basis can be consid- ered for simulations where the improved speed wins over lost accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic properties of low-dimensional systems of Ag with different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bulk moduli (a) and Young’s moduli (b) are shown for all systems, shear moduli (c) and Poisson’s ratio (d) are shown only for 3D and stable 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Units for moduli are GPa nm3−D, where D is the system dimen- sionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=" Agid Ag3D Aghex Aghc Agsq (a) 120 - 100 Bulk modulus 08 60 - 40 - 20- 0 (b) 140 - 120 - 80 Young's r F 09 40 - 20 0 (c) 50 - 40 - modulus Shear r 1 20 0 (d) Fos'O 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 - LCAO-M LCAO-H LCAO-U PW DFTB Basis set11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Effect of basis set on the electronic structure of low-dimensional metals made of Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Heatmap visualizes the number of s-type states (Ns), p-type states (Np), d-type states (Nd), and total number of states (Nt) within a ∼ 1 eV energy window around the Fermi-level [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (19)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Computational cost of different basis sets: Time in seconds to calculate the energy of systems using 24 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The parenthesis contain the number of atoms in the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Systems DFTB LCAO-M LCAO-H LCAO-U PW 1D (32) 10 175 265 310 11890 2D hc (64) 20 215 355 610 13120 2D sq (64) 18 190 300 500 12370 2D hex(64) 17 130 290 655 6885 3D (64) 19 145 855 2220 2050 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Combined scanning of xc functionals and basis sets Above we investigated xc functionals (with PW basis) and basis sets (with PBE functional) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' How- ever, the performance of xc functionals and basis sets can be coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We therefore complement our analy- sis by combined scanning of different xc functionals with different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The bond lengths, cohesive energies, elastic constants, and orbital contributions to DOS ob- tained at different basis set-xc functional -combinations are shown in Tables V, VI, and VII in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For LDA, the choice of basis set did not affect the co- hesion dependence on C (Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Changing the basis set from PW to LCAO increases the cohesive energy for C ≥ 4 and decreases it for C = 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Decreasing the LCAO size also decreases the cohesion, as expected in the light of variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bond lengths with PW, LCAO-U and LCAO-H basis are nearly equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' With LCAO-M bonds are longer for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The elastic properties are nearly basis-independent, with the notable exception of LCAO-M (Table VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Most sensitive to the choice of basis is the electronic structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' all LCAO vari- ants show the same trend, which however differs signifi- cantly from PW (Table VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For GGAs, the performance remains robust upon re- ducing the size of the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In fact, the observations in Subsection III D with PBE are representative for other GGAs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Switching PW to LCAO-U or LCAO-H changes bond lengths and cohesive energies less than 1 %;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' less robust LCAO-M decreases cohesive energies by 4 % and increases bond lengths by ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 % (Table V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Basis set sensitivity is the smallest for PW91 and the largest for RPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic constants follow the accuracy trends similar to those of energetics and geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' PBE shows some basis set sensitivity, especially for the bulk moduli of 2D systems (Table VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For hybrid functionals, the matters are less systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Using LCAO-M in conjunction with unscreened B3LYP and PBE0 functionals results in significant overbinding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' bond lengths are underestimated by more than 10 % (Ta- ble V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' With LCAO-H and LCAO-U basis sets the same xc functionals underestimate bonds only by ≈ 2 %, while increase cohesive energies by ≤ 24 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' B3LYP and PBE0 are thus extremely sensitive to the quality of LCAO ba- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moreover, B3LYP and PBE0 are unable to produce elastic moduli due to persistent numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In con- trast, the screened HSE functionals produced robust ge- ometries, energetics and elastic properties upon changing the size of the LCAO basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The robustness was even better than with PW91 and PBE, although admittedly at a considerable computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The orbital con- tributions to DOS with PW and LCAO basis were dif- ferent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' the same effect was observed for PBE functional (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Among different LCAO variants, LCAO-H and LCAO-U show similar orbital contributions for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In addition to energetic and geometric prop- erties, the peculiarities of B3LYP and PBE0 functionals are observable also in electronic properties (Table VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In general, hybrid functionals in conjunction with LCAO- H and LCAO-U basis requires prohibitive computational resources even for single atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The effect of DFT implementation In addition to DFT attributes, it is important also to be able to rely on the DFT implementation itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For completeness, therefore, we briefly discuss the mag- nitude of differences related to the numerical imple- mentation of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We calculated the cohesive ener- gies, bond lengths, and elastic moduli also with the GPAW code, using plane wave basis with the same 800 eV energy cutoff and default parameters [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The QuantumATK/GPAW cohesive energies were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1671 eV / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1661 eV (1D), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5062 eV / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5054 eV (2D hc), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8293 eV / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8286 eV (2D sq), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0583 eV / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0570 eV (2D hex), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5326 eV / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5323 eV (3D), bond lenghts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6480 ˚A/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6501 ˚A (1D), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6700 ˚A/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6682 ˚A (2D hc), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6998 ˚A/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='700567 ˚A (2D sq), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7877 ˚A/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7894 ˚A (2D hex), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='80 Ag3D Aghex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 Aghc Agid 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 Ag3D Aghex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 ABsq Aghc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 Agid Ag3D Aghex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='80 Agsq Aghc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 Agid Ag3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 Aghex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 Aghc Agid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 LCAO-U PW DFTB LCAO-M LCAO-H Basis set12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9301 ˚A/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9305 ˚A (3D), and bulk moduli 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='32 GPa nm2 /18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='73 GPa nm2 (1D), 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 GPa nm / 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='21 GPa nm (2D hc), 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 GPa nm / 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 GPa nm (2D sq), 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 GPa nm / 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='79 GPa nm (2D hex), 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 GPa / 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 GPa (3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Thus, default parameters without tuning give code- related differences in cohesive energies ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 meV, in bond lengths ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='002 ˚A, and in bulk moduli ≲ 1 % (2D systems) or ≃ 2% (1D and 3D systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Although the comparison used the PBE functional and plane waves, it is reasonable to suspect the level of differences to remain similar also for other functionals and basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Over- all, code-related differences remain considerably smaller than the differences originating from physical attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' SUMMARY AND CONCLUSION In summary, we investigated the performance of vari- ous DFT attributes in the modeling of low-dimensional elemental metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' For future reference, the number of k-points, the size of the vacuum region, and the magni- tude of Fermi-broadening were given tolerance-dependent rules of thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Such rules help choosing combinations of attributes that result in commensurate accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The most robust against the choice of basis set was HSE06, followed by HSE03, PBE, PW91, RPBE and LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The B3LYP produced inaccurate cohesions and bond lengths—with the highest computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Only the electronic structure in B3LYP was in line with other hybrid functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The energetics, geometries, and elastic properties with PW, LCAO-U, and LCAO-H basis sets were in over- all good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The greatest disparities between PW and LCAO methods resided in the orbital contribu- tions to the DOS, although in the total DOS they were moderated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' On a general level, LCAO-U and LCAO- H performed similarly at different xc functionals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' there- fore, for general purposes, LCAO-H should be preferred over LCAO-U due to superior efficiency (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' The LCAO-M basis worked varyingly well in many respects, except when used in conjunction with B3LYP and PBE0 functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' To conclude, in the research of metallic bonding at low dimensions, the best value for a given cost is proba- bly given by semi-local PW91 and PBE xc functionals in conjunction with moderately-sized LCAO-U or LCAO- H basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' These results are encouraging for doing large-scale, high-throughput DFT simulations to gener- ate data for machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' In comparison, DFTB is a very speedy method and is capable of simu- lations unaccessible by DFT [73–75], but the quality of parametrization needs to be ensured first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' We hope that our results and gentle recommendations help lifting 2D metal research to new heights, expedite better interac- tion with experiments, and feed machine learning algo- rithms with quality data to drive further discoveries in low-dimensional metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the Finnish Grid and Cloud Infras- tructure (FGCI) for computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Novoselov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Geim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Morozov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Dubonos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Grigorieva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Firsov, Science 306, 666 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Novoselov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schedin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Booth, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Khotkevich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Morozov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Geim, Proceedings of the National Academy of Sciences 102, 10451 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [3] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kalantar-Zadeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Coleman, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Strano, Nature nanotechnology 7, 699 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Shanmugam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mensah, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Babu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gawusu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chanda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Tu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Neisiany, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' F¨orsth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sas, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Das, Particle & Particle Systems Characterization 39, 2200031 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhou, Advanced Energy Materials 12, 2003841 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Manzeli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ovchinnikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pasquier, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yazyev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kis, Nature Reviews Materials 2, 1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Geng and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, Advanced Materials 30, 1800865 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Deng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bachmatiuk, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sandeep, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Popov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Eckert, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R¨ummeli, Science 343, 1228 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ma, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 2, 456 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Fan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Niu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, Chemical Reviews 118, 6409 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Park, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, Ma- terials Today Advances 8, 100092 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mendes, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Luo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bachmatiuk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gemming, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zeng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Fu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Liu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R¨ummeli, Advanced Science 8, 2100619 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zagler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Reticcioli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mangler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Scheinecker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Franchini, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kotakoski, 2D Materials 7, 045017 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Duan, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Du, Acta Materialia 229, 117844 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hohenberg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kohn, Physical Review 136, B864 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kohn and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sham, Physical Review 140, A1133 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ganz, Physical Chemistry Chemical Physics 17, 19695 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Dornfeld, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ganz, Physical Chemistry Chemical Physics 17, 26036 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ganz, Journal of Nanomaterials 2016, 8429510 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nevalaita and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, Physical Review B 97, 035411 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nevalaita and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, Physical Review B 98, 115433 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nevalaita and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, Nanoscale 11, 22019 13 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nevalaita and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, AIP Advances 10, 065327 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ono, Physical Review B 102, 165424 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ren, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Shao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Huang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Shi, Journal of Materials Chemistry C 9, 4554 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ono, Physical Review Materials 5, 104004 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Anam and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gaston, Journal of Physics: Condensed Matter 33, 125901 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kapoor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sharma, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ahluwalia, Physica E: Low-dimensional Systems and Nanostructures 131, 114745 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kutana, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Altalhi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ruan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Penev, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yakobson, Nanoscale Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 4, 1408 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sangolkar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pawar, Physica status solidi (b) 259, 2100489 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sangolkar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jha, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pawar, Advanced Theory and Simulations 5, 2200057 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [32] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kresse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Furthm¨uller, Physical Review B 54, 11169 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Burke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ernzerhof, Physical Re- view Letter 77, 3865 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Lehtola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Steigemann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Oliveira, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Marques, SoftwareX 7, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [35] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Briggs, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sullivan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bernholc, Physical Review B 54, 14362 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Enkovaara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rostgaard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mortensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Du�lak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ferrighi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gavnholt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Glinsvad, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Haikola, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hansen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kristoffersen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kuisma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Larsen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Lehtovaara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ljung- berg, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Lopez-Acevedo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moses, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ojanen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Olsen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Petzold, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Romero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Stausholm- Møller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Strange, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Tritsaris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vanin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Walter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hammer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H¨akkinen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Madsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Niem- inen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nørskov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Puska, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rantala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schiøtz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Thygesen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jacobsen, Journal of Physics: Condensed Matter 22, 253202 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Soler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Artacho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Garc´ıa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jun- quera, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ordej´on, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S´anchez-Portal, Journal of Physics: Condensed Matter 14, 2745 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Brandbyge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mozos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ordej´on, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Taylor, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Stokbro, Physical Review B 65, 165401 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schmidt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Marques, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Botti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Marques, npj Comput Mater 5 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Mortazavi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Novikov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Podryabinkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Roche, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rabczuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Shapeev, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhuang, Applied Materials Today 20, 100685 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Cai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wei, Nanoscale Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 2, 3115 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schleder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Acosta, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Fazzio, ACS Ap- plied Materials & Interfaces 12, 20149 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ryu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chen, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 51, 1899 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [44] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Canadell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Doublet, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Iung, Orbital Ap- proach to the Electronic Structure of Solids (Oxford Uni- versity Press, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elstner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Porezag, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jungnickel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elsner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Haugk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Suhai, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Seifert, Phys- ical Review B 58, 7260 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Smidstrup, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Markussen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vancraeyveld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wellen- dorff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Schneider, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gunst, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Verstichel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Stradi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Khomyakov, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Vej-Hansen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Chill, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rasmussen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Penazzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Corsetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ojan- per¨a, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Palsgaard, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Martinez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Blom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Brandbyge, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Stokbro, Journal of Physics: Condensed Matter 32, 015901 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Harris, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hodgkinson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pickard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Yates, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zorin, Magnetic Resonance in Chemistry 45, S174 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [48] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Blum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gehrke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hanke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Havu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Havu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Reuter, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Scheffler, Computer Physics Communications 180, 2175 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' van Setten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Giantomassi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bousquet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ver- straete, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hamann, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gonze, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rignanese, Computer Physics Communications 226, 39 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [50] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Liu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nocedal, Mathematical Programming 45, 503 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Monkhorst and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pack, Physical Review B 13, 5188 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [52] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' M¨akinen, Computational Materials Science 47, 237 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [53] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sz˝ucs, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hajnal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Gonz´alez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Or- tega, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P´erez, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Flores, Applied Surface Science 212-213, 861 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sz˝ucs, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hajnal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Scholz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sanna, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauen- heim, Applied Surface Science 234, 173 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, Physical Review B 45, 13244 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Burke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ernzerhof, Physical Re- view Letter 77, 3865 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [57] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hammer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Hansen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nørskov, Physical Review B 59, 7413 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [58] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ziesche, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Eschrig, Electronic struc- ture of solids’ 91 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [59] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Burke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, Derivation of a generalized gradient approximation: The pw91 den- sity functional, in Electronic density functional theory (Springer, 1998) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 81–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [60] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Matsushita, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Nakamura, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Oshiyama, Physical Review B 84, 075205 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [61] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Becke, The Journal of Chemical Physics 98, 5648 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Perdew, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ernzerhof, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Burke, The Journal of Chemical Physics 105, 9982 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [63] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Pela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Marques, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Teles, Journal of Physics: Condensed Matter 27, 505502 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Heyd, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Scuseria, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ernzerhof, The Journal of Chemical Physics 118, 8207 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [65] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bl¨ochl, Physical Review B 50, 17953 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [66] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kittel, Introduction to solid state physics, 8th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (Wi- ley New York, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [67] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Paier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Marsman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Kresse, The Journal of Chemical Physics 127, 024103 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Zhang, Computer Physics Communica- tions 220, 403 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [69] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Tang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Geng, Computer Physics Communications 267, 108033 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [70] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Tschoegl, Australian Journal of Physics 11, 154 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ma´zdziarz, 2D Materials 6, 048001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [72] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jamal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Jalali Asadabadi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Ahmad, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Rahna- maye Aliabad, Computational Materials Science 95, 592 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [73] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Korhonen, Nanoscale 7, 10140 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' [74] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H¨akkinen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Huber, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' von Issendorff, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moseler, Physical Review Letter 98, 015701 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 14 [75] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Koskinen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' H¨akkinen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Seifert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Sanna, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Frauenheim, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Moseler, New Journal of Physics 8, 9 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 15 APPENDIX TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Bond lengths d(�A) and Cohesive energies Ecoh(eV) for each lattice type corresponding to different DFT-attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1D Honeycomb Square Hexagonal 3D DFT-Methods d Ecoh d Ecoh d Ecoh d Ecoh d Ecoh DFTB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='572 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='691 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='562 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='450 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='636 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='804 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='819 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='967 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='008 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='891 LDA-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='584 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='513 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='591 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='012 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='623 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='475 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='712 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='761 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='840 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='547 LDA-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='553 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='563 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='562 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='606 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='563 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='693 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='858 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='827 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='660 LDA-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='587 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='553 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='126 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='598 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='590 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='685 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='887 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='826 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='672 LDA-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='591 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='542 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='138 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='595 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='586 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='682 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='881 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='828 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='638 RPBE-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='959 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='760 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='198 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='764 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='474 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='853 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='677 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='982 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='065 RPBE-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='989 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='731 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='251 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='531 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='831 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='738 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='965 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='130 RPBE-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='691 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='001 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='723 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='262 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='736 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='547 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='824 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='756 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='963 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='143 RPBE-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='992 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='709 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='248 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='734 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='523 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='822 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='732 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='962 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='100 PW91-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='679 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='145 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='470 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='717 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='806 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='807 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='026 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='941 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='529 PW91-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='655 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='171 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='670 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='522 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='703 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='858 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='790 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='083 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='586 PW91-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='186 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='536 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='696 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='876 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='785 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='598 PW91-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='639 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='185 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='659 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='534 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='693 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='862 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='783 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='089 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='928 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='560 PBE-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='690 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='126 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='710 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='441 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='724 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='771 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='814 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='994 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='501 PBE-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='155 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='685 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='497 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='710 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='826 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='797 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='053 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='558 PBE-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='651 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='170 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='677 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='510 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='844 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='790 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='073 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='571 PBE-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='648 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='167 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='670 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='506 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='829 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='788 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='058 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='930 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='533 B3LYP-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='373 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='734 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='410 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='164 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='457 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='029 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='558 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='586 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='725 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='340 B3LYP-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='655 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='067 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='691 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='426 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='714 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='772 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='812 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='977 B3LYP-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='679 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='461 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='705 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='816 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='803 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='025 B3LYP-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='944 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='715 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='211 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='737 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='470 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='830 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='659 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='986 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='963 PBE0-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='322 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='877 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='358 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='807 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='409 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='978 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='512 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='657 PBE0-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='635 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='092 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='654 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='523 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='679 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='970 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='773 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='219 PBE0-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='626 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='128 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='567 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='670 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='764 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='277 PBE0-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='963 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='671 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='288 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='690 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='640 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='879 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='910 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='444 HSE03-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='694 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='030 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='715 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='351 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='729 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='696 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='825 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='919 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='725 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='436 HSE03-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='044 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='693 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='385 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='714 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='728 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='807 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='949 HSE03-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='663 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='058 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='687 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='396 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='710 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='744 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='801 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='966 HSE03-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='651 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='049 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='664 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='392 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='697 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='742 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='787 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='971 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='925 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='484 HSE06-LCAO-M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='697 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='061 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='716 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='358 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='733 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='707 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='829 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='954 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='431 HSE06-LCAO-H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='676 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='693 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='391 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='719 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='738 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='812 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='961 HSE06-LCAO-U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='666 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='088 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='402 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='709 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='753 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='803 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='978 HSE06-PW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='650 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='664 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='396 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='695 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='750 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='786 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='982 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='923 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='479 TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Elastic constants for 1D (GPa nm2), 2D (GPa nm), and 3D (GPa) calculated by using different DFT-attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' 1D Honeycomb Square Hexagonal 3D DFT-Methods C11 C11 C12 C66 C11 C12 C66 C11 C12 C66 C11 C12 C66 DFTB 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 LDA-LCAO-M 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 LDA-LCAO-H 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 LDA-LCAO-U 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 LDA-PW 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 RPBE-LCAO-M 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 RPBE-LCAO-H 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 RPBE-LCAO-U 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 RPBE-PW 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 PW91-LCAO-M 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 PW91-LCAO-H 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 PW91-LCAO-U 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 PW91-PW 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 PBE-LCAO-M 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='63 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 PBE-LCAO-H 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 PBE-LCAO-U 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 16 TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' (Continued) 1D Honeycomb Square Hexagonal 3D DFT-Methods C11 C11 C12 C66 C11 C12 C66 C11 C12 C66 C11 C12 C66 PBE-PW 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 B3LYP-LCAO-M 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 B3LYP-LCAO-H 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 B3LYP-LCAO-U 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 B3LYP-PW 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 PBE0-LCAO-M 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 PBE0-LCAO-H 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 PBE0-LCAO-U 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 PBE0-PW 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 HSE03-LCAO-M 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 HSE03-LCAO-H 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 HSE03-LCAO-U 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 HSE03-PW 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='7 HSE06-LCAO-M 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='8 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 HSE06-LCAO-H 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 HSE06-LCAO-U 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 HSE06-PW 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='4 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content=' Estimation of contribution of s, p, and d orbitals to the density of states by implementing different DFT-attributes 1D Honeycomb Square Hexagonal 3D DFT-Methods Ns Np Nd Ns Np Nd Ns Np Nd Ns Np Nd Ns Np Nd DFTB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 LDA-LCAO-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 LDA-LCAO-H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 LDA-LCAO-U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 LDA-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 RPBE-LCAO-M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 RPBE-LCAO-H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 RPBE-LCAO-U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 RPBE-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 PW91-LCAO-M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 PW91-LCAO-H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 PW91-LCAO-U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 PW91-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 PBE-LCAO-M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 PBE-LCAO-H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 PBE-LCAO-U 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 PBE-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 B3LYP-LCAO-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 B3LYP-LCAO-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 B3LYP-LCAO-U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 B3LYP-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 PBE0-LCAO-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 PBE0-LCAO-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 PBE0-LCAO-U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='00 PBE0-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 HSE03-LCAO-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 HSE03-LCAO-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='06 HSE03-LCAO-U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='06 HSE03-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 HSE06-LCAO-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='06 HSE06-LCAO-H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 HSE06-LCAO-U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='05 HSE06-PW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} +page_content='01' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQf_P6F/content/2301.01945v1.pdf'} diff --git a/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf b/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f2455e1924448eca41d667b4c6f57e053929698e --- /dev/null +++ b/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2e43e242e82e9a21edb2013bfa5d1b31040b17036f7c12271d84a34632d3e3f +size 1228283 diff --git a/-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss b/-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cfc7e78393886f9eb90ff2622ab14bb1838902d9 --- /dev/null +++ b/-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:31e28e94a6e54e320298551e25c8c34e9a6fe55d6fa5d4104292ed27c856d46f +size 3080237 diff --git a/-dAyT4oBgHgl3EQf3fmN/vector_store/index.pkl b/-dAyT4oBgHgl3EQf3fmN/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a36101f8bb8c00a7ed53aad1e939a7c9d75d49e3 --- /dev/null +++ b/-dAyT4oBgHgl3EQf3fmN/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b014f782a61584d447e791f9ed652f7923830dcdf169b18989c1e01fe60a28b6 +size 102790 diff --git a/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/2301.13440v1.pdf.txt b/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/2301.13440v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d51ce623beb80478f975f41626cfdcf4762628b9 --- /dev/null +++ b/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/2301.13440v1.pdf.txt @@ -0,0 +1,748 @@ +arXiv:2301.13440v1 [math.RT] 31 Jan 2023 +Real characters in nilpotent blocks +Benjamin Sambale∗ +February 1, 2023 +Dedicated to Pham Huu Tiep on the occasion of his 60th birthday. +Abstract +We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally +determined. We further conjecture that the Frobenius–Schur indicators of those characters can be +computed for p = 2 in terms of the extended defect group. We derive this from a more general conjecture +on the Frobenius–Schur indicator of projective indecomposable characters of 2-blocks with one simple +module. This extends results of Murray on 2-blocks with cyclic and dihedral defect groups. +Keywords: real characters; Frobenius–Schur indicators; nilpotent blocks +AMS classification: 20C15, 20C20 +1 Introduction +An important task in representation theory is to determine global invariants of a finite group G by means of +local subgroups. Dade’s conjecture, for instance, predicts the number of irreducible characters χ ∈ Irr(G) +such that the p-part χ(1)p is a given power of a prime p (see [23, Conjecture 9.25]). Since Gow’s work [7], +there has been an increasing interest in counting real (i. e. real-valued) characters and more generally +characters with a given field of values. +The quaternion group Q8 testifies that a real irreducible character χ is not always afforded by a repre- +sentation over the real numbers. The precise behavior is encoded by the Frobenius–Schur indicator (F-S +indicator, for short) +ǫ(χ) := +1 +|G| +� +g∈G +χ(g2) = + + + + + +0 +if χ ̸= χ, +1 +if χ is realized by a real representation, +−1 +if χ is real, but not realized by a real representation. +(1) +A new interpretation of the F-S indicator in terms of superalgebras has been given recently in [13]. The case +of the dihedral group D8 shows that ǫ(χ) is not determined by the character table of G. The computation +∗Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Han- +nover, Germany, sambale@math.uni-hannover.de +1 + +of F-S indicators can be a surprisingly difficult task, which has not been fully completed for the simple +groups of Lie type, for instance (see [25]). Problem 14 on Brauer’s famous list [2] asks for a group-theoretical +interpretation of the number of χ ∈ Irr(G) with ǫ(χ) = 1. +To obtain deeper insights, we fix a prime p and assume that χ lies in a p-block B of G with defect group +D. By complex conjugation we obtain another block B of G. If B ̸= B, then clearly ǫ(χ) = 0 for all +χ ∈ Irr(B). Hence, we assume that B is real, i. e. B = B. John Murray [18, 19] has computed the F-S +indicators when D is a cyclic 2-group or a dihedral 2-group (including the Klein four-group). His results +depend on the fusion system of B, on Erdmann’s classification of tame blocks and on the structure of the +so-called extended defect group E of B (see Definition 7 below). For p > 2 and D cyclic, he obtained in +[20] partial information on the F-S indicators in terms of the Brauer tree of B. +The starting point of my investigation is the well-known fact that 2-blocks with cyclic defect groups are +nilpotent. Assume that B is nilpotent and real. If B is the principal block, then G = Op′(G)D and +Irr(B) = Irr(G/Op′(G)) = Irr(D). In this case the F-S indicators of B are determined by D alone. Thus, +suppose that B is non-principal. By Broué–Puig [4], there exists a height-preserving bijection Irr(D) → +Irr(B), λ �→ λ ∗ χ0 where χ0 ∈ Irr(B) is a fixed character of height 0 (see also [16, Definition 8.10.2]). +However, this bijection does not in general preserve F-S indicators. For instance, the dihedral group D24 +has a nilpotent 2-block with defect group C4 and a nilpotent 3-block with defect group C3, although every +character of D24 is real. Our main theorem asserts that the number of real characters in a nilpotent block +is nevertheless locally determined. To state it, we introduce the extended inertial group +NG(D, bD)∗ := +� +g ∈ NG(D) : bg +D ∈ {bD, bD} +� +where bD is a Brauer correspondent of B in DCG(D). +Theorem A. Let B be a real, nilpotent p-block of a finite group G with defect group D. Let bD be a Brauer +correspondent of B in DCG(D). Then the number of real characters in Irr(B) of height h coincides with +the number of characters λ ∈ Irr(D) of degree ph such that λt = λ where +NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩. +If p > 2, then all real characters in Irr(B) have the same F-S indicator. +In contrast to arbitrary blocks, Theorem A implies that nilpotent real blocks have at least one real character +(cf. [20, p. 92] and [8, Theorem 5.3]). If bD = bD, then B and D have the same number of real characters, +because NG(D, bD) = DCG(D). This recovers a result of Murray [18, Lemma 2.2]. As another consequence, +we will derive in Proposition 5 a real version of Eaton’s conjecture [5] for nilpotent blocks as put forward +by Héthelyi–Horváth–Szabó [12]. +The F-S indicators of real characters in nilpotent blocks seem to lie somewhat deeper. We still conjecture +that they are locally determined by a defect pair (see Definition 7) for p = 2 as follows. +Conjecture B. Let B be a real, nilpotent, non-principal 2-block of a finite group G with defect pair (D, E). +Then there exists a height preserving bijection Γ : Irr(D) → Irr(B) such that +ǫ(Γ(λ)) = +1 +|D| +� +e∈E\D +λ(e2) +(2) +for all λ ∈ Irr(D). +2 + +The right hand side of (2) was introduced and studied by Gow [8, Lemma 2.1] more generally for any +groups D ≤ E with |E : D| = 2. This invariant was later coined the Gow indicator by Murray [20, Eq. +(2)]. For 2-blocks of defect 0, Conjecture B confirms the known fact that real characters of 2-defect 0 +have F-S indicator 1 (see [8, Theorem 5.1]). There is no such result for odd primes p. As a matter of fact, +every real character has p-defect 0 whenever p does not divide |G|. In Theorem 10 we prove Conjecture B +for abelian defect groups D. Then it also holds for all quasisimple groups G by work of An–Eaton [1]. +Murray’s results mentioned above, imply Conjecture B also for dihedral D. +For p > 2, the common F-S indicator in the situation of Theorem A is not locally determined. For instance, +G = Q8⋊C9 = SmallGroup(72, 3) has a non-principal real 3-block with D ∼= C9 and common F-S indicator +−1, while its Brauer correspondent in NG(D) ∼= C18 has common F-S indicator 1. Nevertheless, for cyclic +defect groups D we find another way to compute this F-S indicator in Theorem 3 below. +Our second conjecture applies more generally to blocks with only one simple module. +Conjecture C. Let B be a real, non-principal 2-block with defect pair (D, E) and a unique projective +indecomposable character Φ. Then +ǫ(Φ) = |{x ∈ E \ D : x2 = 1}|. +Here ǫ(Φ) is defined by extending (1) linearly. If ǫ(Φ) = 0, then E does not split over D and Conjecture C +holds (see Proposition 8 below). Conjecture C implies a stronger, but more technical statement on 2-blocks +with a Brauer correspondent with one simple module (see Theorem 13 below). This allows us to prove the +following. +Theorem D. Conjecture C implies Conjecture B. +We remark that our proof of Theorem D does not work block-by-block. For solvable groups we offer a +purely group-theoretical version of Conjecture C at the end of Section 4. +Theorem E. Conjectures B and C hold for all nilpotent 2-blocks of solvable groups. +We have checked Conjectures B and C with GAP [6] in many examples using the libraries of small groups, +perfect groups and primitive groups. +2 Theorem A and its consequences +Our notation follows closely Navarro’s book [22]. Let B be a p-block of a finite group G with defect group +D. Recall that a B-subsection is a pair (u, b) where u ∈ D and b is a Brauer correspondent of B in CG(u). +For χ ∈ Irr(B) and ϕ ∈ IBr(b) we denote the corresponding generalized decomposition number by du +χϕ. If +u = 1, we obtain the (ordinary) decomposition number dχϕ = d1 +χϕ. We put l(b) = |IBr(b)| as usual. +Following [22, p. 114], we define a class function χ(u,b) by +χ(u,b)(us) := +� +ϕ∈IBr(b) +du +χϕϕ(s) +3 + +for s ∈ CG(u)0 and χ(u,b)(x) = 0 whenever x is outside the p-section of u. If R is a set of representatives +for the G-conjugacy classes of B-subsections, then χ = � +(u,b)∈R χ(u,b) by Brauer’s second main theorem +(see [22, Problem 5.3]). Now suppose that B is nilpotent and λ ∈ Irr(D). By [16, Proposition 8.11.4], each +Brauer correspondent b of B is nilpotent and in particular l(b) = 1. Broué–Puig [4] have shown that, if χ +has height 0, then +λ ∗ χ := +� +(u,b)∈R +λ(u)χ(u,b) ∈ Irr(B) +and (λ ∗ χ)(1) = λ(1)χ(1). Note also that du +λ∗χ,ϕ = λ(u)du +χϕ. +Proof of Theorem A. Let R be a set of representatives for the G-conjugacy classes of B-subsections +(u, bu) ≤ (D, bB) (see [22, p. 219]). Since B is nilpotent, we have IBr(bu) = {ϕu} for all (u, bu) ∈ R. +Since the Brauer correspondence is compatible with complex conjugation, (u, bu)t ≤ (D, bD)t = (D, bD) +where NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩. Thus, (u, bu)t is D-conjugate to some (u′, bu′) ∈ R. +If p > 2, there exists a unique p-rational character χ0 ∈ Irr(B) of height 0, which must be real by +uniqueness (see [4, Remark after Theorem 1.2]). If p = 2, there is a 2-rational real character χ0 ∈ Irr(B) +of height 0 by [8, Theorem 5.1]. Then du +χ0,ϕu = du +χ0,ϕu ∈ Z and +χ(u,bu) +0 += χ(u,bu) +0 += χ(u,bu)t +0 += χ(u′,bu′) +0 +. +Now let λ ∈ Irr(D). Then +λ ∗ χ0 = +� +(u,bu)∈R +λ(u)χ(u,bu) +0 += +� +(u,bu)∈R +λ(u)χ(u′,bu′) +0 +. +Since the class functions χ(u,b) +0 +have disjoint support, they are linearly independent. Therefore, λ ∗ χ0 is +real if and only if λ(ut) = λ(u′) = λ(u) for all (u, bu) ∈ R. Since every conjugacy class of D is represented +by some u with (u, bu) ∈ R, we conclude that λ ∗ χ0 is real if and only λt = λ. Moreover, if λ(1) = ph, +then λ ∗ χ0 has height h. This proves the first claim. +To prove the second claim, let p > 2 and IBr(B) = {ϕ}. Then the decomposition numbers dλ∗χ0,ϕ = λ(1) +are powers of p; in particular they are odd. A theorem of Thompson and Willems (see [26, Theorem 2.8]) +states that all real characters χ with dχ,ϕ odd have the same F-S indicator. So in our situation all real +characters in Irr(B) have the same F-S indicator. +Since the automorphism group of a p-group is “almost always” a p-group (see [11]), the following conse- +quence is of interest. +Corollary 1. Let B be a real, nilpotent p-block with defect group D such that p and |Aut(D)| are odd. +Then B has a unique real character. +Proof. The hypothesis on Aut(D) implies that NG(D, bD)∗ = DCG(D). Hence by Theorem A, the number +of real characters in Irr(B) is the number of real characters in D. Since p > 2, the trivial character is the +only real character of D. +4 + +The next lemma is a consequence of Brauer’s second main theorem and the fact that |{g ∈ G : g2 = x}| = +|{g ∈ CG(x) : g2 = x}| is locally determined for g, x ∈ G. +Lemma 2 (Brauer). For every p-block B of G and every B-subsection (u, b) with ϕ ∈ IBr(b) we have +� +χ∈Irr(B) +ǫ(χ)du +χϕ = +� +ψ∈Irr(b) +ǫ(ψ)du +ψϕ = +� +ψ∈Irr(b) +ǫ(ψ)ψ(u) +ψ(1) dψϕ. +If l(b) = 1, then +� +χ∈Irr(B) +ǫ(χ)du +χϕ = +1 +ϕ(1) +� +ψ∈Irr(b) +ǫ(ψ)ψ(u). +Proof. The first equality is [3, Theorem 4A]. The second follows from u ∈ Z(CG(u)). If l(b) = 1, then +ψ(1) = dψϕϕ(1) for ψ ∈ Irr(b) and the last claim follows. +Recall that a canonical character of B is a character θ ∈ Irr(DCG(D)) lying in a Brauer correspondent of +B such that D ≤ Ker(θ) (see [22, Theorem 9.12]). We define the extended stabilizer +NG(D)∗ +θ := +� +g ∈ NG(D) : θg ∈ {θ, θ} +� +. +The following results adds some detail to the nilpotent case of [20, Theorem 1]. +Theorem 3. Let B be a real, nilpotent p-block with cyclic defect group D = ⟨u⟩ and p > 2. Let θ ∈ +Irr(CG(D)) be a canonical character of B and set T := NG(D)∗ +θ. Then one of the following holds: +(1) θ ̸= θ. All characters in Irr(B) are real with F-S indicator ǫ(θT ). +(2) θ = θ. The unique non-exceptional character χ0 ∈ Irr(B) is the only real character in Irr(B) and +ǫ(χ0) = sgn(χ0(u))ǫ(θ) where sgn(χ0(u)) is the sign of χ0(u). +Proof. Let bD be a Brauer correspondent of B in CG(D) containing θ. Then T = NG(D, bD)∗. If θ ̸= θ, +then T inverts the elements of D since p > 2. Thus, Theorem A implies that all characters in Irr(B) are +real. By [20, Theorem 1(v)], the common F-S indicator is the Gow indicator of θ with respect to T. This +is easily seen to be ǫ(θT ) (see [20, after Eq. (2)]). +Now assume that θ = θ. Here Theorem A implies that the unique p-rational character χ0 ∈ Irr(B) is the +only real character. In particular, χ0 must be the unique non-exceptional character. Note that (u, bD) is +a B-subsection and IBr(bD) = {ϕ}. Since χ0 is p-rational, du +χ0ϕ = ±1. Since all Brauer correspondents of +B in CG(u) are conjugate under NG(D), the generalized decomposition numbers are Galois conjugate, in +particular du +χ0ϕ does not depend on the choice of bD. Hence, +χ0(u) = |NG(D) : NG(D)θ|du +χ0ϕϕ(1) +and du +χ0ϕ = sgn(χ0(u)). Moreover, θ is the unique non-exceptional character of bD and θ(u) = θ(1). By +Lemma 2, we obtain +ǫ(χ0) = sgn(χ0(u)) +� +χ∈Irr(B) +ǫ(χ)du +χϕ = sgn(χ0(u)) +ϕ(1) +� +ψ∈Irr(bD) +ǫ(ψ)ψ(u) = sgn(χ0(u))ǫ(θ). +5 + +If B is a nilpotent block with canonical character θ ̸= θ, the common F-S indicator of the real characters +in Irr(B) is not always ǫ(θT ) as in Theorem 3. A counterexample is given by a certain 3-block of G = +SmallGroup(288, 924) with defect group D ∼= C3 × C3. +We now restrict ourselves to 2-blocks. Héthelyi–Horváth–Szabó [12] introduced four conjectures, which +are real versions of Brauer’s conjecture, Olsson’s conjecture and Eaton’s conjecture. We only state the +strongest of them, which implies the remaining three. Let D(0) := D and D(k+1) := [D(k), D(k)] for k ≥ 0 +be the members of the derived series of D. +Conjecture 4 (Héthelyi–Horváth–Szabó). Let B be a 2-block with defect group D. For every h ≥ 0, the +number of real characters in Irr(B) of height ≤ h is bounded by the number of elements of D/D(h+1) which +are real in NG(D)/D(h+1). +A conjugacy class K of G is called real if K = K−1 := {x−1 : x ∈ K}. A conjugacy class K of a normal +subgroup N ⊴ G is called real under G if there exists g ∈ G such that Kg = K−1. +Proposition 5. Let B be a nilpotent 2-block with defect group D and Brauer correspondent bD in DCG(D). +Then the number of real characters in Irr(B) of height ≤ h is bounded by the number of conjugacy classes +of D/D(h+1) which are real under NG(D, bD)∗/D(h+1). In particular, Conjecture 4 holds for B. +Proof. We may assume that B is real. As in the proof of Theorem A, we fix some 2-rational real character +χ0 ∈ Irr(B) of height 0. Now λ ∗ χ0 has height ≤ h if and only if λ(1) ≤ ph for λ ∈ Irr(B). By [14, +Theorem 5.12], the characters of degree ≤ ph in Irr(D) lie in Irr(D/D(h+1)). By Theorem A, λ ∗ χ0 is +real if and only if λt = λ. By Brauer’s permutation lemma (see [23, Theorem 2.3]), the number of those +characters λ coincides with the number of conjugacy classes K of D/D(h+1) such that Kt = K−1. Now +Conjecture 4 follows from NG(D, bD)∗ ≤ NG(D). +3 Extended defect groups +We continue to assume that p = 2. As usual we choose a complete discrete valuation ring O such that +F := O/J(O) is an algebraically closed field of characteristic 2. Let Cl(G) be the set of conjugacy classes +of G. For K ∈ Cl(G) let K+ := � +x∈K x ∈ Z(FG) be the class sum of K. We fix a 2-block B of FG with +block idempotent 1B = � +K∈Cl(G) aKK+ where aK ∈ F. The central character of B is defined by +λB : Z(FG) → F, +K+ �→ +�|K|χ(g) +χ(1) +�∗ +where g ∈ K, χ ∈ Irr(B) and ∗ denotes the canonical reduction O → F (see [22, Chapter 2]). +Since λB(1B) = 1, there exists K ∈ Cl(G) such that aK ̸= 0 ̸= λB(K+). We call K a defect class of +B. By [22, Corollary 3.8], K consists of elements of odd order. According to [22, Corollary 4.5], a Sylow +2-subgroup D of CG(x) where x ∈ K is a defect group of B. For x ∈ K let +CG(x)∗ := {g ∈ G : gxg−1 = x±1} ≤ G +be the extended centralizer of x. +6 + +Proposition 6 (Gow, Murray). Every real 2-block B has a real defect class K. Let x ∈ K. Choose a +Sylow 2-subgroup E of CG(x)∗ and put D := E ∩ CG(x). Then the G-conjugacy class of the pair (D, E) +does not depend on the choice of K or x. +Proof. For the principal block (which is always real since it contains the trivial character), K = {1} is +a real defect class and E = D is a Sylow 2-subgroup of G. Hence, the uniqueness follows from Sylow’s +theorem. Now suppose that B is non-principal. The existence of K was first shown in [8, Theorem 5.5]. +Let L be another real defect class of B and choose y ∈ L. By [9, Corollary 2.2], we may assume after +conjugation that E is also a Sylow 2-subgroup of CG(y)∗. Let Dx := E ∩ CG(x) and Dy := E ∩ CG(y). +We may assume that |E : Dx| = 2 = |E : Dy| (cf. the remark after the proof). +We now introduce some notation in order to apply [17, Proposition 14]. Let Σ = ⟨σ⟩ ∼= C2. We consider +FG as an F[G × Σ]-module where G acts by conjugation and gσ = g−1 for g ∈ G (observe that these +actions indeed commute). For H ≤ G × Σ let +TrG×Σ +H +: (FG)H → (FG)G×Σ, α �→ +� +x∈R +αx +be the relative trace with respect to H, where R denotes a set of representatives of the right cosets of H +in G × Σ. By [17, Proposition 14], we have 1B ∈ TrG×Σ +Ex +(FG) where Ex := Dx⟨exσ⟩ for some ex ∈ E \ Dx. +By the same result we also obtain that Dy⟨eyσ⟩ with ey ∈ E \ Dy is G-conjugate to Ex. This implies that +Dy is conjugate to Dx inside NG(E). In particular, (Dx, E) and (Dy, E) are G-conjugate as desired. +Definition 7. In the situation of Proposition 6 we call E an extended defect group and (D, E) a defect +pair of B. +We stress that real 2-blocks can have non-real defect classes and non-real blocks can have real defect classes +(see [10, Theorem 3.5]). +It is easy to show that non-principal real 2-blocks cannot have maximal defect (see [22, Problem 3.8]). +In particular, the trivial class cannot be a defect class and consequently, |E : D| = 2 in those cases. +For non-real blocks we define the extended defect group by E := D for convenience. Every given pair of +2-groups D ≤ E with |E : D| = 2 occurs as a defect pair of a real (nilpotent) block. To see this, let Q ∼= C3 +and G = Q ⋊ E with CE(Q) = D. Then G has a unique non-principal block with defect pair (D, E). +We recall from [14, p. 49] that +� +χ∈Irr(G) +ǫ(χ)χ(g) = |{x ∈ G : x2 = g}| +(3) +for all g ∈ G. The following proposition provides some interesting properties of defect pairs. +Proposition 8 (Gow, Murray). Let B be a real 2-block with defect pair (D, E). Let bD be a Brauer +correspondent of B in DCG(D). Then the following holds: +(i) NG(D, bD)∗ = NG(D, bD)E. In particular, bD is real if and only if E = DCE(D). +(ii) For u ∈ D, we have � +χ∈Irr(B) ǫ(χ)χ(u) ≥ 0 with strict inequality if and only if u is G-conjugate to +e2 for some e ∈ E \ D. In particular, E splits over D if and only if � +χ∈Irr(B) ǫ(χ)χ(1) > 0. +7 + +(iii) E/D′ splits over D/D′ if and only if all height zero characters in Irr(B) have non-negative F-S +indicator. +Proof. +(i) See [19, Lemma 1.8] and [18, Theorem 1.4]. +(ii) See [19, Lemma 1.3]. +(iii) See [8, Theorem 5.6]. +The next proposition extends [18, Lemma 1.3]. +Corollary 9. Suppose that B is a 2-block with defect pair (D, E) where D is abelian. Then E splits over +D if and only if all characters in Irr(B) have non-negative F-S indicator. +Proof. If B is non-real, then E = D splits over D and all characters in Irr(B) have F-S indicator 0. Hence, +let B = B. By Kessar–Malle [15], all characters in Irr(B) have height 0. Hence, the claim follows from +Proposition 8(iii). +Theorem 10. Let B be a real, nilpotent 2-block with defect pair (D, E) where D is abelian. If E splits over +D, then all real characters in Irr(B) have F-S indicator 1. Otherwise exactly half of the real characters +have F-S indicator 1. In either case, Conjecture B holds for B. +Proof. If E splits over D, then all real characters in Irr(B) have F-S indicator 1 by Corollary 9. Otherwise +we have � +χ∈Irr(B) ǫ(χ) = 0 by Proposition 8(ii), because all characters in Irr(B) have the same degree. +Hence, exactly half of the real characters have F-S indicator 1. Using Theorem A we can determine the +number of characters for each F-S indicator. For the last claim, we may therefore replace B by the unique +non-principal block of G = Q ⋊ E where Q ∼= C3 and CE(Q) = D (mentioned above). In this case +Conjecture B follows from Gow [8, Lemma 2.2] or Theorem E. +Example 11. Let B be a real block with defect group D ∼= C4 × C2. Then B is nilpotent since Aut(D) +is a 2-group and D is abelian. Moreover |Irr(B)| = 8. The F-S indicators depend not only on E, but also +on the way D embeds into E. The following cases can occur (here M16 denotes the modular group and +[16, 3] refers to the small group library): +F-S indicators +E ++ + + + + + ++ +D8 × C2 ++ + + + − − −− +Q8 × C2, C4 ⋊ C4 with Φ(D) = E′ ++ + + + 0 0 0 0 +D, D × C2, D8 ∗ C4, [16, 3] ++ + − − 0 0 0 0 +C2 +4, C8 × C2, M16, C4 ⋊ C4 with Φ(D) ̸= E′ +The F-S indicator ǫ(Φ) appearing in Conjecture C has an interesting interpretation as follows. Let Ω := +{g ∈ G : g2 = 1}. The conjugation action of G on Ω turns FΩ into an FG-module, called the involution +module. +8 + +Lemma 12 (Murray). Let B be a real 2-block and ϕ ∈ IBr(B). Then ǫ(Φϕ) is the multiplicity of ϕ as a +constituent of the Brauer character of FΩ. +Proof. See [18, Lemma 2.6]. +Next we develop a local version of Conjecture C. Let B be a real 2-block with defect pair (D, E) and B- +subsection (u, b). If E = DCE(u), then b is real and (CD(u), CE(u)) is a defect pair of b by [19, Lemma 2.6] +applied to the subpair (⟨u⟩, b). Conversely, if b is real, we may assume that (CD(u), CE(u)) is a defect +pair of b by [19, Theorem 2.7]. If b is non-real, we may assume that (CD(u), CD(u)) = (CD(u), CE(u)) is +a defect pair of b. +Theorem 13. Let B be 2-block of a finite group G with defect pair (D, E). Suppose that Conjecture C +holds for all 2-blocks of sections of G. Let (u, b) be a B-subsection with defect pair (CD(u), CE(u)) such +that IBr(b) = {ϕ}. Then +� +χ∈Irr(B) +ǫ(χ)du +χϕ = +� +|{x ∈ D : x2 = u}| +if B is the principal block, +|{x ∈ E \ D : x2 = u}| +otherwise. +Proof. If B is not real, then B is non-principal and E = D. It follows that ǫ(χ) = 0 for all χ ∈ Irr(B) and +|{x ∈ E \ D : x2 = u}| = 0. +Hence, we may assume that B is real. By Lemma 2, we have +� +χ∈Irr(B) +ǫ(χ)du +χϕ = +� +ψ∈Irr(b) +ǫ(ψ)du +ψϕ = +1 +ϕ(1) +� +ψ∈Irr(b) +ǫ(ψ)ψ(u). +(4) +Suppose that B is the principal block. Then b is the principal block of CG(u) by Brauer’s third main +theorem (see [22, Theorem 6.7]). The hypothesis l(b) = 1 implies that ϕ = 1CG(u) and CG(u) has a normal +2-complement N (see [22, Corollary 6.13]). It follows that Irr(b) = Irr(CG(u)/N) = Irr(CD(u)) and +� +ψ∈Irr(b) +ǫ(ψ)du +ψϕ = +� +λ∈Irr(CD(u)) +ǫ(λ)λ(u) = |{x ∈ CD(u) : x2 = u}| +by (3). Since every x ∈ D with x2 = u lies in CD(u), we are done in this case. +Now let B be a non-principal real 2-block. If b is not real, then (4) shows that � +χ∈Irr(B) ǫ(χ)du +χϕ = 0. On +the other hand, we have CE(u) = CD(u) ≤ D and |{x ∈ E \ D : x2 = u}| = 0. Hence, we may assume +that b is real. Since every x ∈ E with x2 = u lies in CE(u), we may assume that u ∈ Z(G) by (4). +Then χ(u) = du +χϕϕ(1) for all χ ∈ Irr(B). If u2 /∈ Ker(χ), then χ(u) /∈ R and ǫ(χ) = 0. Thus, it suffices to +sum over χ with du +χϕ = ±dχϕ. Let Z := ⟨u⟩ ≤ Z(G) and G := G/Z. Let ˆB be the unique (real) block of +9 + +G dominated by B. By [19, Lemma 1.7], (D, E) is a defect pair for ˆB. Then, using [14, Lemma 4.7] and +Conjecture C for B and ˆB, we obtain +� +χ∈Irr(B) +ǫ(χ)du +χϕ = +� +χ∈Irr(B) +ǫ(χ)(dχϕ + du +χϕ) − +� +χ∈Irr(B) +ǫ(χ)dχϕ += 2 +� +χ∈Irr( ˆB) +ǫ(χ)dχϕ − +� +χ∈Irr(B) +ǫ(χ)dχϕ += 2|{x ∈ E \ D : x2 = 1}| − |{x ∈ E \ D : x2 = 1}| += +� +λ∈Irr(E) +ǫ(λ)(λ(1) + λ(u)) − +� +λ∈Irr(D) +ǫ(λ)(λ(1) + λ(u)) +− +� +λ∈Irr(E) +ǫ(λ)λ(1) + +� +λ∈Irr(D) +ǫ(λ)λ(1) += +� +λ∈Irr(E) +ǫ(λ)λ(u) − +� +λ∈Irr(D) +ǫ(λ)λ(u) = |{x ∈ E \ D : x2 = u}|. +4 Theorems D and E +The following result implies Theorem D. +Theorem 14. Suppose that B is a real, nilpotent, non-principal 2-block fulfilling the statement of Theorem 13. +Then Conjecture B holds for B. +Proof. Let (D, E) be defect pair of B. By Gow [8, Theorem 5.1], there exists a 2-rational character +χ0 ∈ Irr(B) of height 0 and ǫ(χ0) = 1. Let +Γ : Irr(D) → Irr(B), +λ �→ λ ∗ χ0 +be the Broué–Puig bijection. Let (u1, b1), . . . , (uk, bk) be representatives for the conjugacy classes of B- +subsections. Since B is nilpotent, we may assume that u1, . . . , uk ∈ D represent the conjugacy classes of +D. Let IBr(bi) = {ϕi} for i = 1, . . . , k. Since χ0 is 2-rational, we have σi := du +χ0,ϕi ∈ {±1} for i = 1, . . . , k. +Hence, the generalized decomposition matrix of B has the form +Q = (λ(ui)σi : λ ∈ Irr(D), i = 1, . . . , k) +(see [16, Section 8.10]). Let v := (ǫ(Γ(λ)) : λ ∈ Irr(D)) and w := (w1, . . . , wk) where wi := |{x ∈ E \ D : +x2 = ui}|. Then Theorem 13 reads as vQ = w. +Let di := |CD(ui)| and d = (d1, . . . , dk). Then the second orthogonality relation yields QtQ = diag(d) +where Qt denotes the transpose of Q. It follows that Q−1 = diag(d)−1Q +t and +v = w diag(d)−1Q +t = w diag(d)−1Qt, +10 + +because v = v. Since wi = |{x ∈ E \ D : x2 = uy +i }| for every y ∈ D, we obtain �k +i=1 wi|D : CD(ui)| = +|E \ D| = |D|. In particular, +1 = ǫ(χ0) = +k +� +i=1 +wiσi +|CD(ui)| ≤ +k +� +i=1 +wi|σi| +|CD(ui)| = 1. +Therefore, σi = 1 or wi = 0 for each i. This means that the signs σi have no impact on the solution of the +linear system xQ = w. Hence, we may assume that Q = (λ(ui)) is just the character table of D. Since Q +has full rank, v is the only solution of xQ = w. Setting µ(λ) := +1 +|D| +� +e∈E\D λ(e2), it suffices to show that +(µ(λ) : λ ∈ Irr(D)) is another solution of xQ = w. Indeed, +� +λ∈Irr(D) +λ(ui) +|D| +� +e∈E\D +λ(e2) = +1 +|D| +� +e∈E\D +� +λ∈Irr(D) +λ(ui)λ(e2) += +1 +|D| +� +e∈E\D +e2=u−1 +i +|D : CD(ui)||CD(ui)| = wi +for i = 1, . . . , k. +Theorem E. Conjectures B and C hold for all nilpotent 2-blocks of solvable groups. +Proof. Let B be a real, nilpotent, non-principal 2-block of a solvable group G with defect pair (D, E). +We first prove Conjecture C for B. Since all sections of G are solvable and all blocks dominated by B- +subsections are nilpotent, Conjecture C holds for those blocks as well. Hence, the hypothesis of Theorem 13 +is fulfilled for B. Now by Theorem 14, Conjecture B holds for B. +Let N := O2′(G) and let θ ∈ Irr(N) such that the block {θ} is covered by B. Since B is non-principal, +θ ̸= 1N and therefore θ ̸= θ as N has odd order. Since B also lies over θ, it follow that Gθ < G. Let b +be the Fong–Reynolds correspondent of B in the extended stabilizer G∗ +θ. By [22, Theorem 9.14] and [20, +p. 94], the Clifford correspondence Irr(b) → Irr(B), ψ �→ ψG preserves decomposition numbers and F-S +indicators. Thus, we need to show that b has defect pair (D, E). Let β be the Fong–Reynolds correspondent +of B in Gθ. By [22, Theorem 10.20], β is the unique block over θ. In particular, the block idempotents +1β = 1θ are the same (we identify θ with the block {θ}). Since b is also the unique block of G∗ +θ over θ, we +have 1b = 1θ + 1θ = � +x∈N αxx for some αx ∈ F. Let S be a set of representatives for the cosets G/G∗ +θ. +Then +1B = +� +s∈S +(1θ + 1θ)s = +� +s∈S +1s +b = +� +g∈N +�� +s∈S +αgs−1 +� +g. +Hence, there exists a real defect class K of B such that αgs−1 ̸= 0 for some g ∈ K and s ∈ S. Of course +we can assume that g = gs−1. Then 1b does not vanish on g. By [22, Theorem 9.1], the central characters +λB, λb and λθ agree on N. It follows that K is also a real defect class of b. Hence, we may assume that +(D, E) is a defect pair of b. +It remains to consider G = G∗ +θ and B = b. Then D is a Sylow 2-subgroup of Gθ by [22, Theorem 10.20] +and E is a Sylow 2-subgroup of G. Since |G : Gθ| = 2, it follows that Gθ ⊴ G and N = O2′(Gθ). By +11 + +[21, Lemma 1 and 2], β is nilpotent and Gθ is 2-nilpotent, i. e. Gθ = N ⋊ D and G = N ⋊ E. Let +�Φ := � +χ∈Irr(B) χ(1)χ = ϕ(1)Φ where IBr(B) = {ϕ}. We need to show that +ǫ(�Φ) = ϕ(1)|{x ∈ E \ D : x2 = 1}|. +Note that χN = χ(1) +2θ(1)(θ + θ). By Frobenius reciprocity, it follows that �Φ = 2θ(1)θG and +�ΦN = |G : N|θ(1)(θ + θ). +Since Φ vanishes on elements of even order, �Φ vanishes outside N. Since �ΦGθ is a sum of non-real characters +in β, we have +ǫ(�Φ) = +1 +|G| +� +g∈Gθ +�Φ(g2) + 1 +|G| +� +g∈G\Gθ +�Φ(g2) = +1 +|G| +� +g∈G\Gθ +�Φ(g2). +Every g ∈ G \ Gθ = NE \ ND with g2 ∈ N is N-conjugate to a unique element of the form xy where +x ∈ E \ D is an involution and y ∈ CN(x) (Sylow’s theorem). Setting ∆ := {x ∈ E \ D : x2 = 1}, we +obtain +ǫ(�Φ) = θ(1) +|N| +� +x∈∆ +|N : CN(x)| +� +y∈CN (x) +(θ(y) + θ(y)) = 2θ(1) +� +x∈∆ +1 +|CN(x)| +� +y∈CN(x) +θ(y). +(5) +For x ∈ ∆ let Hx := N⟨x⟩. Again by Sylow’s theorem, the N-orbit of x is the set of involutions in Hx. +From θx = θ we see that θHx is an irreducible character of 2-defect 0. By [8, Theorem 5.1], we have +ǫ(θHx) = 1. Now applying the same argument as before, it follows that +1 = ǫ(θHx) = +1 +|N| +� +g∈Hx\N +θHx(g2) = +2 +|CN(x)| +� +y∈CN(x) +θ(y). +Combined with (5), this yields ǫ(�Φ) = 2θ(1)|∆|. By Green’s theorem (see [22, Theorem 8.11]), ϕN = θ + θ +and ǫ(�Φ) = ϕ(1)|∆| as desired. +For non-principal blocks B of solvable groups with l(B) = 1 it is not true in general that Gθ is 2- +nilpotent in the situation of Theorem E. For example, a (non-real) 2-block of a triple cover of A4 × A4 +has a unique simple module. Extending this group by an automorphism of order 2, we obtain the group +G = SmallGroup(864, 3988), which fulfills the assumptions with D ∼= C4 +2, N ∼= C3 and |G : NE| = 9. +In order to prove Conjecture C for arbitrary 2-blocks of solvable groups, we may follow the steps in the +proof above and invoke a result on fully ramified Brauer characters [24, Theorem 2.1]. The claim then +boils down to a purely group-theoretical statement: Let B be a real, non-principal 2-block of a solvable +group G with defect pair (D, E) and l(B) = 1. Let G := G/O2′(G). Then +|{x ∈ G \ Gθ : x2 = 1}| = |{x ∈ E \ D : x2 = 1}| +� +|G : EN|. +Unfortunately, I am unable to prove this. +Acknowledgment +I thank Gabriel Navarro for providing some arguments for Theorem E from his paper [21]. John Mur- +ray and three anonymous referees have made many valuable comments, which improved the quality of the +manuscript. The work is supported by the German Research Foundation (SA 2864/3-1 and SA 2864/4-1). +12 + +References +[1] J. An and C. W. Eaton, Nilpotent Blocks of Quasisimple Groups for the Prime Two, Algebr. Represent. +Theory 16 (2013), 1–28. +[2] R. Brauer, Representations of finite groups, in: Lectures on Modern Mathematics, Vol. I, 133–175, +Wiley, New York, 1963. +[3] R. Brauer, Some applications of the theory of blocks of characters of finite groups. III, J. Algebra 3 +(1966), 225–255. +[4] M. Broué and L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128. +[5] C. W. Eaton, Generalisations of conjectures of Brauer and Olsson, Arch. Math. (Basel) 81 (2003), +621–626. +[6] The +GAP +Group, +GAP +– +Groups, +Algorithms, +and +Programming, +Version +4.12.0; +2022, +(http://www.gap-system.org). +[7] R. Gow, Real-valued characters and the Schur index, J. Algebra 40 (1976), 258–270. +[8] R. Gow, Real-valued and 2-rational group characters, J. Algebra 61 (1979), 388–413. +[9] R. Gow, Real 2-blocks of characters of finite groups, Osaka J. Math. 25 (1988), 135–147. +[10] R. Gow and J. Murray, Real 2-regular classes and 2-blocks, J. Algebra 230 (2000), 455–473. +[11] G. T. Helleloid and U. Martin, The automorphism group of a finite p-group is almost always a p-group, +J. Algebra 312 (2007), 294–329. +[12] L. Héthelyi, E. Horváth and E. Szabó, Real characters in blocks, Osaka J. Math. 49 (2012), 613–623. +[13] T. Ichikawa and Y. Tachikawa, The Super Frobenius–Schur Indicator and Finite Group Gauge Theories +on Pin− Surfaces, to appear in Commun. Math. Phys., DOI: 10.1007/s00220-022-04601-9. +[14] I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. +[15] R. Kessar and G. Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2) +178 (2013), 321–384. +[16] M. Linckelmann, The block theory of finite group algebras. Vol. II, London Mathematical Society +Student Texts, Vol. 92, Cambridge University Press, Cambridge, 2018. +[17] J. Murray, Strongly real 2-blocks and the Frobenius-Schur indicator, Osaka J. Math. 43 (2006), 201– +213. +[18] J. Murray, Components of the involution module in blocks with cyclic or Klein-four defect group, J. +Group Theory 11 (2008), 43–62. +[19] J. Murray, Real subpairs and Frobenius-Schur indicators of characters in 2-blocks, J. Algebra 322 +(2009), 489–513. +[20] J. Murray, Frobenius-Schur indicators of characters in blocks with cyclic defect, J. Algebra 533 (2019), +90–105. +13 + +[21] G. Navarro, Nilpotent characters, Pacific J. Math. 169 (1995), 343–351. +[22] G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, +Vol. 250, Cambridge University Press, Cambridge, 1998. +[23] G. Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathemat- +ics, Vol. 175, Cambridge University Press, Cambridge, 2018. +[24] G. Navarro, B. Späth and P. H. Tiep, On fully ramified Brauer characters, Adv. Math. 257 (2014), +248–265. +[25] S. Trefethen and C. R. Vinroot, A computational approach to the Frobenius-Schur indicators of finite +exceptional groups, Internat. J. Algebra Comput. 30 (2020), 141–166. +[26] W. Willems, Duality and forms in representation theory, in: Representation theory of finite groups +and finite-dimensional algebras (Bielefeld, 1991), 509–520, Progr. Math., Vol. 95, Birkhäuser, Basel, +1991. +14 + diff --git a/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/load_file.txt b/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..004adad3c98810035e5adb3729822719d1fe525e --- /dev/null +++ b/-dFQT4oBgHgl3EQf6zaC/content/tmp_files/load_file.txt @@ -0,0 +1,579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf,len=578 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='13440v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='RT] 31 Jan 2023 Real characters in nilpotent blocks Benjamin Sambale∗ February 1, 2023 Dedicated to Pham Huu Tiep on the occasion of his 60th birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Abstract We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We further conjecture that the Frobenius–Schur indicators of those characters can be computed for p = 2 in terms of the extended defect group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We derive this from a more general conjecture on the Frobenius–Schur indicator of projective indecomposable characters of 2-blocks with one simple module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This extends results of Murray on 2-blocks with cyclic and dihedral defect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Keywords: real characters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Frobenius–Schur indicators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' nilpotent blocks AMS classification: 20C15, 20C20 1 Introduction An important task in representation theory is to determine global invariants of a finite group G by means of local subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Dade’s conjecture, for instance, predicts the number of irreducible characters χ ∈ Irr(G) such that the p-part χ(1)p is a given power of a prime p (see [23, Conjecture 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since Gow’s work [7], there has been an increasing interest in counting real (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' real-valued) characters and more generally characters with a given field of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The quaternion group Q8 testifies that a real irreducible character χ is not always afforded by a repre- sentation over the real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The precise behavior is encoded by the Frobenius–Schur indicator (F-S indicator, for short) ǫ(χ) := 1 |G| � g∈G χ(g2) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if χ ̸= χ, 1 if χ is realized by a real representation, −1 if χ is real, but not realized by a real representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (1) A new interpretation of the F-S indicator in terms of superalgebras has been given recently in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The case of the dihedral group D8 shows that ǫ(χ) is not determined by the character table of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The computation ∗Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Han- nover, Germany, sambale@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='uni-hannover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='de 1 of F-S indicators can be a surprisingly difficult task, which has not been fully completed for the simple groups of Lie type, for instance (see [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Problem 14 on Brauer’s famous list [2] asks for a group-theoretical interpretation of the number of χ ∈ Irr(G) with ǫ(χ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' To obtain deeper insights, we fix a prime p and assume that χ lies in a p-block B of G with defect group D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By complex conjugation we obtain another block B of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If B ̸= B, then clearly ǫ(χ) = 0 for all χ ∈ Irr(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, we assume that B is real, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' B = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' John Murray [18, 19] has computed the F-S indicators when D is a cyclic 2-group or a dihedral 2-group (including the Klein four-group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' His results depend on the fusion system of B, on Erdmann’s classification of tame blocks and on the structure of the so-called extended defect group E of B (see Definition 7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For p > 2 and D cyclic, he obtained in [20] partial information on the F-S indicators in terms of the Brauer tree of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The starting point of my investigation is the well-known fact that 2-blocks with cyclic defect groups are nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Assume that B is nilpotent and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If B is the principal block, then G = Op′(G)D and Irr(B) = Irr(G/Op′(G)) = Irr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In this case the F-S indicators of B are determined by D alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Thus, suppose that B is non-principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Broué–Puig [4], there exists a height-preserving bijection Irr(D) → Irr(B), λ �→ λ ∗ χ0 where χ0 ∈ Irr(B) is a fixed character of height 0 (see also [16, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' However, this bijection does not in general preserve F-S indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For instance, the dihedral group D24 has a nilpotent 2-block with defect group C4 and a nilpotent 3-block with defect group C3, although every character of D24 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Our main theorem asserts that the number of real characters in a nilpotent block is nevertheless locally determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' To state it, we introduce the extended inertial group NG(D, bD)∗ := � g ∈ NG(D) : bg D ∈ {bD, bD} � where bD is a Brauer correspondent of B in DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent p-block of a finite group G with defect group D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let bD be a Brauer correspondent of B in DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the number of real characters in Irr(B) of height h coincides with the number of characters λ ∈ Irr(D) of degree ph such that λt = λ where NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If p > 2, then all real characters in Irr(B) have the same F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In contrast to arbitrary blocks, Theorem A implies that nilpotent real blocks have at least one real character (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 92] and [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If bD = bD, then B and D have the same number of real characters, because NG(D, bD) = DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This recovers a result of Murray [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' As another consequence, we will derive in Proposition 5 a real version of Eaton’s conjecture [5] for nilpotent blocks as put forward by Héthelyi–Horváth–Szabó [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The F-S indicators of real characters in nilpotent blocks seem to lie somewhat deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We still conjecture that they are locally determined by a defect pair (see Definition 7) for p = 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjecture B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent, non-principal 2-block of a finite group G with defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then there exists a height preserving bijection Γ : Irr(D) → Irr(B) such that ǫ(Γ(λ)) = 1 |D| � e∈E\\D λ(e2) (2) for all λ ∈ Irr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 2 The right hand side of (2) was introduced and studied by Gow [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1] more generally for any groups D ≤ E with |E : D| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This invariant was later coined the Gow indicator by Murray [20, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For 2-blocks of defect 0, Conjecture B confirms the known fact that real characters of 2-defect 0 have F-S indicator 1 (see [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' There is no such result for odd primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' As a matter of fact, every real character has p-defect 0 whenever p does not divide |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In Theorem 10 we prove Conjecture B for abelian defect groups D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then it also holds for all quasisimple groups G by work of An–Eaton [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray’s results mentioned above, imply Conjecture B also for dihedral D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For p > 2, the common F-S indicator in the situation of Theorem A is not locally determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For instance, G = Q8⋊C9 = SmallGroup(72, 3) has a non-principal real 3-block with D ∼= C9 and common F-S indicator −1, while its Brauer correspondent in NG(D) ∼= C18 has common F-S indicator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Nevertheless, for cyclic defect groups D we find another way to compute this F-S indicator in Theorem 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Our second conjecture applies more generally to blocks with only one simple module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjecture C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, non-principal 2-block with defect pair (D, E) and a unique projective indecomposable character Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then ǫ(Φ) = |{x ∈ E \\ D : x2 = 1}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Here ǫ(Φ) is defined by extending (1) linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If ǫ(Φ) = 0, then E does not split over D and Conjecture C holds (see Proposition 8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjecture C implies a stronger, but more technical statement on 2-blocks with a Brauer correspondent with one simple module (see Theorem 13 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This allows us to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjecture C implies Conjecture B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We remark that our proof of Theorem D does not work block-by-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For solvable groups we offer a purely group-theoretical version of Conjecture C at the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We have checked Conjectures B and C with GAP [6] in many examples using the libraries of small groups, perfect groups and primitive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 2 Theorem A and its consequences Our notation follows closely Navarro’s book [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a p-block of a finite group G with defect group D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Recall that a B-subsection is a pair (u, b) where u ∈ D and b is a Brauer correspondent of B in CG(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For χ ∈ Irr(B) and ϕ ∈ IBr(b) we denote the corresponding generalized decomposition number by du χϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If u = 1, we obtain the (ordinary) decomposition number dχϕ = d1 χϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We put l(b) = |IBr(b)| as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Following [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 114], we define a class function χ(u,b) by χ(u,b)(us) := � ϕ∈IBr(b) du χϕϕ(s) 3 for s ∈ CG(u)0 and χ(u,b)(x) = 0 whenever x is outside the p-section of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If R is a set of representatives for the G-conjugacy classes of B-subsections, then χ = � (u,b)∈R χ(u,b) by Brauer’s second main theorem (see [22, Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now suppose that B is nilpotent and λ ∈ Irr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [16, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='4], each Brauer correspondent b of B is nilpotent and in particular l(b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Broué–Puig [4] have shown that, if χ has height 0, then λ ∗ χ := � (u,b)∈R λ(u)χ(u,b) ∈ Irr(B) and (λ ∗ χ)(1) = λ(1)χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Note also that du λ∗χ,ϕ = λ(u)du χϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let R be a set of representatives for the G-conjugacy classes of B-subsections (u, bu) ≤ (D, bB) (see [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 219]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since B is nilpotent, we have IBr(bu) = {ϕu} for all (u, bu) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since the Brauer correspondence is compatible with complex conjugation, (u, bu)t ≤ (D, bD)t = (D, bD) where NG(D, bD)∗/DCG(D) = ⟨tDCG(D)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Thus, (u, bu)t is D-conjugate to some (u′, bu′) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If p > 2, there exists a unique p-rational character χ0 ∈ Irr(B) of height 0, which must be real by uniqueness (see [4, Remark after Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If p = 2, there is a 2-rational real character χ0 ∈ Irr(B) of height 0 by [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then du χ0,ϕu = du χ0,ϕu ∈ Z and χ(u,bu) 0 = χ(u,bu) 0 = χ(u,bu)t 0 = χ(u′,bu′) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now let λ ∈ Irr(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then λ ∗ χ0 = � (u,bu)∈R λ(u)χ(u,bu) 0 = � (u,bu)∈R λ(u)χ(u′,bu′) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since the class functions χ(u,b) 0 have disjoint support, they are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Therefore, λ ∗ χ0 is real if and only if λ(ut) = λ(u′) = λ(u) for all (u, bu) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since every conjugacy class of D is represented by some u with (u, bu) ∈ R, we conclude that λ ∗ χ0 is real if and only λt = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Moreover, if λ(1) = ph, then λ ∗ χ0 has height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' To prove the second claim, let p > 2 and IBr(B) = {ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the decomposition numbers dλ∗χ0,ϕ = λ(1) are powers of p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' in particular they are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' A theorem of Thompson and Willems (see [26, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='8]) states that all real characters χ with dχ,ϕ odd have the same F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' So in our situation all real characters in Irr(B) have the same F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since the automorphism group of a p-group is “almost always” a p-group (see [11]), the following conse- quence is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent p-block with defect group D such that p and |Aut(D)| are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then B has a unique real character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The hypothesis on Aut(D) implies that NG(D, bD)∗ = DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence by Theorem A, the number of real characters in Irr(B) is the number of real characters in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since p > 2, the trivial character is the only real character of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 4 The next lemma is a consequence of Brauer’s second main theorem and the fact that |{g ∈ G : g2 = x}| = |{g ∈ CG(x) : g2 = x}| is locally determined for g, x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Lemma 2 (Brauer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For every p-block B of G and every B-subsection (u, b) with ϕ ∈ IBr(b) we have � χ∈Irr(B) ǫ(χ)du χϕ = � ψ∈Irr(b) ǫ(ψ)du ψϕ = � ψ∈Irr(b) ǫ(ψ)ψ(u) ψ(1) dψϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If l(b) = 1, then � χ∈Irr(B) ǫ(χ)du χϕ = 1 ϕ(1) � ψ∈Irr(b) ǫ(ψ)ψ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The first equality is [3, Theorem 4A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The second follows from u ∈ Z(CG(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If l(b) = 1, then ψ(1) = dψϕϕ(1) for ψ ∈ Irr(b) and the last claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Recall that a canonical character of B is a character θ ∈ Irr(DCG(D)) lying in a Brauer correspondent of B such that D ≤ Ker(θ) (see [22, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We define the extended stabilizer NG(D)∗ θ := � g ∈ NG(D) : θg ∈ {θ, θ} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The following results adds some detail to the nilpotent case of [20, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent p-block with cyclic defect group D = ⟨u⟩ and p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let θ ∈ Irr(CG(D)) be a canonical character of B and set T := NG(D)∗ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then one of the following holds: (1) θ ̸= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' All characters in Irr(B) are real with F-S indicator ǫ(θT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (2) θ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The unique non-exceptional character χ0 ∈ Irr(B) is the only real character in Irr(B) and ǫ(χ0) = sgn(χ0(u))ǫ(θ) where sgn(χ0(u)) is the sign of χ0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let bD be a Brauer correspondent of B in CG(D) containing θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then T = NG(D, bD)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If θ ̸= θ, then T inverts the elements of D since p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Thus, Theorem A implies that all characters in Irr(B) are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [20, Theorem 1(v)], the common F-S indicator is the Gow indicator of θ with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This is easily seen to be ǫ(θT ) (see [20, after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (2)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now assume that θ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Here Theorem A implies that the unique p-rational character χ0 ∈ Irr(B) is the only real character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, χ0 must be the unique non-exceptional character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Note that (u, bD) is a B-subsection and IBr(bD) = {ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since χ0 is p-rational, du χ0ϕ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since all Brauer correspondents of B in CG(u) are conjugate under NG(D), the generalized decomposition numbers are Galois conjugate, in particular du χ0ϕ does not depend on the choice of bD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, χ0(u) = |NG(D) : NG(D)θ|du χ0ϕϕ(1) and du χ0ϕ = sgn(χ0(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Moreover, θ is the unique non-exceptional character of bD and θ(u) = θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Lemma 2, we obtain ǫ(χ0) = sgn(χ0(u)) � χ∈Irr(B) ǫ(χ)du χϕ = sgn(χ0(u)) ϕ(1) � ψ∈Irr(bD) ǫ(ψ)ψ(u) = sgn(χ0(u))ǫ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 5 If B is a nilpotent block with canonical character θ ̸= θ, the common F-S indicator of the real characters in Irr(B) is not always ǫ(θT ) as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' A counterexample is given by a certain 3-block of G = SmallGroup(288, 924) with defect group D ∼= C3 × C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We now restrict ourselves to 2-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Héthelyi–Horváth–Szabó [12] introduced four conjectures, which are real versions of Brauer’s conjecture, Olsson’s conjecture and Eaton’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We only state the strongest of them, which implies the remaining three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let D(0) := D and D(k+1) := [D(k), D(k)] for k ≥ 0 be the members of the derived series of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjecture 4 (Héthelyi–Horváth–Szabó).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a 2-block with defect group D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For every h ≥ 0, the number of real characters in Irr(B) of height ≤ h is bounded by the number of elements of D/D(h+1) which are real in NG(D)/D(h+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' A conjugacy class K of G is called real if K = K−1 := {x−1 : x ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' A conjugacy class K of a normal subgroup N ⊴ G is called real under G if there exists g ∈ G such that Kg = K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a nilpotent 2-block with defect group D and Brauer correspondent bD in DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the number of real characters in Irr(B) of height ≤ h is bounded by the number of conjugacy classes of D/D(h+1) which are real under NG(D, bD)∗/D(h+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, Conjecture 4 holds for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We may assume that B is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' As in the proof of Theorem A, we fix some 2-rational real character χ0 ∈ Irr(B) of height 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now λ ∗ χ0 has height ≤ h if and only if λ(1) ≤ ph for λ ∈ Irr(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='12], the characters of degree ≤ ph in Irr(D) lie in Irr(D/D(h+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Theorem A, λ ∗ χ0 is real if and only if λt = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Brauer’s permutation lemma (see [23, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='3]), the number of those characters λ coincides with the number of conjugacy classes K of D/D(h+1) such that Kt = K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now Conjecture 4 follows from NG(D, bD)∗ ≤ NG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 3 Extended defect groups We continue to assume that p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' As usual we choose a complete discrete valuation ring O such that F := O/J(O) is an algebraically closed field of characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Cl(G) be the set of conjugacy classes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For K ∈ Cl(G) let K+ := � x∈K x ∈ Z(FG) be the class sum of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We fix a 2-block B of FG with block idempotent 1B = � K∈Cl(G) aKK+ where aK ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The central character of B is defined by λB : Z(FG) → F, K+ �→ �|K|χ(g) χ(1) �∗ where g ∈ K, χ ∈ Irr(B) and ∗ denotes the canonical reduction O → F (see [22, Chapter 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since λB(1B) = 1, there exists K ∈ Cl(G) such that aK ̸= 0 ̸= λB(K+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We call K a defect class of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [22, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='8], K consists of elements of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' According to [22, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='5], a Sylow 2-subgroup D of CG(x) where x ∈ K is a defect group of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For x ∈ K let CG(x)∗ := {g ∈ G : gxg−1 = x±1} ≤ G be the extended centralizer of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 6 Proposition 6 (Gow, Murray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Every real 2-block B has a real defect class K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Choose a Sylow 2-subgroup E of CG(x)∗ and put D := E ∩ CG(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the G-conjugacy class of the pair (D, E) does not depend on the choice of K or x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For the principal block (which is always real since it contains the trivial character), K = {1} is a real defect class and E = D is a Sylow 2-subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, the uniqueness follows from Sylow’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now suppose that B is non-principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The existence of K was first shown in [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let L be another real defect class of B and choose y ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [9, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='2], we may assume after conjugation that E is also a Sylow 2-subgroup of CG(y)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Dx := E ∩ CG(x) and Dy := E ∩ CG(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We may assume that |E : Dx| = 2 = |E : Dy| (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' the remark after the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We now introduce some notation in order to apply [17, Proposition 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Σ = ⟨σ⟩ ∼= C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We consider FG as an F[G × Σ]-module where G acts by conjugation and gσ = g−1 for g ∈ G (observe that these actions indeed commute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For H ≤ G × Σ let TrG×Σ H : (FG)H → (FG)G×Σ, α �→ � x∈R αx be the relative trace with respect to H, where R denotes a set of representatives of the right cosets of H in G × Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [17, Proposition 14], we have 1B ∈ TrG×Σ Ex (FG) where Ex := Dx⟨exσ⟩ for some ex ∈ E \\ Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By the same result we also obtain that Dy⟨eyσ⟩ with ey ∈ E \\ Dy is G-conjugate to Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This implies that Dy is conjugate to Dx inside NG(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, (Dx, E) and (Dy, E) are G-conjugate as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In the situation of Proposition 6 we call E an extended defect group and (D, E) a defect pair of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We stress that real 2-blocks can have non-real defect classes and non-real blocks can have real defect classes (see [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It is easy to show that non-principal real 2-blocks cannot have maximal defect (see [22, Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, the trivial class cannot be a defect class and consequently, |E : D| = 2 in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For non-real blocks we define the extended defect group by E := D for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Every given pair of 2-groups D ≤ E with |E : D| = 2 occurs as a defect pair of a real (nilpotent) block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' To see this, let Q ∼= C3 and G = Q ⋊ E with CE(Q) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then G has a unique non-principal block with defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We recall from [14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 49] that � χ∈Irr(G) ǫ(χ)χ(g) = |{x ∈ G : x2 = g}| (3) for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The following proposition provides some interesting properties of defect pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proposition 8 (Gow, Murray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real 2-block with defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let bD be a Brauer correspondent of B in DCG(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the following holds: (i) NG(D, bD)∗ = NG(D, bD)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, bD is real if and only if E = DCE(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (ii) For u ∈ D, we have � χ∈Irr(B) ǫ(χ)χ(u) ≥ 0 with strict inequality if and only if u is G-conjugate to e2 for some e ∈ E \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, E splits over D if and only if � χ∈Irr(B) ǫ(χ)χ(1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 7 (iii) E/D′ splits over D/D′ if and only if all height zero characters in Irr(B) have non-negative F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (i) See [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='8] and [18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (ii) See [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (iii) See [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The next proposition extends [18, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Suppose that B is a 2-block with defect pair (D, E) where D is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then E splits over D if and only if all characters in Irr(B) have non-negative F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If B is non-real, then E = D splits over D and all characters in Irr(B) have F-S indicator 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, let B = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Kessar–Malle [15], all characters in Irr(B) have height 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, the claim follows from Proposition 8(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent 2-block with defect pair (D, E) where D is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If E splits over D, then all real characters in Irr(B) have F-S indicator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Otherwise exactly half of the real characters have F-S indicator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In either case, Conjecture B holds for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If E splits over D, then all real characters in Irr(B) have F-S indicator 1 by Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Otherwise we have � χ∈Irr(B) ǫ(χ) = 0 by Proposition 8(ii), because all characters in Irr(B) have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, exactly half of the real characters have F-S indicator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Using Theorem A we can determine the number of characters for each F-S indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For the last claim, we may therefore replace B by the unique non-principal block of G = Q ⋊ E where Q ∼= C3 and CE(Q) = D (mentioned above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In this case Conjecture B follows from Gow [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='2] or Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real block with defect group D ∼= C4 × C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then B is nilpotent since Aut(D) is a 2-group and D is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Moreover |Irr(B)| = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The F-S indicators depend not only on E, but also on the way D embeds into E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The following cases can occur (here M16 denotes the modular group and [16, 3] refers to the small group library): F-S indicators E + + + + + + ++ D8 × C2 + + + + − − −− Q8 × C2, C4 ⋊ C4 with Φ(D) = E′ + + + + 0 0 0 0 D, D × C2, D8 ∗ C4, [16, 3] + + − − 0 0 0 0 C2 4, C8 × C2, M16, C4 ⋊ C4 with Φ(D) ̸= E′ The F-S indicator ǫ(Φ) appearing in Conjecture C has an interesting interpretation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Ω := {g ∈ G : g2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The conjugation action of G on Ω turns FΩ into an FG-module, called the involution module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 8 Lemma 12 (Murray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real 2-block and ϕ ∈ IBr(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then ǫ(Φϕ) is the multiplicity of ϕ as a constituent of the Brauer character of FΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' See [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Next we develop a local version of Conjecture C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real 2-block with defect pair (D, E) and B- subsection (u, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If E = DCE(u), then b is real and (CD(u), CE(u)) is a defect pair of b by [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='6] applied to the subpair (⟨u⟩, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conversely, if b is real, we may assume that (CD(u), CE(u)) is a defect pair of b by [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If b is non-real, we may assume that (CD(u), CD(u)) = (CD(u), CE(u)) is a defect pair of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be 2-block of a finite group G with defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Suppose that Conjecture C holds for all 2-blocks of sections of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let (u, b) be a B-subsection with defect pair (CD(u), CE(u)) such that IBr(b) = {ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then � χ∈Irr(B) ǫ(χ)du χϕ = � |{x ∈ D : x2 = u}| if B is the principal block, |{x ∈ E \\ D : x2 = u}| otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If B is not real, then B is non-principal and E = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It follows that ǫ(χ) = 0 for all χ ∈ Irr(B) and |{x ∈ E \\ D : x2 = u}| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, we may assume that B is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Lemma 2, we have � χ∈Irr(B) ǫ(χ)du χϕ = � ψ∈Irr(b) ǫ(ψ)du ψϕ = 1 ϕ(1) � ψ∈Irr(b) ǫ(ψ)ψ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (4) Suppose that B is the principal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then b is the principal block of CG(u) by Brauer’s third main theorem (see [22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The hypothesis l(b) = 1 implies that ϕ = 1CG(u) and CG(u) has a normal 2-complement N (see [22, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It follows that Irr(b) = Irr(CG(u)/N) = Irr(CD(u)) and � ψ∈Irr(b) ǫ(ψ)du ψϕ = � λ∈Irr(CD(u)) ǫ(λ)λ(u) = |{x ∈ CD(u) : x2 = u}| by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since every x ∈ D with x2 = u lies in CD(u), we are done in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now let B be a non-principal real 2-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If b is not real, then (4) shows that � χ∈Irr(B) ǫ(χ)du χϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' On the other hand, we have CE(u) = CD(u) ≤ D and |{x ∈ E \\ D : x2 = u}| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, we may assume that b is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since every x ∈ E with x2 = u lies in CE(u), we may assume that u ∈ Z(G) by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then χ(u) = du χϕϕ(1) for all χ ∈ Irr(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' If u2 /∈ Ker(χ), then χ(u) /∈ R and ǫ(χ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Thus, it suffices to sum over χ with du χϕ = ±dχϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Z := ⟨u⟩ ≤ Z(G) and G := G/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let ˆB be the unique (real) block of 9 G dominated by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='7], (D, E) is a defect pair for ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then, using [14, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='7] and Conjecture C for B and ˆB, we obtain � χ∈Irr(B) ǫ(χ)du χϕ = � χ∈Irr(B) ǫ(χ)(dχϕ + du χϕ) − � χ∈Irr(B) ǫ(χ)dχϕ = 2 � χ∈Irr( ˆB) ǫ(χ)dχϕ − � χ∈Irr(B) ǫ(χ)dχϕ = 2|{x ∈ E \\ D : x2 = 1}| − |{x ∈ E \\ D : x2 = 1}| = � λ∈Irr(E) ǫ(λ)(λ(1) + λ(u)) − � λ∈Irr(D) ǫ(λ)(λ(1) + λ(u)) − � λ∈Irr(E) ǫ(λ)λ(1) + � λ∈Irr(D) ǫ(λ)λ(1) = � λ∈Irr(E) ǫ(λ)λ(u) − � λ∈Irr(D) ǫ(λ)λ(u) = |{x ∈ E \\ D : x2 = u}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 4 Theorems D and E The following result implies Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Suppose that B is a real, nilpotent, non-principal 2-block fulfilling the statement of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then Conjecture B holds for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let (D, E) be defect pair of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Gow [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1], there exists a 2-rational character χ0 ∈ Irr(B) of height 0 and ǫ(χ0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let Γ : Irr(D) → Irr(B), λ �→ λ ∗ χ0 be the Broué–Puig bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let (u1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , (uk, bk) be representatives for the conjugacy classes of B- subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since B is nilpotent, we may assume that u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , uk ∈ D represent the conjugacy classes of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let IBr(bi) = {ϕi} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since χ0 is 2-rational, we have σi := du χ0,ϕi ∈ {±1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, the generalized decomposition matrix of B has the form Q = (λ(ui)σi : λ ∈ Irr(D), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , k) (see [16, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let v := (ǫ(Γ(λ)) : λ ∈ Irr(D)) and w := (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , wk) where wi := |{x ∈ E \\ D : x2 = ui}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then Theorem 13 reads as vQ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let di := |CD(ui)| and d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then the second orthogonality relation yields QtQ = diag(d) where Qt denotes the transpose of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It follows that Q−1 = diag(d)−1Q t and v = w diag(d)−1Q t = w diag(d)−1Qt, 10 because v = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since wi = |{x ∈ E \\ D : x2 = uy i }| for every y ∈ D, we obtain �k i=1 wi|D : CD(ui)| = |E \\ D| = |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, 1 = ǫ(χ0) = k � i=1 wiσi |CD(ui)| ≤ k � i=1 wi|σi| |CD(ui)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Therefore, σi = 1 or wi = 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' This means that the signs σi have no impact on the solution of the linear system xQ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, we may assume that Q = (λ(ui)) is just the character table of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since Q has full rank, v is the only solution of xQ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Setting µ(λ) := 1 |D| � e∈E\\D λ(e2), it suffices to show that (µ(λ) : λ ∈ Irr(D)) is another solution of xQ = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Indeed, � λ∈Irr(D) λ(ui) |D| � e∈E\\D λ(e2) = 1 |D| � e∈E\\D � λ∈Irr(D) λ(ui)λ(e2) = 1 |D| � e∈E\\D e2=u−1 i |D : CD(ui)||CD(ui)| = wi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let B be a real, nilpotent, non-principal 2-block of a solvable group G with defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We first prove Conjecture C for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since all sections of G are solvable and all blocks dominated by B- subsections are nilpotent, Conjecture C holds for those blocks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, the hypothesis of Theorem 13 is fulfilled for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now by Theorem 14, Conjecture B holds for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let N := O2′(G) and let θ ∈ Irr(N) such that the block {θ} is covered by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since B is non-principal, θ ̸= 1N and therefore θ ̸= θ as N has odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since B also lies over θ, it follow that Gθ < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let b be the Fong–Reynolds correspondent of B in the extended stabilizer G∗ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [22, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='14] and [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 94], the Clifford correspondence Irr(b) → Irr(B), ψ �→ ψG preserves decomposition numbers and F-S indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Thus, we need to show that b has defect pair (D, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let β be the Fong–Reynolds correspondent of B in Gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [22, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='20], β is the unique block over θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In particular, the block idempotents 1β = 1θ are the same (we identify θ with the block {θ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since b is also the unique block of G∗ θ over θ, we have 1b = 1θ + 1θ = � x∈N αxx for some αx ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let S be a set of representatives for the cosets G/G∗ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then 1B = � s∈S (1θ + 1θ)s = � s∈S 1s b = � g∈N �� s∈S αgs−1 � g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, there exists a real defect class K of B such that αgs−1 ̸= 0 for some g ∈ K and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Of course we can assume that g = gs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then 1b does not vanish on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [22, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1], the central characters λB, λb and λθ agree on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It follows that K is also a real defect class of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Hence, we may assume that (D, E) is a defect pair of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' It remains to consider G = G∗ θ and B = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then D is a Sylow 2-subgroup of Gθ by [22, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='20] and E is a Sylow 2-subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since |G : Gθ| = 2, it follows that Gθ ⊴ G and N = O2′(Gθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By 11 [21, Lemma 1 and 2], β is nilpotent and Gθ is 2-nilpotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Gθ = N ⋊ D and G = N ⋊ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let �Φ := � χ∈Irr(B) χ(1)χ = ϕ(1)Φ where IBr(B) = {ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' We need to show that ǫ(�Φ) = ϕ(1)|{x ∈ E \\ D : x2 = 1}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Note that χN = χ(1) 2θ(1)(θ + θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Frobenius reciprocity, it follows that �Φ = 2θ(1)θG and �ΦN = |G : N|θ(1)(θ + θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since Φ vanishes on elements of even order, �Φ vanishes outside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Since �ΦGθ is a sum of non-real characters in β, we have ǫ(�Φ) = 1 |G| � g∈Gθ �Φ(g2) + 1 |G| � g∈G\\Gθ �Φ(g2) = 1 |G| � g∈G\\Gθ �Φ(g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Every g ∈ G \\ Gθ = NE \\ ND with g2 ∈ N is N-conjugate to a unique element of the form xy where x ∈ E \\ D is an involution and y ∈ CN(x) (Sylow’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Setting ∆ := {x ∈ E \\ D : x2 = 1}, we obtain ǫ(�Φ) = θ(1) |N| � x∈∆ |N : CN(x)| � y∈CN (x) (θ(y) + θ(y)) = 2θ(1) � x∈∆ 1 |CN(x)| � y∈CN(x) θ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (5) For x ∈ ∆ let Hx := N⟨x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Again by Sylow’s theorem, the N-orbit of x is the set of involutions in Hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' From θx = θ we see that θHx is an irreducible character of 2-defect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By [8, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1], we have ǫ(θHx) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Now applying the same argument as before, it follows that 1 = ǫ(θHx) = 1 |N| � g∈Hx\\N θHx(g2) = 2 |CN(x)| � y∈CN(x) θ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Combined with (5), this yields ǫ(�Φ) = 2θ(1)|∆|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' By Green’s theorem (see [22, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='11]), ϕN = θ + θ and ǫ(�Φ) = ϕ(1)|∆| as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For non-principal blocks B of solvable groups with l(B) = 1 it is not true in general that Gθ is 2- nilpotent in the situation of Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' For example, a (non-real) 2-block of a triple cover of A4 × A4 has a unique simple module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Extending this group by an automorphism of order 2, we obtain the group G = SmallGroup(864, 3988), which fulfills the assumptions with D ∼= C4 2, N ∼= C3 and |G : NE| = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' In order to prove Conjecture C for arbitrary 2-blocks of solvable groups, we may follow the steps in the proof above and invoke a result on fully ramified Brauer characters [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The claim then boils down to a purely group-theoretical statement: Let B be a real, non-principal 2-block of a solvable group G with defect pair (D, E) and l(B) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Let G := G/O2′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Then |{x ∈ G \\ Gθ : x2 = 1}| = |{x ∈ E \\ D : x2 = 1}| � |G : EN|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Unfortunately, I am unable to prove this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Acknowledgment I thank Gabriel Navarro for providing some arguments for Theorem E from his paper [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' John Mur- ray and three anonymous referees have made many valuable comments, which improved the quality of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' The work is supported by the German Research Foundation (SA 2864/3-1 and SA 2864/4-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 12 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' An and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Eaton, Nilpotent Blocks of Quasisimple Groups for the Prime Two, Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Theory 16 (2013), 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Brauer, Representations of finite groups, in: Lectures on Modern Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' I, 133–175, Wiley, New York, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Brauer, Some applications of the theory of blocks of characters of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' III, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 3 (1966), 225–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Broué and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Puig, A Frobenius theorem for blocks, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 56 (1980), 117–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Eaton, Generalisations of conjectures of Brauer and Olsson, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (Basel) 81 (2003), 621–626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [6] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 2022, (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='gap-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Gow, Real-valued characters and the Schur index, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 40 (1976), 258–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Gow, Real-valued and 2-rational group characters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 61 (1979), 388–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Gow, Real 2-blocks of characters of finite groups, Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 25 (1988), 135–147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Gow and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray, Real 2-regular classes and 2-blocks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 230 (2000), 455–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Helleloid and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Martin, The automorphism group of a finite p-group is almost always a p-group, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 312 (2007), 294–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Héthelyi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Horváth and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Szabó, Real characters in blocks, Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 49 (2012), 613–623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Ichikawa and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Tachikawa, The Super Frobenius–Schur Indicator and Finite Group Gauge Theories on Pin− Surfaces, to appear in Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=', DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content='1007/s00220-022-04601-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [14] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Kessar and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' (2) 178 (2013), 321–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Linckelmann, The block theory of finite group algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' II, London Mathematical Society Student Texts, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 92, Cambridge University Press, Cambridge, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray, Strongly real 2-blocks and the Frobenius-Schur indicator, Osaka J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 43 (2006), 201– 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray, Components of the involution module in blocks with cyclic or Klein-four defect group, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Group Theory 11 (2008), 43–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray, Real subpairs and Frobenius-Schur indicators of characters in 2-blocks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 322 (2009), 489–513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Murray, Frobenius-Schur indicators of characters in blocks with cyclic defect, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra 533 (2019), 90–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 13 [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Navarro, Nilpotent characters, Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 169 (1995), 343–351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 250, Cambridge University Press, Cambridge, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathemat- ics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 175, Cambridge University Press, Cambridge, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Navarro, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Späth and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Tiep, On fully ramified Brauer characters, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 257 (2014), 248–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Trefethen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Vinroot, A computational approach to the Frobenius-Schur indicators of finite exceptional groups, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Algebra Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 30 (2020), 141–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Willems, Duality and forms in representation theory, in: Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 509–520, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 95, Birkhäuser, Basel, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFQT4oBgHgl3EQf6zaC/content/2301.13440v1.pdf'} diff --git a/.gitattributes b/.gitattributes index b58bdf3ba23a1f1a0c1118bf416735575f1877ae..c39c6b6619f11b91d46e2180888ac2b8aca69331 100644 --- a/.gitattributes +++ b/.gitattributes @@ -6209,3 +6209,88 @@ xtFJT4oBgHgl3EQfhCzP/content/2301.11564v1.pdf filter=lfs diff=lfs merge=lfs -tex 79FAT4oBgHgl3EQfoh2y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 99AzT4oBgHgl3EQfSvs8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf filter=lfs diff=lfs merge=lfs -text +OdFLT4oBgHgl3EQfPC8X/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf filter=lfs diff=lfs merge=lfs -text +ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf filter=lfs diff=lfs merge=lfs -text +q9FPT4oBgHgl3EQf9TVT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +f9AzT4oBgHgl3EQfMfsT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +xtFJT4oBgHgl3EQfhCzP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +I9AyT4oBgHgl3EQfsPkA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +p9FPT4oBgHgl3EQf8DXy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf filter=lfs diff=lfs merge=lfs -text +KNFRT4oBgHgl3EQfDTe1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +gNAyT4oBgHgl3EQfj_gw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +HNFJT4oBgHgl3EQfuC2G/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +LdE4T4oBgHgl3EQfig1d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf filter=lfs diff=lfs merge=lfs -text +aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf filter=lfs diff=lfs merge=lfs -text +gtE0T4oBgHgl3EQf6QKw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +PdE0T4oBgHgl3EQfTwBe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +uNE0T4oBgHgl3EQf9gI3/content/2301.02801v1.pdf filter=lfs diff=lfs merge=lfs -text +7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf filter=lfs diff=lfs merge=lfs -text +ktFPT4oBgHgl3EQf2TWb/content/2301.13186v1.pdf filter=lfs diff=lfs merge=lfs -text +sNE2T4oBgHgl3EQffAfS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf filter=lfs diff=lfs merge=lfs -text +7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf filter=lfs diff=lfs merge=lfs -text +eNAyT4oBgHgl3EQfwvnx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +r9FJT4oBgHgl3EQfbyya/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ptE3T4oBgHgl3EQf8As1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +SNFGT4oBgHgl3EQfgygC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf filter=lfs diff=lfs merge=lfs -text +eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf filter=lfs diff=lfs merge=lfs -text +kb_47/content/kb_47.pdf filter=lfs diff=lfs merge=lfs -text +b9E4T4oBgHgl3EQfPgyg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf filter=lfs diff=lfs merge=lfs -text +c9FPT4oBgHgl3EQfyDV1/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +cNE5T4oBgHgl3EQfEg7V/content/2301.05415v1.pdf filter=lfs diff=lfs merge=lfs -text +jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf filter=lfs diff=lfs merge=lfs -text +btE1T4oBgHgl3EQfKwNx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf filter=lfs diff=lfs merge=lfs -text +-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf filter=lfs diff=lfs merge=lfs -text +kb_47/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf filter=lfs diff=lfs merge=lfs -text +u9AzT4oBgHgl3EQfP_sX/content/2301.01191v1.pdf filter=lfs diff=lfs merge=lfs -text +09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf filter=lfs diff=lfs merge=lfs -text +K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf filter=lfs diff=lfs merge=lfs -text +3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf filter=lfs diff=lfs merge=lfs -text +-dAyT4oBgHgl3EQf3fmN/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +PNFRT4oBgHgl3EQfIjco/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf filter=lfs diff=lfs merge=lfs -text +4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf filter=lfs diff=lfs merge=lfs -text +GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf filter=lfs diff=lfs merge=lfs -text +bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf filter=lfs diff=lfs merge=lfs -text +Z9E2T4oBgHgl3EQfEwaS/content/2301.03639v1.pdf filter=lfs diff=lfs merge=lfs -text +3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf filter=lfs diff=lfs merge=lfs -text +9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf filter=lfs diff=lfs merge=lfs -text +btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf filter=lfs diff=lfs merge=lfs -text +gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf filter=lfs diff=lfs merge=lfs -text +hdE4T4oBgHgl3EQfSAzu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +YdA0T4oBgHgl3EQfFv8F/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf filter=lfs diff=lfs merge=lfs -text +o9E3T4oBgHgl3EQfLgkW/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf filter=lfs diff=lfs merge=lfs -text +stFJT4oBgHgl3EQfcixn/content/2301.11544v1.pdf filter=lfs diff=lfs merge=lfs -text +7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf filter=lfs diff=lfs merge=lfs -text +_tE1T4oBgHgl3EQfDAJg/content/2301.02871v1.pdf filter=lfs diff=lfs merge=lfs -text +LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf filter=lfs diff=lfs merge=lfs -text +stFJT4oBgHgl3EQfcixn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf filter=lfs diff=lfs merge=lfs -text +K9E2T4oBgHgl3EQfAwYv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf filter=lfs diff=lfs merge=lfs -text +bNE2T4oBgHgl3EQfwAiL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf filter=lfs diff=lfs merge=lfs -text +PNE4T4oBgHgl3EQf-A73/content/2301.05361v1.pdf filter=lfs diff=lfs merge=lfs -text +3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +_tE1T4oBgHgl3EQfDAJg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +q9E1T4oBgHgl3EQf2wUi/content/2301.03481v1.pdf filter=lfs diff=lfs merge=lfs -text +TNE2T4oBgHgl3EQfWwfK/content/2301.03838v1.pdf filter=lfs diff=lfs merge=lfs -text +9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf b/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..966e017387f662b53fa13b50338173f9daa90cab --- /dev/null +++ b/09FAT4oBgHgl3EQfjh3M/content/2301.08606v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b695af2616285d98273841018a18aa78094c4b40106abf6bf4f7a3f24cd300d4 +size 943062 diff --git a/1tAyT4oBgHgl3EQfofi7/content/tmp_files/2301.00509v1.pdf.txt b/1tAyT4oBgHgl3EQfofi7/content/tmp_files/2301.00509v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9aac399a800091f7a9dbde6d72cd4abce26b06a4 --- /dev/null +++ b/1tAyT4oBgHgl3EQfofi7/content/tmp_files/2301.00509v1.pdf.txt @@ -0,0 +1,2567 @@ +Time-Varying Coefficient DAR Model and +Stability Measures for Stablecoin Prices: An +Application to Tether +Antoine Djogbenou,∗ Emre Inan,† Joann Jasiak‡ +This Version: January 3, 2023 +Abstract +This paper examines the dynamics of Tether, the stablecoin with the largest +market capitalization. We show that the distributional and dynamic proper- +ties of Tether/USD rates have been evolving from 2017 to 2021. We use local +analysis methods to detect and describe the local patterns, such as short-lived +trends, time-varying volatility and persistence. To accommodate these pat- +terns, we consider a time varying parameter Double Autoregressive tvDAR(1) +model under the assumption of local stationarity of Tether/USD rates. We es- +timate the tvDAR model non-parametrically and test hypotheses on the func- +tional parameters. In the application to Tether, the model provides a good fit +and reliable out-of-sample forecasts at short horizons, while being robust to +time-varying persistence and volatility. In addition, the model yields a simple +plug-in measure of stability for Tether and other stablecoins for assessing and +comparing their stability. +Keywords: +Stablecoins, Tether, Prices, DAR Model, Persistence, Time- +Varying Parameters, Conditional Heteroskedasticity, Local Stationarity. +JEL number: C58, C13. +∗York University, Canada, e-mail: daa@yorku.ca +†York University, Canada, e-mail: emreynan@yorku.ca +‡York University, Canada, e-mail: jasiakj@yorku.ca. +The authors thank C. Gourieroux and H. Kim and the participants of CMStatistics 2022 and Canadian +Economic Association (CEA) 2022 meetings for helpful comments. This project was supported by the +Digital Currency Research Clusters Initiative, the Natural Sciences and Engineering Research Council of +Canada (NSERC), and the Social Sciences and Humanities Research Council of Canada (SSHRC). +arXiv:2301.00509v1 [econ.EM] 2 Jan 2023 + +THIS VERSION: January 3, 2023 +1 +1 +Introduction +The total market capitalization of cryptocurrencies is currently over 1 trillion U.S. dollar, +with the top three cryptocurrencies in terms of market capitalization being Bitcoin (BTC), +Ethereum (ETH), and Tether (USDT). While Bitcoin and Euthereum are characterized +by high price volatility, Tether is a stablecoin, i.e. a cryptocurrency designed to maintain +a stable price compared to other cryptocurrencies such as Bitcoin and Ethereum. It is the +first and by far the largest stablecoin in the market with the highest daily volume of over +$100 billion. In order to achieve price stability, the value of Tether is pegged 1-to-1 with +the U.S. dollar. There also exist other stablecoins with values to other currency or gold +and managed by either a single authority (usually the service provider) or a network of +participants (the whole protocol). +Allen, Gu, and Jagtiani (2022) recently discussed how stable cryptocurrencies provide +alternative financial instruments for market participants and how appropriately regulated +crypto markets could allow increased public confidence and lead to growth and innova- +tion. +The November 2021 report by the US President’s Working Group on Financial +Markets (PWG), the Federal Deposit Insurance Corporation (FDIC), and the Office of +the Comptroller of the Currency (OCC) highlight various risks that need to be addressed. +These include user protection and run risk, payment system risk, systemic risk and con- +centration of economic power. +They provided various recommendations, including the +requirement for stablecoin issuers to be insured depository institutions. See President’s +Working Group (2021) for more details. Furthermore, Li and Mayer (2021) show that +collateralized stablecoins like Tether could create systemic risk if the issuer does not have +enough reserve to maintain its stability. More recently, Chen, Qin, and Zhang (2022, page +5) pointed out the important role of Tether in the trading volume of Bitcoin compared +to US dollars since 2017 and noted the limited reserve of Tether according to anecdotal +evidence. +Despite the increased interest in stablecoins and the recommendations of more scrutiny +by regulators, these crypto assets’ stability is still ineffective. For example, TerraUSD, +an algorithmic stablecoin, collapsed in May 2022. +This situation posits the need for +predictability of stablecoin prices and easy tools for proactive assessment of stability. +To address those issues, this paper made the following contributions. First, we analyze + +THIS VERSION: January 3, 2023 +2 +the local Tether price from historical data and pin down important features in its dynamic. +These features include the local pattern of the mean and the conditional pattern of Tether +price as well as the role of specific events in this dynamic. Second, we develop a time- +varying model for Tether price that incorporates these specificities. Third, we propose, +based on the model, measures that can be used to assess the stability of stablecoins and +mitigate risks. +More specifically, we examine the dynamics of Tether/USD rates and documents the +time varying distributional properties of this series. We apply local analysis methods to +reveal the time varying mean, volatility and persistence. In particular, we observe periods +when Tether rates deviate from the peg, which are often combined with increased volatility. +During those episodes, local persistence measures increase, suggesting unit root dynamics +of Tether. +Based on these findings, we consider an extension of the Double-Autoregressive (DAR) +model, called the dynamic time-varying parameter Double-Autoregressive (tvDAR). The +DAR model [Ling (2004)] accommodates the conditional heteroscedasticity and nests the +ARCH and the autoregressive of order one AR(1) models, including the unit root model +with the autoregressive coefficient equal to 1. More specifically, the DAR is a nonlinear +Markov 1 process, which becomes a stationary martingale when the autoregressive coef- +ficient is equal to 1 [Gourieroux, Jasiak (2019)]. The DAR model, unlike the traditional +autoregressive AR(1)-ARCH process, provides valid inference and consistent parameter +estimators for the autoregressive coefficient values including 1. The proposed extension to +a deterministic time-varying parameters model relies on the assumption of local strict sta- +tionarity of the process, following the approach of Dahlhaus (2000) and Dahlhaus, Richter, +Wu (2019). Then, during the episodes of unit root dynamics, the process satisfies locally +the stationary martingale condition. The time varying tvDAR model provides a good fit +to the Tether/USD rates and gives reliable one step ahead out-of-sample predictions. To +obtain the empirical results, we employed a rectangular kernel and an Epanechnikov ker- +nel. The first is an asymmetric kernel, which permits the incorporation of past information +in a pre-specified window and could be used for out-of-sample prediction. The second is a +symmetric kernel that uses information around any time period and is more suitable for +inference on the parameters in the model. +Moreover, the tvDAR model provides a simple plug-in measure of stability for sta- + +THIS VERSION: January 3, 2023 +3 +blecoins, based on the Lyapunov exponent. This measure is commonly used to assess +the stability of deterministic dynamical systems and to test for chaos [see, e.g., Sprott +(2014)]. The Lyapunov exponent for the AR(1)-ARCH model has been determined by +Borkovec and Kluppenberg (2001), and shown to be the condition of strict stationarity +of that process [see also Borkovec (2000)]. It has been also considered by Nelson (1990) +in the context of the IGARCH model and by Cline and Pu (2004) in a non-parametric +framework. The Lyapunov exponent was also used as a stability measure in application +to the Vector Autoregressive VAR model by Dechert and Gencay (1992). Those authors +have introduced an alternative stability measure based on the noise-to-signal ratio for +linear dynamic models [see also LeBaron (1994) for introduction to chaos]. +In this paper, the sample Lyapunov exponent is computed from the model parameter +estimates and proposed as a measure of stability for stablecoins. A more conservative +measure, based on the condition of second-order stationarity is also introduced. +Both +measures can be computed locally and used to assess the stability of a stablecoin over +time, or to compare the stability of different stablecoins. +The time-varying coefficient approach based on the assumption of local stationarity dis- +tinguishes our approach from the literature that relies on the assumption of global strong +stationarity of the series. For instance, Baum¨ohl and Vyrost (2022) use high frequency +data to compute a spectral density-based quantile dependence measure under a strict +stationarity condition, which does not seem to be satisfied by Tether. Bianchi, Rossini, +and Iacopini (2022) estimate a Bayesian VAR with stochastic volatility and Student-t dis- +tributed shocks (BVAR-SV-t). However, the conditional volatility equation is constrained +to unit root dynamics, which is inconsistent with the empirical evidence provided in this +paper. +The paper is organized as follows. Section 2 describes the stablecoins. Section 3 dis- +cusses the local dynamic analysis of the price of Tether. Section 4 discusses the modelling +approach, estimation procedures and stability measures. Section 5 presents the empiri- +cal results based on the estimation and inference on the DAR(1) and tvDAR(1) models, +including the sample stability measures. Section 6 concludes. Appendix A contains the +technical results. Simulation and additional empirical results on stability measures are +relegated to Appendices B and C, respectively. + +THIS VERSION: January 3, 2023 +4 +2 +Stablecoins +This section defines stablecoins and discusses their classification, issuance, and redemption +mechanisms. In addition, we discuss how the market prices of stablecoins are determined. +2.1 +Definition and Classification of Stablecoins +Stablecoins are a type of cryptocurrency designed to maintain a stable price and reduced +volatility, compared to other cryptocurrencies such as Bitcoin and Ethereum. Conven- +tionally, stablecoin companies peg the value of their coins to that of a physical asset such +as a fiat currency or gold with the assumption that the market price of their coins will +eventually stabilize, establishing equivalency with the reference asset. The strategies used +to achieve price stability of stablecoins are discussed below. +There is currently no standard in place that private enterprises should comply with +to qualify as a legitimate stablecoin company. This leaves stablecoin enterprises with un- +limited design options to choose from to differentiate their business models. Currently, +business models of stablecoin companies differ in their economic design, the quality of +backing they maintain, stability assumptions they rely on, and legal protection they pro- +vide for coin holders (Catalini and de Gortari, 2021). +While the underlining business models may be diverse and complex, there is interest +in the elements of such models to understand their economic implications. For example, +one element of interest is the mechanism stabelcoins rely on to stabilize price and another +is how the responsibilities are distributed over stablecoin protocols. +There exist two alternative mechanisms used by stablecoin companies to achieve price +stability. They either hold collaterals in their reserves to back the value of their coins +or they adjust the supply of coins through software codes to restore the peg with the +reference asset. When the market value of a cryptocurrency is backed by collaterals, the +cryptocurrency is referred to as a collateralized stablecoin. Conventionally, collateralized +stablecoins are split into two sub-categories including off-chain collateralized stablecoins +and on-chain collateralized stablecoins. Off-chain collateralized stablecoins are backed by +a set of collaterals that have an economic value outside of the blockchain. The reserves of +this type of stablecoins usually consist of a fiat currency such as the US dollar for Tether +(USDT) or a commodity such as gold for PAX Gold (PAXG). Stablecoins are labelled as + +THIS VERSION: January 3, 2023 +5 +on-chain collateralized if the underlying collaterals are composed of crypto assets that are +created in a digital form and recorded on a distributed ledger. For instance, Dai (DAI), +the largest on-chain stablecoin project, supports 18 collateral assets including not only +cryptocurriencies such as Ethereum (ETH) and Chainlink (LINK) but also stablecoins +such as Tether (USDT), USD Coin (USDC), TrueUSD (TUSD) and PAX dollar (USDP). +Some projects opt for developing software codes to minimize price fluctuations instead +of collateralizing their coins. This type of cryptocurrencies is called an algorithmic stable- +coin as they try to stabilize their price around the peg by contracting or expanding the +coin supply with the help of computer algorithms embedded in their design. TerraUSD +(UST) was until May 2022 the only example of an algortihmic stablecoin that has a market +capitalization over a billion US dollar. +In terms of distribution of responsibilities, stablecoins can be categorized as centralized +or decentralized. Centralized stablecoins rely on a single legal entity to maintain the price +stability, to manage and protect the collaterals, and to fulfill its obligations to users. For +instance, Tether Limited is the legal entity that has the authority as well as the respon- +sibility over every Tether in the circulation. Unlike centralized stablecoins, decentralized +stablecoins distribute these responsibilities within their network through smart contracts. +This allows network participants to take an active role in determining the rules of the +stablecoin protocol such as the set of eligible collaterals and the minimum collateral re- +quirements. The decentralized stablecoin DAI grants users who hold its governance token +Maker (MKR) the right to vote on the changes to its protocol. +Li and Mayer (2021) noted that the introduction of stablecoins is comparable to “the +unregulated creation of safe assets to meet agents’ transactional demands” known as +shadow banking. Unlike stablecoin issuers, shadow banks must play the role of credit +guarantees in the case of insolvency. However, as we will see later, the observed prices of +stablecoins tend to deviate from the peg. In addition, these crypto assets face multiple +risks including the risk of liquidation. +For stablecoins designed to be on par with a fiat currency, the use of reserve allows +the stablecoins issuers to sell or buy the currency to achieve its price stability. +This +mechanism helps stablecoin companies to underpin the market value of their coins and +protect against the highly volatile nature of the cryptocurrency markets. It resembles fixed +exchange rate regimes currently implemented in Panama, Qatar, and Saudi Arabia. In the + +THIS VERSION: January 3, 2023 +6 +fixed exchange rate regime, the central bank also uses its foreign reserves to buy or sell its +domestic currency to maintain the fixed parity with the currency peg. When the reserve +system fails, the domestic currency can be devaluated. +Lyons and Viswanath-Natraj +(2020) documented that contrary to central banks with some macroeconomic mandate, +including keeping inflation around its target, stablecoin issuers do not have any policy +functions. In addition, stablecoin companies cannot use the interest rate or devaluation +policy to control the exchange rate. For our analysis, we focus on Tether, which is by +far the primarily traded stablecoin in terms of market capitalization. As we will show +later, Tether price tends to be noticeably affected by events in the crypto world, leading +to deviations from the peg despite using reserves. +Given the aforementioned significant risks for stablecoin holders, there is a need to +develop appropriate tools to assess their predictability and stability. This paper develops +a model based on the properties of Tether price and uses it to propose tests for its sta- +bility. Before discussing the specificity of the cryptocurrency of interest and the modeling +strategy, we provide further explanation on the issuance and redemption of stablecoins. +2.2 +Issuance and Redemption +Issuance and redemption are the two fundamental market activities that determine the +equilibrium quantity of a stablecoin in the market. +The equilibrium quantity goes up +when new coins are issued, and it goes down when existing coins are redeemed. While the +equilibrium quantity changes with issuance and redemption, the price of a stablecoin is +held constant during these transactions by the service provider. This constant price policy +is the result of the pegging strategy explained in the previous section. +Issuance and redemption of stablecoins are presumably initiated by users.1 How these +transactions are executed depends on whether stablecoin has a centralized or decentralized +structure. Issuance takes place following the transfer of funds by user to the stablecoin +enterprise. Depending on whether stablecoin is centralized or decentralized, these funds +are deposited either into banking accounts of a custodian or into a cryptographical vault. +For example, an individual or a business who wants to buy Tether should transfer the +funds, specifically the US dollar, to Tether Limited’s accounts at Cathay Bank and Hwatai +1See Griffin and Shams (2020) for further discussion on whether Tether issuances are supply-driven or +demand-driven. + +THIS VERSION: January 3, 2023 +7 +Bank in Taiwan. The collection of these funds constitutes the reserves of the stablecoin +enterprise, and they are meant to be kept as a collateral to back the value of every coin in +the circulation. Once the funds are successfully deposited, stablecoins are issued through +smart contracts and credited into the user’s wallet. For centralized stablecoins, it is the +issuer or the agent that authorizes the issuance of the coins whereas it is done automatically +by the blockchain technology for decentralized stablecoins. +Redemption of stablecoins is also initiated by user but the difference is that the trans- +actions take place in the reserve order. To redeem stablecoins, users place an order on the +blockchain to exchange their stablecoins for the collateral. Upon the order, the stablecoin +enterprise becomes obliged to withdraw the stablecoins from circulation and give the user +the corresponding amount of collateral in return. The stablecoins that the user redeems +are destroyed subsequently from the protocol. Stablecoin projects pledge in their whitepa- +pers that their coins are 100% redeemable and redemptions can be performed any time +users want at the predetermined price. Hence, one can argue that redeemability becomes +the liability of stablecoin enterprises and plays a key role in the sustainability of their +projects. It is the issuer that is liable to users for redemptions in centralized stablecoins. +Decentralized stablecoins, on the other hand, have no single legal entity that shoulders the +responsibility. It is the whole network that is responsible for undertaking redemptions. +2.3 +Market Price of Stablecoins +Stablecoin prices are usually fluctuating around the target value. While their value in +terms of the reference asset is fixed during issuance and redemptions, they are often +traded at a premium or a discount on the exchange platforms.2 This splits the market +for stablecoins between the primary market and the secondary market, which can be +considered analogous to the market for traditional securities. The primary market for +stablecoins is where stablecoins are created (issuance) or destroyed (redemption) at the +fixed exchange rate predetermined by the stablecoin initiative. +The secondary market +is where the users trade stablecoins within and across cryptocurrency exchanges such +as Binance, Coinbase Exchange, Kraken and Bitfinex. The price of stablecoins in the +secondary market is determined as a result of the market activities. According to Griffin +2See Lyons and Viswanath-Natraj (2020) for the detailed analysis of premium and discount on stablecoin +prices. + +THIS VERSION: January 3, 2023 +8 +and Shams (2020), the secondary market activities account for most of the aggregate +Tether flow from 2014 to 2018. Hence, the price of Tether and other stablecoins in the +secondary market can be considered as the effective rate at which individuals or businesses +can buy and sell stablecoins on a day-to-day basis. +At first glance, the design elements stands out as the primary mechanism through which +stablecoin projects try to achieve the price stability. However, the price stabilization in +the stablecoin market could be multifaceted. For instance, Lyons and Viswanath-Natraj +(2020) argue that the price gap between the primary market and the secondary market, +which corresponds to the deviation from the peg, can be mitigated also by arbitrage +activities. As long as investors have access to the primary market, the price deviations in +the secondary market creates an opportunity for them to make profit. When the price of +a stablecoin in the secondary market is above the peg, the arbitrager can buy the coin at +the target exchange rate from the primary market and sell it in the secondary market to +make profit. This increases the supply of the stablecoin in the secondary market, so it puts +downward pressure on its price. Similarly, when the price of a stablecoin in the secondary +market is below the parity, an arbitrager can buy the coin from the secondary market +and redeem it at the peg ratio in the primary market. In this case, the demand from the +arbitrager puts upward pressure on the secondary market price. Hence, one could expect +that the price of stablecoins across cryptocurrency exchanges would stabilize around the +peg through arbitrage activities. For example, introduction of Tether to the Ethereum +blockchain in April 2019, which is associated with increased direct access of investors to +the primary market, is found to have a stabilizing effect on the price of Tether in the +secondary market (Lyons and Viswanath-Natraj, 2020). +Bullman et al. (2019) provides the list of alternative tools that each type of stablecoins +can adopt to maintain the peg. The list consists of fees, redemption limits, and penalty +fees to name a few. Fees and redemption limits could be used by collateralized stablecoins +to limit the users’ transactions and prevent sudden liquidations while penalty fees can help +maintain the minimum level of collateralization. For example, Tether Limited imposes the +minimum amount of 100,000 USD required for a fiat withdrawal or deposit and charges +the greater of $1,000 or 0.1% fee per fiat withdrawal and per fiat deposit.3 +The stabilization strategies of Tether have not always been fully successful. The next +3https://tether.to/fees/ + +THIS VERSION: January 3, 2023 +9 +section presents empirical evidence based on the dynamic analysis of the data. +3 +Tether Dynamics +This section examines the patterns in Tether dynamics in the sample of T = 1361 daily +closing prices recorded between November 9, 2017, and July 31, 2021. +Figure 1 displays the evolution of daily closing rates of the Tether against US Dollar. +We observe the episodes of explosive dynamics mixed with more stable periods as well as +the convergence of the process at the end of the trajectory to a smooth process taking +values close to 1. The convergence of Tether towards the peg and its reduced range after +2021 are associated with increased volume displayed in Figure 2. +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0.96 +0.98 +1 +1.02 +1.04 +1.06 +1.08 +Daily Price +Figure 1: Tether/USD daily closing rates +During the sampling period, the lowest and highest price were 0.9666 and 1.0779, +respectively. Although 0.0334 and 0.0779 deviations from the one US dollar parity may +look small, they can provide important arbitrage opportunities if the investor is holding +a large position in the crypto asset. While the mean over this period is 1.022 and close +to one, as expected, the volatility around the mean is 0.066. The evolution of the price +shows an alternate of relatively large and small deviation in the stablecoin price due to +changes in its demand and lags in intervention by Tether to maintain price stability. + +THIS VERSION: January 3, 2023 +10 +The fluctuation of the Tether price around the one-dollar peg can be connected with +the European snake in the tunnel currency system created in April 1972 by an agreement. +To increase the convergence among the different currencies in the European Economic +Community (EEC), the agreement objective was to create a single currency band within +which all the EEC currencies could fluctuate and not deviate too much from a peg. The +peg was defined using first gold and, later on, the US dollar. More details on the system +can be found in Day (1976). To achieve stability around the peg, central banks had to use +their reserve to intervene by buying or selling local currencies. The system was difficult +to sustain as several currencies left the agreement. Although stablecoin issuers do not +have a macroeconomic policy function as central banks, the difficulties in maintaining the +snake currency system also speak to the challenge of maintaining stablecoin prices around +its peg using the reserve system discussed above. To understand the price movements of +Tether, we first discuss factors that affect its demand during the sampling period. +Figure 2: Time varying volume of Tether +The daily volume series in Figure 2 exhibits larger fluctuations during the year 2021 +of the sampling period. Although the overall trend was positive, the daily volume varied + +Volume in million USD (USDT) +2.5 +1.5 +0.5 +Jui 2017 +Jan 2018 + Jul 2018 + Jan 2019 + Jul 2019 + Jan 2020 + Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022THIS VERSION: January 3, 2023 +11 +roughly between $15.4 billion and $279 billion. The highest daily volume of $279 billion +was achieved on May 19, 2021, when the Chinese government cracked down on its domestic +market for cryptocurrencies. Later, the daily volume plunged to as low as $33.7 billion in +July 2021. +The high levels of daily volume in 2020 and 2021 could be explained by an increasing +interest from investors as the market for cryptocurrencies grew substantially during the +initial stages of the Covid-19 pandemic. Tether’s daily volume increased drastically from +less than $10 billion in 2018 to as high as $279 billion in 2021. While the daily volume +is observed to be increasing almost steadily in 2018 and 2019, it exhibits rather a volatile +pattern in 2020 and 2021. For instance, in early 2021, the daily volume of Tether more +than doubled in a matter of a few months and reached a peak of 99.3 billion USD on the +March 13th, a day after the infamous “Crypto Black Thursday”. However, the pattern +in Figure 2 indicates that the daily volume of Tether decreased between May and July of +2021 and returned to its pre-pandemic level. +The convergence to reduced range and small variation around the constant value of 1 +occurs first in Tether in May 2018 and is interrupted by the end of September 2018. In +the environnement of Tether, the convergence is observed simultaneously for other mostly +traded traded stablecoins such as USD Coin, Binance USD, True USD, and Pax Dollar +starting from July 2020 and in Dai starting from December 2020. During this period, +these stablecoins displayed a period of improved stability towards the end of the sampling +period. Also, the Bitcoin and Etheureum prices in US Dollars have increased. This period +of stability overlaps with the period of bullish run in the cryptocurrency market. The +cryptocurrency market indices such as the S&P Cryptocurrency Broad Digital Market +(BDM) Index recorded more than fivefold increase between September 2020 and May +2021.4 The strong demand for cryptocurrencies also benefited the stablecoin companies +as the total market capitalization of the top 10 stablecoins went up from approximately +$20 billion in September 2020 to slightly over $100 billion in July 2021.5 +The stability is interrupted again for all stablecoins when the cryptocurrency market +shrunk by over $300 billion on April 17, 2021 in less than 24 hours.6 While the waves +4https://www.spglobal.com/spdji/en/indices/digital-assets/sp-cryptocurrency-broad-digital-market- +index/#overview +5https://www.statista.com/statistics/1255835/stablecoin-market-capitalization/ +6https://www.forbes.com/sites/jonathanponciano/2021/04/18/crypto-flash-crash-wiped-out-300- + +THIS VERSION: January 3, 2023 +12 +of sell-off caused the price of Bitcoin to plummet by 10.5%, the price of all stablecoins +increased simultaneously. This can be evidence in favor of the previous studies which +suggest that stablecoins could provide hedging opportunities for cryptocurrency investors +against Bitcoin’s volatility, e.g., Wang and Wu (2020). However, one could also argue that +the risk mitigating properties of stablecoins, which are closely linked to the comovements +between the price of stablecoins and that of Bitcoin, could be changing locally. For exam- +ple, stablecoins showed resilience against even a much stronger market crash in mid-May +2021. On May 19, 2021, the Chinese government announced that the banks in China are +banned from providing cryptocurrency services to their clients.7 The market reacted to +this news almost immediately as Bitcoin shed 30% of its value over the course of the day. +During the crash, all stablecoins except for Terra USD managed to keep their price stable +around the one-dollar peg. +3.1 +Local Analysis of Tether Price +This subsection analyzes the local dynamics of Tether price series. We examine its local +means, variances, and autocorrelations. +(a) +Local mean and variance +This section studies the evolution of Tether/USD rates and identifies local patterns in this +series by considering time varying descriptive statistics computed by rolling over a window +of 50 days. +billion-in-less-than-24-hours-spurring-massive-bitcoin-liquidations/?sh=7d60735b2c89 +7https://www.theguardian.com/technology/2021/may/19/bitcoin-falls-30-after-china-crackdown + +THIS VERSION: January 3, 2023 +13 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0.98 +0.99 +1 +1.01 +1.02 +Local Mean +=1 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0 +1 +2 +3 +Local Variance +10 +-4 +Figure 3: Local mean and variance for the price of Tether +The locally estimated marginal mean µt and variance σ2 +t are displayed in panels a) and +b) of Figure 3.8 +The figure’s top panel reports the local mean of the price series and shows its evolution +over the sampling period. We observe that: the local mean of Tether varies across sub- +periods, and it displays local trends. Especially, in the first half of the sampling period, +a strong local trend is observed, which is interrupted by a return of the series to values +close to 1. For example, in August 2019, the local mean increases, which is akin to the +pattern of financial bubbles observed in stock prices (rational stochastic bubbles as in +Blanchard and Watson (1982) or intrinsic bubbles as in Froot and Obstfeld (1991)). See +Kortian (1995) for more details. These patterns, however, disappear towards the end of +the sampling period and the local mean becomes more stable and close to the target value +of 1. Moreover, there are periods where the target value of 1 falls within the confidence +intervals of local means: April 19, 2018 to May 5, 2018, May 9, 2018 to June 16, 2018, +8The lower and upper bounds of the confidence interval at the 95% level are calculated under the iid +assumption as ˆµt ∓ 1.96 +� +ˆσt +n for each window of n days. + +THIS VERSION: January 3, 2023 +14 +October 10, 2018 to October 17, 2018, January 2, 2019 to January 3, 2019, and April +2, 2020 to May 1, 2020. The local variance of the price series is plotted in the bottom +panel of the figure. It varies over time, and its variation is much higher in the first half +of the sampling period. For example, in 2018, Tether had periods of high volatility from +January to March as well as periods of low volatility such as from April to November. +On the other hand, Tether is less volatile during the second half of the sampling period +as the local variance takes smaller values except for a short period of increased volatility +between mid-March and early May of 2020. +Although the rolling window approach helps detect local trends, it needs to be inter- +preted with caution. For a window size of n days, the first n-1 observations in the dataset +are eliminated due to rolling. In addition, using a longer rolling window (e.g., 100 days) +may over-smooth the changes in the mean and variance as compared to a shorter window +(e.g., 50 days). Therefore, we use the window of 50 days for further computations.9 +The distributional changes in Tether also concern the range and quantiles of the series. +Overall, we observe that: +1. The local mean is changing over time and is close to 1 between April and June 2018, +and after January 2021. +2. The variance is time varying and diminishes over time. +In Section 3.2, we identify a series of events that are closely related to Tether, and +provide a detailed explanation of the reason why those events could be the driving force +behind the changes we observe in the local statistics of Tether. +3.2 +Event Analysis +The dynamics of Tether are strongly influenced by events, which can be used to distinguish +the episodes of distinct patterns in the local mean and variance. +Figure 4 shows the evolution of the local mean and the local variance of Tether along +with the series of events that can be important for the dynamics of Tether, which can +9Also, note that n=50 is large enough to estimate parameters within each window consistently. Fur- +thermore, n=50 divided by the sample size of T=1361 is the bandwidth bT = n/T = 0.0367 in our local +analysis, which will be discussed later. In the literature, an optimal choice should satisfy Tb3 +T = o(1). In +our case, we have Tb3 +T = 0.0675, which is relatively small. See Dahlhaus, Richter, Wu (2019, page 1039) +for more details. + +THIS VERSION: January 3, 2023 +15 +help explain the trend reversals in its local statistics. Total of 11 such events are identified +including 7 events for the local mean and 4 events for the local variance. +In 2018, the local mean shows a downward trend for the most part of the year, except +for a brief period of recovery between early May and Late September. This should come as +no surprise because 2018 was a very tumultuous year for Tether Limited and its business +partners. Tether Limited was being scrutinized by the media and scholars for the quality of +its reserves and its close ties to the cryptocurrency exchange Bitfinex. More specifically, +Tether was publicly accused for not holding enough reserves to back all of its coins in +circulation and for manipulating the price of Bitcoin by pumping unbacked supply of +Tether into the market through Bitfinex to buy Bitcoins. Amid these controversies, the +local mean of Tether is found to be decreasing for the most part of the year, which could +be linked directly or indirectly to a shift in investor sentiment towards Tether. +In 2018, there is also a short period of a slight upward trend in the local mean roughly +between early May and late September. +In early May, the owners of Tether Limited +made their first significant attempt to show their willingness to address the investors‘ +concerns about the accountability of their business and that of Bitfinex. +On May 7, +2018, the cryptocurrency exchange Bitfinex officially announced that Peter Warrack, who +worked previously at RBC Royal Bank for 20 years as an anti-money laundering specialist, +joins their team as the Chief Compliance Officer. +Upon this news, the local mean of +Tether enjoys a period of recovery and hovers above the one-dollar peg. Nevertheless, the +local mean of Tether starts to decrease once again in September and reaches its lowest +level in November. The downfall of Tether during this period could be triggered by the +introduction of USD Coin (USDC) on September 26, 2018. +USD Coin, which is also +designed to be on par with the US dollar, relies on the business principles similar to that +of Tether but it claims to offer its users an improved transparency in its business activities. + +THIS VERSION: January 3, 2023 +16 +Figure 4: Important events for the local mean and the local variance of Tether +Note: This note provides a description of the events. +Peter Warrack: Peter Warrack was hired by Bitfinex as the Chief Compliance Officer on May 7, +2018. +USDC launched: USD Coin was launched on September 26, 2018. +Partial Backing: Bloomberg suggested on December 12, 2018 that Tether could be fully backed. +Partial Backing 2: On March 14, 2019, Tether made changes to its backing policy on its official +website. +Tether wins appellate: Bitfinex won a motion in the New York Supreme Court to delay sub- +mission of its business documents. +WHO Covid: WHO made an announcement on Twitter on December 31, 2019 to acknowledge +the cases of pneumonia in Wuhan, China. +Coinbase Outage: Bitcoin shed over 10% of its value in a matter of minutes on May 9, 2020, +which was followed by an outage in the cryptocurrency exchange Coinbase. +Wire Deposits: Bitfinex “temporarily paused” EUR, USD, JPY, and GBP wire deposits on +October 11, 2018. +TRON: Tether went live on the Tron network on April 17, 2019. +Black Thursday: Black Thursday: Bitcoin’s price reduced by around 50% in less than a day on +March 12, 2020. +Chain Swap: On June 22, 2020, Tether announced on Twitter that they would implement a +chain swap for a sizable amount of USDT from Tron TRC20 to ERC20 protocol on June 29th. + +Local Mean (USDT) +1.02 +Warrack +PIAOS OHM +6 +uedde +1.015 +no +ieie. +1.01 +1.005 +0.995 +0.99 +0.985 + Jui 2017 + Jan 2018 + Jul 2018 + Jan 2019 + Jul 2019 +Jan 2020 +Jul 2020 + Jan 2021 + Jul 2021 +Jan 2022 +×10-4 +Local Variance(USDT) +TRON. +Deposits +2.5 +1.5 +0.5 +Jui 2017 +Jan 2018 +Jul 2018 + Jan 2019 + Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 + Jul 2021 +Jan 2022THIS VERSION: January 3, 2023 +17 +Tether makes up for the loses quickly as the local mean increases remarkably from +its lowest level in November 2018 to its highest level in January 2019 just under two +months. The surge in the mean price of Tether could be a sign of positive reaction from +the investors as the bank statements obtained by Bloomberg showed that on the contrary +to the allegations, Tether Limited could be holding enough reserves to back its coins in the +circulation.10 However, this positive attitude towards Tether does not seem to last long as +the local mean diminishes once again starting in January 2019. The downward pressure +on the local mean was so strong during this period that it persisted up until August +2019 despite Tether’s effort to be more transparent about the composition of its reserves. +According to coindesk.com,11 Tether changed the terms on its website in Mid-February +implying that while every Tether is always 100% backed by their reserves, the reserves are +not necessarily composed of only fiat currency as they claimed before but also includes +cash equivalents, other assets, and receivables from loans. This update, however, did not +stop the New York State Attorney General (NYSAG) sue Bitfinex and Tether Limited on +April 24, 2019 on the basis of an ongoing fraud.12 +As the downward trend in the local mean dies out in August 2019, another episode +of increasing local trend is observed subsequently. The trend becomes more noticeable +around September 24, 2019 when iFinex Inc, the parent company of Bitfinex and Tether +Limited, won a motion in the court against NYSAG meaning that the company does not +have to hand over the documents related to its business activities until further notice.13 +Following this small victory in the court, the local mean of Tether diverges from the peg +and keeps increasing until the end of the year. +On December 31, 2019, the World Health Organization (WHO) announced via its +official Twitter account that they were informed of cases of pneumonia of unknown cause +in Wuhan City, China.14 With this announcement, WHO acknowledged the problem of +an epidemic in China, which would soon turn into a global health and economic crisis of +10https://www.bloomberg.com/news/articles/2018-12-18/crypto-mystery-clues-suggest-tether-has-the- +billions-it-promised +11https://www.coindesk.com/markets/2019/03/14/tether-says-its-usdt-stablecoin-may-not-be-backed- +by-fiat-alone/ +12https://ag.ny.gov/press-release/2019/attorney-general-james-announces-court-order-against-crypto- +currency-company +13https://www.forbes.com/sites/michaeldelcastillo/2019/09/24/bitfinex-and-tether-win-appeal-from- +new-york-supreme-court-in-900-million-case/?sh=7c8bf41132bc +14see https://twitter.com/who/status/1213795226072109058?lang=en for the original tweet from WHO + +THIS VERSION: January 3, 2023 +18 +Figure 5: Autocorrelation functions of Tether over different periods +COVID-19 pandemic. Subsequently, the local mean of Tether starts to decline cancelling +out the gains from the last quarter of 2019. +3.3 +Persistence in Tether Series +Let us now examine the serial correlation in Tether. The autocorrelation function (ACF) +of Tether is computed from the entire sampling period 2017-2021 as well as from each +calendar year separately. Figure 5 presents the computed ACF functions: the ACF over +the entire period (panel (a)), and in years 2017-2021 in panels (b) to (e), consecutively. +The ACF calculated from the entire sample exhibits a long range persistence. However, +the subperiod analysis reveals that the persistence in the ACF of Tether is strong up to +and including 2019,15 whereas the series has a short memory in 2020 and 2021. +When combined with the results from the local statistics, it can be inferred that the +period of long-range persistence in Tether coincides with the period of level shifts and high +volatility as documented in Figure 1. Likewise, when the variation is small and the local +mean stabilizes around the one-dollar peg as in 2020 and 2021, Tether displays a short +memory. +15In 2017, there are only 53 observations, which could be the reason for the weak evidence for the +persistence. + +Panel (a): 2017-2021 +Panel (b): 2017 +Panel (c): 2018 +0.8 +0.6 +0.6 + 0.4 +0.2 +0.2 +0.2 +-0.2 +20 +10 +15 +20 +10 +15 +10 +Lag +Lag +Lag +Panel (d): 2019 +Panel (e): 2020 +Panel (f): 2021 +1 +0.8 +0.8 +0.6 +4 0.4 +00.4 +0.2 +0.2 +0.2 +0.2 +-0.2 +29 +10 +15 +20 +5 +10 +15 +10 +15 +Lag +Lag +LagTHIS VERSION: January 3, 2023 +19 +In brief, the empirical results show that the analysis of Tether based on global statistics +would provide unreliable results, especially concerning the serial correlation of the series. +For example, different values and range of serial correlation are obtained in year 2021, as +compared to years 2018-2019. +Let us now focus on the autocorrelation values. More specifically, the autocorrela- +tion at lag one of the series can be estimated from the autoregressive coefficient of an +autoregressive of order 1 (AR(1)) model. We first consider the autoregressive coefficient +estimated from the AR(1) model fitted to the whole sample of demeaned Tether prices +xt = yt − µ +xt = ρxt−1 + σeet, +(3.1) +where et is assumed to be a white noise with mean 0 and variance 1. The AR(1) process +is stationary when |ρ| < 1 and nonstationary and explosive when ρ = 1. Model (3.1) is +estimated globally by the OLS, or equivalently by maximizing the Gaussian Maximum +Likelihood as follows: +ˆθT = Argmaxθ +T +� +t=1 +l(xt|xt−1; θ) = Argmaxθ +T +� +t=2 +−1 +2 +� +log +� +2πσ2 +e +� ++ (xt − ρxt−1)2 +σ2e +� +where θ = (ρ, σ2 +e). +The estimated parameter values are ˆρT = 0.674 and ˆσ2 +e,T = 0.545. Next, model (3.1) is +fitted locally and estimated by rolling with a window of length 50 and displayed in Figure +6. +The top panel of Figure 6 shows the autoregressive coefficient/correlation at lag 1 +estimates. We observe that the autoregressive coefficient is close to 1 during the explosive +episodes in 2018 and 2019, which violates the stationary condition of the AR(1) process. +The unit root dynamics of Tether resembles the stock prices and exchange rates. +It +suggests that Tether is then locally efficient in financial terms. The autocorrelation at +lag 1 is close to 0.5 at the end of the sampling period. In comparison with the results +presented in Section 3.1, estimating the autocorrelation at lag one based on the AR(1) +model gives us greater flexibility to assess the change in the persistence of Tether as we +are able to examine its evolution at a daily frequency rather than on a yearly basis. For + +THIS VERSION: January 3, 2023 +20 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +Figure 6: AR(1) parameter estimates and conditional volatility for xt +example, the estimated values of the AR(1) coefficient ρ suggest that the autocorrelation +at lag one ranges between -0.20 and 1.02 in 2018 whereas in Section 3.1, the ACF at lag +1 was estimated to be slightly over 0.76 for the entire year. +The bottom panel shows the conditional volatility ˆσe(t) = +� +1 +50 +�t +τ=t−49(xτ − ˆρ(τ) xτ−1)2 +of the price of Tether under the AR(1) assumption. The figure shows that this price ex- +hibits periods of low volatility and high volatility. Especially from October 2020 onwards, +the conditional volatility decreases remarkably and becomes very close to 0. This result is +consistent with the period of stability we observed in the price series of Tether during the +cryptocurrency bull market explained in Section 3. If the volatility was computed over +the full sample we would have a constant estimate which does not capture the changes in +volatility over time. To accommodate this feature, we consider the time-varying volatil- +ity model which also allows us to have valid inference when the estimated correlation +parameter is close to one unlike the AR(1) model. + +THIS VERSION: January 3, 2023 +21 +3.4 +Properties of rolling estimators +Let us now examine the properties of the rolling estimators of time varying parameters +written as deterministic functions of time. First, we estimated by rolling the time varying +marginal mean and variance functions, hoping to approximate m(t) and σ2(t) in a simple +model +yt = m(t) + σ(t)ut, +under the simplifying assumption of Normally distributed i.i.d. process ut with mean 0 +and variance 1. Then, in finite sample +ˆm(t) = 1 +50 +t +� +τ=t−49 +yτ ∼ N +�m(t) + · · · + m(t − 49) +50 +, σ2(t) + · · · + σ2(t − 49) +250 +� +We see that ˆm(t) is biased of m(t) towards an integrated mean. By the same argument it +can be shown that ˆσ2(t) is biased of both σ2(t) and the integrated variance. +Let us now consider the time varying parameters θ(t) = (ρ(t), σ2 +e(t)) of a time varying +parameter AR(1) model: +xt = ρ(t)xt−1 + σe(t)et, +where xt = yt −m(t). Then, the normality-based MLE estimator ˆθ∗(t) obtained by rolling +can be written as the following kernel MLE estimator (see Fan et al. (1998) for the method +of local kernel-weighted likelihood estimation using local polynomial fitting). +ˆθ∗ +T (t) += +Argmaxθ +1 +50 +t +� +τ=t−49 +l(xτ|xτ−1; θ) += +Argmaxθ +1 +50 +T +� +τ=1 +1t−49≤τ≤t l(xτ|xτ−1; θ) += +Argmaxθ +T +� +τ=1 +� 1 +501−49≤τ−t≤0 l(xτ|xτ−1; θ) +� += +Argmaxθ +T +� +τ=1 +1 +50 1−49/50≤(τ−t)/50≤0 l(xτ|xτ−1; θ) += +Argmaxθ +T +� +τ=1 +1 +50 K +�τ − t +50 +� +l(xτ|xτ−1; θ), t = 1, ..., T + +THIS VERSION: January 3, 2023 +22 +with the kernel K(u) = 1[−1,0](u). +It is easy to see that the dimension of the parameter of interest [θ(1), ..., θ(T)] depends on +the number of observations T. To circumvent this difficulty, we can replace the functional +parameter [θ∗(1), ..., θ∗(T)] with t ∈ N by an alternative functional parameter θ(c), c ∈ +(0, 1) on [0,1], such that θ∗(t) = θ(t/T). The rolling MLE is such that +ˆθ∗ +T (t) = ˆθT (t/T) = Argmaxθ +T +� +τ=1 +T +50 K +�τ/T − t/T +50/T +� +l(xτ|xτ−1; θ), t = 1, ..., T. +This formula can be extended to any value of argument c ∈ [0, 1]: +ˆθT (c) = Argmaxθ +T +� +τ=1 +T +50 K +�τ/T − c +50/T +� +l(xτ|xτ−1; θ), c ∈ [0, 1]. +Dahlhaus (2000) and Dahlhaus, Richter, Wu (2019) show that under regularity conditions +the functional parameters θ(c) of a locally stationary process can be consistently estimated. +Instead of considering the time varying parameters in calendar time, we have now defined +the functional parameters in a deformed time t → t/T that depends on T. The functional +parameter is now independent of the observations, while the effect of T is introduced by +considering a triangular array approach. This leads to a sequence of models indexed by +T: +xt,T = ρ(t/T)xt−1,T + σe(t/T)et,T , +where xt,T = yt,T − m(t/T). +This approach motivates our modelling approach presented in the next section. +4 +DAR(1) Model for Tether Price +To account for time-varying conditional mean and volatility, we introduce the Double +Autoregressive (tvDAR) process of order 1 with time varying parameters. The first part +of this section recalls the constant parameter DAR model. Next, the estimation of both +type of models is discussed. The last part of this section presents the stability measures +and their estimators. + +THIS VERSION: January 3, 2023 +23 +4.1 +Model with Constant Parameters +We consider the DAR process of order 1 for the demeaned Tether price series: +xt = φxt−1 + ηt +� +ω + αx2 +t−1, +(4.2) +where ω > 0, α > 0, and ηt, t = 1, ..., T is an independent and identically distributed (i.i.d.) +sequence with mean 0 and variance 1. The parameter φ captures the conditional mean +dependence. Parameter α represents the past dependence in the conditional variance. The +model is semi-parametric and conditionally heteroskedastic.16 Borkovec and Kluppenberg +(2001), Ling (2004) show that there exists a unique strictly stationary and ergodic solution +to model (4.2) when the following assumptions hold: +Assumption A.1: ηt has a symmetric and continuous density with mean 0 and variance +1. +Assumption A.2: The parameter space is Θ = {θ = (φ, ω, α) : E(ln|φ+ηt +√α|) < 0 with +|φ| ≤ ˜φ, ω ≤ ω ≤ ˜ω, α ≤ α ≤ ˜α where ˜φ, ω, ˜ω, α, ˜α are positive constants. +Assumption A.1 is not a stringent assumption. It is satisfied in particular if ηt, t = +1, ..., T are normally distributed. +Assumption A.2 is the existence and negativity of +the Lyapunov exponent ensuring the existence and uniqueness of a stationary solution +[Borkovec and Kluppenberg (2001)]. The region of φ, α that satisfy the negativity condi- +tion is displayed in Figure 1, p. 64 Ling (2004) and Figure 1, p. 191 Chen et al. (2014). +It includes cases when φ ≥ 1 as well as E(x2 +t ) = ∞. The processes xt that satisfy As- +sumption 2 are strictly stationary. Some of these processes are also weakly (second-order) +stationary and satisfy additionally the condition φ2 + α < 1 ensuring that E(x2 +t ) < ∞. +Thus, the marginal variance of those processes is finite. +When φ = 1, and E(ln |1 + ηt +√α|) < 0, the process xt is a strictly stationary martin- +gale process with volatility induced “mean-reversion” [Gourieroux, Jasiak (2019)]. Model +(4.2) is non-stationary when the Lyapunov exponent is non-negative. In particular, it is +nonstationary at the boundary points (φ, α) = (±1, 0) and nests the standard unit root +models at these two points. When φ = 0, the process is an ARCH(1) model. Moreover, +process (xt) is strictly stationary when x0 is drawn from a stationary distribution. +16More on the models with conditional heteroscedasticity and their applications in finance can be found +in Gourieroux (1997). Zakoian (1994) also proposed maximum likelihood and least squares estimators for +conditionally heteroscedastic model with threshold. + +THIS VERSION: January 3, 2023 +24 +Under assumptions A.1 and A.2, the parameter space Θ is compact and there exists +a unique strictly stationary solution of the model for any θ ∈ Θ. In addition, we assume +that: +Assumption A.3: The model is well-specified, i.e. the process satisfies equation (4.2) +for the true value of parameter θ0 = (φ0, w0, α0) and the true density ψ0 of η. The true +parameter value θ0 is an interior point in Θ. +Assumption A.4: The observed process is the unique, strictly stationary solution asso- +ciated with (θ0, ψ0). +These two conditions are introduced for the identification of the model and parameter +estimation. +4.2 +Model with Time-Varying Parameters +The DAR(1) model can be extended to a time-varying parameter model by using the +triangular array approach for locally stationary processes [Dahlhaus (2000), Dahlhaus, +Richter, Wu (2019)]. The time varying tvDAR(1) model is written for locally demeaned +observations xt,T , t = 1, ..., T indexed by t and T (triangular array) and defined by: +xt,T = φ(t/T)xt−1,T + ηt,T +� +ω(t/T) + α(t/T)x2 +t−1,T , +(4.3) +where for each time T, (ηt,T ) is a strong (i.i.d) white noise with mean zero, unit variance +and a symmetric distribution invariant in T. +φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] are +deterministic functions. We assume that these functions are smooth. +Assumption a.1: The functions φ(.), ω(.), α(.), are positive, deterministic and twice +differentiable on [0, 1]. +Moreover, the trajectories of the process have to be little responsive to small changes of +the parameters, which is ensured by a Lipschitz condition. More precisely, let us consider +process xt(c) defined by: +xt(c) = φ(c)xt−1(c) + ηt(c) +� +ω(c) + α(c)xt−1(c)2, +(4.4) +We assume that the following condition holds: +Assumption a.2: + +THIS VERSION: January 3, 2023 +25 +Let the Lq norm for q > 0 be denoted by ||.||q. Then, +i) For each c ∈ (0, 1), process {xt(c)} is stationary and ergodic. +ii) c → xt(c) is continuous for any t and ||supcxt(c)||q < ∞. +iii) There exists α, 1 ≥ α > 0 and CB > 0, such that +||xt(c) − xt(c′)||q < CB|c − c′|α uniformly in t and c, c′ ∈ (0, 1). +Under Assumptions a.1 and a.2, if T is large and t/T in a small interval (c−ϵ, c), then +the parameters are almost constant over that interval and locally model (4.4) is close to +model (4.3) with φ = φ(c), ω = ω(c), α = α(c). This explains the local stationarity. +When all the observations xt,T , t = 1, ..., T are considered, the variation of the pa- +rameters prevents the DAR process from being globally stationary. However, it is locally +stationary, if Assumption A.2 is locally satisfied, i.e. E[ln|φ(c) + ηt(c)α(c)|] < 0 for any c. +This is the condition on the negativity of the local Lyapunov exponent. +4.3 +Estimation +4.3.1 Estimation of the Model with Constant Parameters +The parameter estimates of model (4.2) are obtained by maximizing the quasi-maximum +likelihood (QML) objective function, i.e. +the log-likelihood function for normally dis- +tributed ηt. +LT (θ) = −1 +2 +T +� +t=2 +ln +� +ω + αx2 +t−1 +� +− 1 +2 +T +� +t=2 +(xt − φxt−1)2 +� +ω + αx2 +t−1 +� , +(4.5) +where θ = [φ, ω, α]′. The QML estimators of Model (4.2): +ˆθT = Argmaxθ∈ΘLT (θ) +are consistent under Assumptions A.1 to A.4, and the vector of QMLE estimators ˆθT = +[ˆφT ˆωT ˆαT ]′ → θ0 in probability, where θ0 = [φ0 ω0 α0]′ [Ling (2004,2007)]. Moreover, if +the following assumption: +Assumption A.5: E(η4 +t ) < ∞ +is satisfied, Li and Ling (2008) and Chen, Li and Ling (2014) show that the Quasi Maxi- +mum Likelihood estimators (QMLE) of θ are also asymptotically normal when |φ| ≥ 1 17 +as well as E(x2 +t ) = ∞. +17The ML/OLS estimators of φ from a linear autoregressive AR(1) model with constant parameters are +not asymptotically normal when φ = 1. + +THIS VERSION: January 3, 2023 +26 +√ +T(ˆθT − θ0) → N(0, diag(Σ−1, κΩ−1)) +where this convergence is in distribution, Σ = E0[x2 +t−1/(ω0 + α0x2 +t−1)] +Ω = E0 +� +1 +(ω0 + α0x2 +t−1)2 +� +1 +x2 +t−1 +x2 +t−1 +x4 +t−1 +�� +, +diag(Σ−1, κΩ−1) denotes the block-diagonal matrix with Σ−1 as the upper left block and +κΩ−1 as the bottom right block and κ is the kurtosis less 1 of the distribution of η. In +particular, κ = 2 when ηt is normal. +These asymptotic results are valid for any true +distribution of η, not necessarily a Gaussian distribution. The consistent estimators of Σ +and Ω are +ˆΣT = +1 +T − 1 +T +� +t=2 +[x2 +t−1/(ˆωT + ˆαT x2 +t−1)], +ˆΩT = +1 +T − 1 +T +� +t=2 +1 +(ˆωT + ˆαT x2 +t−1)2 +� +1 +x2 +t−1 +x2 +t−1 +x4 +t−1 +� +. +The model residuals are defined as: +ˆηt,T = (xt − ˆφT xt−1)/ +� +ˆωT + ˆαT x2 +t−1. +The model residuals ˆηt,T , t = 1, ..., T allow us to estimate non-parametrically the error +density to verify ex-post the symmetry assumption. The parameter κ is estimated by +ˆκT = +1 +T−1 +�T +t=2 ˆη2 +t,T − 1 = +1 +T−1 +�T +t=2 +(xt−ˆφxt−1) +4 +(ˆω+ˆαx2 +t−1) +2 − 1 allowing us to accommodate the +heavy tailed distribution of the stablecoin prices. +4.3.2 Estimation of the Model with Time-Varying Parameters +Let us consider the locally stationary tvDAR model. +The dynamic model (4.3) of +triangular arrays xt,T , t = 1, ..., T is non-parametric and depends on the functional param- +eters φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] and on the density function of the noise ηt,T . The +estimation of φ(.), ω(.), α(.) can be done by the local-in-time QML estimators. We con- +sider a kernel K defined on [−1/2, 1/2] and bandwith bT , bT > 0 (following the notation + +THIS VERSION: January 3, 2023 +27 +used in Dahlhaus, Richter, Wu (2019), p. 1035). The local negative log-conditional quasi +likelihood is +LT,b(c, φ, ω, α) = +1 +TbT +T +� +t=2 +K +�t/T − c +bT +� � +�−1 +2 ln +� +ω + αx2 +t−1,T +� +− 1 +2 +(xt,T − φxt−1,T )2 +� +ω + αx2 +t−1,T +� +� +� , +(4.6) +Then, the local negative QML estimator of θ(c) = [φ(c), ω(c), α(c)] is: +ˆθT,b(c) = ArgmaxθLT,b(c, φ, ω, α). +(4.7) +Under suitable regularity conditions given in [Dahlhaus, Richter, Wu (2019)], this +functional QML estimator ˆθT,bT (c), c ∈ [0, 1] is consistent of θ(c), c ∈ [0, 1] and its limiting +distribution is normal: +� +TbT (ˆθb(c) − θ0(c)) → N(0, +� +K(y)2dy J(c)−1I(c)J(c)−1), +where → denotes the weak convergence of processes indexed by c, J(c) is the Hessian +matrix and I(c) is the outer product of scores, both evaluated at c. Under the symmetry +assumption on ηt and for a strictly stationary and ergodic xt, the information matrix I(c) +simplifies to a block diagonal matrix [Ling (2004), Remark 1]. +The regularity conditions concern the functional parameter, the distribution of noise +ηt, the kernel K and the bandwidth bT . They can be found in Dahlhaus, Richter, Wu +(2019), since the DAR model is a nonlinear autoregressive model discussed in Dahlhaus, +Richter, Wu (2019), Example 5.5, p. 1039. In particular, the bandwidth has to satisfy the +conditions bT → 0, TbT → ∞, Tb3 +T → 0 when T tends to infinity. +4.4 +Stability measures +The Lyapunov exponent measures the average logarithmic rate of separation or conver- +gence of initially close trajectories in chaotic systems, and the sensitivity to initial con- +ditions, in general. A negative value of the Lyapunov exponent indicates the stability +of the dynamical system, while a positive value indicates chaos. The more negative the +Lyapunov exponent, the more stable the system. Therefore, it is used for testing for chaos + +THIS VERSION: January 3, 2023 +28 +[Sprott (2003)]. In this section, the Lyapunov exponent is proposed as a measure of stabil- +ity of Tether and other stable coins. In the framework of the DAR model, the Lyapunov +exponent is: +λ = E(ln |φ + η√α|). +The more negative the Lyapunov exponent, the less explosive the process18 and more likely +its marginal variance is finite. +The behavior of E(ln |φ+ηt +√α|) as a function of φ, α can be examined analytically for +selected densities of η, and/or simulated and illustrated graphically [see, Liu et al. (2018) +for graphical illustration]. Appendix A.1 shows the analytical formula of λ for a uniformly +distributed sequence {ηt}. More precisely: +Proposition 1: If η ∼ U[−1,1], the Lyapunov exponent is given by: +λ(φ, α) = +1 +2√α +� +(|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α| + ||φ| − √α| +� +Proof: See Appendix A.1. +This example clarifies that the Lyapunov exponent is a continuous function of φ, α, +although with points of non-differentiability. +Moreover, it is easy to show that that +E(ln |φ + ηt +√α|) is always an even function of φ, i.e. it takes the same value for φ and +−φ. To see that, consider a symmetric density function ψ(η). Then, +E(ln | − φ + ηt +√α|) = +� +(ln | − φ + ηt +√α|)ψ(η)dη. +Because ψ(η) is symmetric, we can change the variable η → −η: +E(ln|−φ+ηt +√α|) = +� +(ln |−φ−ηt +√α|)ψ(−η)dη = +� +(ln |φ+ηt +√α|)ψ(η)dη = E(ln |φ+ηt +√α|). +This proves that the Lyapunov exponent is even in φ. The Lyapunov exponent can be +computed by plug-in from the parameter estimates. Let us first consider the constant +parameter DAR model. The following estimators can be considered: +a) Suppose that the true density function ψ = ψ0 is known. Then, the estimator ˆλ1,T +of the Lyapunov exponent is: +ˆλ1,T = +� +ln |ˆφT + η +� +ˆαT | ψ0(η) dη. +18A strictly stationary process can have infinite moments. + +THIS VERSION: January 3, 2023 +29 +Proposition 2: +When the density ψ(η) = ψ0(η) is known, then under assumptions A.1 to A.4 and the +following condition: +(A.6) +∃δ > 0, such that +� +sup φ0 − δ < φ < φ0 + δ +α0 − δ < α < α0 + δ +| ln |φ + η√α||ψ0(η)dη < ∞ +the estimator ˆλ1,T converges in probability to the true value λ0 = +� +ln |φ0 +η√α0|ψ0(η)dη +of the Lyapunov exponent when T → ∞. +Proof: See Appendix A.2 +b) When the density ψ(η) is unknown, the model is semi-parametric. The Lyapunov +exponent can be estimated by the estimator ˆλ2,T such that: +ˆλ2,T = 1 +T +T +� +t=1 +ln |ˆφT + ˆηt,T +� +ˆαT |. +Therefore this estimator is equal to : +ˆλ2,T = 1 +T +T +� +t=1 +ln +������ +ˆφT + +xt − ˆφT xt−1 +� +ˆωT + ˆαT x2 +t−1 +� +ˆαT +������ += 1 +T +T +� +t=1 +g(xt, xt−1; ˆθT ) +where g(xt, xt−1; θ) = ln +����φ + +xt−φxt−1 +√ +ω+αx2 +t−1 +√α +����. We also introduce the notation GT (θ) = +1 +T +�T +t=1 g(xt, xt−1; θ). +Proposition 3: +Let us introduce the additional conditions: +(A.7) Eθ0g(xt, xt−1; θ) < ∞, ∀θ ∈ Θ. +(A.8) Sufficient Lipschitz condition for stochastic equicontinuity: There exists a stochas- +tic sequence BT with BT = Op(1) and an increasing function h from [0, ∞) to [0, ∞), +continuous at 0 with h(0) = 0, such that for all ˜θ, θ ∈ Θ, |GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)). +Then, under assumptions A.1-A.4 and conditions A.7, A.8 the estimator ˆλ2,T = GT (ˆθT ) → +G(θ0) = λ0 in probability. +Proof: See Appendix A.3. +The asymptotic distributions of estimators ˆλ1,T , ˆλ2,T cannot be obtained asymptot- +ically from the Taylor series expansion because function φ, α → +� +ln |φ + η√α|ψ(η)dη + +THIS VERSION: January 3, 2023 +30 +does not satisfy the necessary differentiability assumption, as pointed out in Proposition +1. However, the distribution of ˆλ1,T , ˆλ2,T can be determined by simulations and used for +hypothesis testing. +c) An alternative stability measure is ξ = φ2+α, which depicts the region of parameter +space ξ < 1 where the marginal variance of xt remains finite, so that the process is both +strictly and weakly stationary. In that sense ξ is a more conservative measure of stability +than the Lyapunov exponent because there is a region of parameter values φ, α where +the condition ξ < 1 no longer holds, while the condition λ < 0 remains satisfied. The +estimator ˆξT of ξ is: +ˆξT = ˆφ2 +T + ˆαT . +Proposition 4: +Under Assumptions A.1-A.5, the estimator ˆξT converges in probability to ξ0 when +T → ∞ and it is asymptotically Normally distributed: +√ +T(ˆξT − ξ0) A∼ N(0, Vξ), +where the formula of the asymptotic variance Vξ is given in Appendix A.4. +The asymptotic variance provides the asymptotically valid standard errors that can be +used to test the null hypothesis ξ < ξ0 using a Wald test statistic. The interpretation of +measure ξ is similar to that of λ: the smaller ξ, the more stable xt. +4.4.1 Model with time varying parameters +The Lyapunov exponent λ2(c) can be estimated locally by computing ˆλ2,T (c) from the +plugged in parameter estimates and residual values of the tvDAR model. For example, +the Lyapunov exponent can be estimated from a kernel-weighted formula +ˆλ2,T (c) = +1 +TbT +T +� +t=1 +K +�t/T − c +bT +� +(ln|ˆφ(t/T) + ˆηt,T +� +ˆα(t/T)|), +with the Epanechnikov kernel K(c) = 3 +2(1−(2c)2) for c ∈ [−1/2, 1/2] and K(c) = 0 other- +wise, which satisfies the regularity condition for localizing kernel [see Dahlhaus, Richter, +Wu (2019, Assumption 2.6)]. + +THIS VERSION: January 3, 2023 +31 +A similar approach can be used to estimate locally the measure ξ(c) = φ2(c) + α(c). +The local plug-in estimator ˆλ2,T (c) is illustrated in the next section. +5 +Empirical Results +This section presents the parameter estimates for the constant parameter DAR model and +the time varying parametr tvDAR model. The time varying parameters DAR model is +estimated first by rolling, which is equivalent to the use of an asymmetric rectangular +kernel, as shown in the previous section. Next, the model is estimated by using a kernel +which assigns higher weights to the observations close to the estimation date, providing +consistent and asymptotically normally distributed estimates, which are used for testing +hypotheses on the constancy of parameters. The estimators of stability measures are also +computed and illustrated graphically. +The estimation of the DAR(1) process with constant parameters is straightforward. +First, we demean the price series of Tether by subtracting the total mean of 1.0022 and +then fit the model 4.2 to the entire series of the demeaned prices, which gives the following +result +Table 1: Estimation of the DAR(1) model using the entire +sample +DAR(1) parameters +φ +ω +α +Estimates +0.699 +9.102e−06 +0.484 +Standard deviation +(0.034) +(2.174e−06) +(0.205) +where the standard deviations of the parameters are obtained using the asymptotic dis- +tribution described in Section 5.3.1. +In the following sections, the model 4.3 is estimated by using two types of kernels. The +first one is the asymmetric rectangular kernel, equivalent to the rolling estimation over the +window of 50. This window ensures good properties of the estimators, while preventing +the over-smoothing. + +THIS VERSION: January 3, 2023 +32 +The second approach relies on the Epanechnikov kernel and produces consistent and +asymptotically normally distributed parameter estimates used for hypotheses testing. +5.1 +Rectangular kernel +The tvDAR(1) is estimated from the demeaned Tether price by rolling, which is a common +practice in applied literature, and a window of length 50 days. This is equivalent to using +an asymmetric rectangular kernel, K(u) = 1(−1,0)(u), bT = 50/T which is computationally +simple, but does not satisfy the smoothness conditions ensuring locally the validity of +asymptotic distribution of the QMLE. The rolling estimate and the confidence interval of +the parameter of interest φ(t/T) is displayed in the first paned of Figure 7 below. +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-1 +-0.5 +0 +0.5 +1 +1.5 +=1 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +Conditional Volatility +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-6 +-5 +-4 +-3 +-2 +-1 +0 +Figure 7: tvDAR(1) parameter φ(t/T), conditional volatility and Lyapunov exponent +λ2(t/T) + +THIS VERSION: January 3, 2023 +33 +The second panel shows the estimated conditional volatility +� +ˆω(t/T) + ˆα(t/T)x2 +t for +Tether price. The third panel in Figure 7 presents the local estimates ˆλ2,T (t/T) of the +Lyapunov exponent computed by plugging in the local parameter estimators. +We observe that there are periods when the autoregressive coefficient is not signifi- +cantly different from 1. For instance, this is the case between October and December 2018 +and between February and May 2019. These results from the tvDAR (1) model confirm +our initial observation in Section 3.1 that the price series of Tether shows strong persis- +tence in 2018 and 2019 whilst allowing us to pinpoint its exact timing. The estimated +conditional volatility based on the estimated parameters has a pattern consistent with the +local variance estimator in Figure 3. The results in the third panel suggests that Assump- +tion 2 holds and the Lyapunov exponent remains negative even for the highest recorded +persistence. The sample Lyapunov exponent varies across time becoming on average more +negative before the end of the sampling period. This indicates that Tether achieves higher +stability over that period. +Full Sample +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0.96 +0.98 +1 +1.02 +1.04 +1.06 +1.08 +y=1 +Predicted E(yt+1|yt) +y +October 2018 to October 2019 +Jul 2018 +Oct 2018 +Jan 2019 +Apr 2019 +Jul 2019 +Oct 2019 +Jan 2020 +0.96 +0.97 +0.98 +0.99 +1 +1.01 +1.02 +1.03 +1.04 +y=1 +Predicted E(yt+1|yt) +y +Figure 8: Tether prices compared to one-step ahead out-of-sample forecasts + +THIS VERSION: January 3, 2023 +34 +The tvDAR(1) model estimated with the asymmetric rectangular kernel can be used for +forecasting at short horizons, under the assumption that the parameter functions remain +constant and equal to the last estimated value. Figure 8 presents the observed Tether price +and the estimate ˆyt+1 of the one-day-ahead conditional mean E(yt+1|yt, . . .) of the price of +Tether using a rolling window. To get ˆyt+1, we add the local mean of Tether price to the +estimated ˆφ using data over 50 days up to date t times xt. The figure shows a close match +between Tether price and its best prediction based on the previous day price. In addition, +the computed mean square prediction error is 1.7428 × 10−5 which is very small. Under +the assumption of locally constant parameters, the 95 % asymptotically valid prediction +intervals in Figure 9 are given by +� +ˆyt+1 − 1.96 +√ +T +� +x2 +t ˆΣ−1, ˆyt+1 + 1.96 +√ +T +� +x2 +t ˆΣ−1 +� +. +Full Sample +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0.95 +0.96 +0.97 +0.98 +0.99 +1 +1.01 +1.02 +1.03 +1.04 +1.05 +y=1 +October 2018 to October 2019 +Jul 2018 +Oct 2018 +Jan 2019 +Apr 2019 +Jul 2019 +Oct 2019 +Jan 2020 +0.95 +0.96 +0.97 +0.98 +0.99 +1 +1.01 +1.02 +1.03 +1.04 +1.05 +y=1 +Figure 9: One-step ahead out-of-sample predicted Tether prices and prediction intervals +Next, we conduct further investigations to analyze the goodness of fit of the tvDAR(1). +For this analysis, we keep the estimation window of 50 and perform the Ljung-Box test of + +THIS VERSION: January 3, 2023 +35 +white noise on ˆηt and ˆη2 +t , while rolling the sample used for the test [see Li (1992), and Li +and Mak (1994)]. Because we consider all subsamples of 50 consecutive dates, it is likely +that at some dates the serial correlation is not fully captured by the tvDAR(1) model. +Our results show that for most periods, residuals ˆηt and ˆη2 +t are serially uncorrelated. More +precisely, we reject the null of no serial correlation for ˆηt only for 1.52% of the subsamples, +while we reject the null of no serial correlation for ˆη2 +t only for 8.08% of the subsamples. +The results suggest that the model captures most of the nonlinear serial correlation in +Tether prices. +5.2 +Epanechnikov kernel +We now use a symmetric Epanechnikov kernel producing consistent and asymptotically +normally distributed estimates for hypothesis testing. +The first panel of Figure 10 plots the estimated time-varying autoregressive coefficient +and its confidence band, while the second panel presents the time-varying estimates for +of the Lyapunov exponent using the ˆλ2,T (t/T) estimator. For the bandwidth, we choose +bT = 50/T using the same window as before. +The results confirm that when the estimation is conducted locally over each period, +the Lyapunov exponent displayed in the second panel of Figure 10 is negative. Hence the +critical validity condition holds for all the dates. Moreover, the Lyapunov exponent is +on average more negative at the end of the sampling period, confirming that Tether has +achieved higher stability at the end of the sampling period. +Furthermore, the estimated dynamic of estimated parameter φ for Tether price in the +first panel of of Figure 10 remains similar to that of Figure 7. These findings confirm +that Tether price has causal dynamics, as the autoregressive parameter and its confidence +interval are mostly between −1 and 1. However, there are few dates when this estimate +is not statistically different from 1, which suggests strong persistence in Tether prices. + +THIS VERSION: January 3, 2023 +36 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +=1 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0 +Figure 10: Kernel-based parameter φ estimates and Lyapunov exponent λ(t/T) +To detect the periods of strong persistence, we test the null hypothesis H0 : φ = 1 +using the series of ˆφ(t/T) and its 95% confidence intervals for each t = 1, ..., T. +We +conduct the hypothesis test using the series of ˆφ(t/T) obtained from the estimation with +the Epanechnikov kernel and report the results in Table 2. +The results suggest that although Tether price is predictable most of the time, there +are intervals of time periods where this tends not to be the case. During these episodes +presented in Table 2, we find high persistence in Tether price. However, as explained +above, the proposed tvDAR approach remains valid and accommodates strong persistence +in the price series. In light of the results in Figure 4, we further inspect the identified +periods. +We observed that the identified episodes in 2018 and 2019 overlap with the +periods of high volatility observed in Figure 4, which ended in February 2020, but started +around the introduction in September 2018 of USD Coin, another stablecoin designed to +maintain price equivalence to the U.S. dollar. Moreover, the episodes in 2020 match with +a small rise in Tether price volatility by the end of July 2020, while the episodes in 2021 + +THIS VERSION: January 3, 2023 +37 +can be associated with the period of increased volatility at the end of our sample. +Table 2: Episodes of high persistence in Tether characterized by φ = 1. +Year +From +To +2018 +September 24 +October 29 +December 3 +October 5 +2019 +January 11 +January 30 +February 9 +February 11 +February 16 +February 21 +2020 +July 25 +August 8 +August 16 +August 19 +2021 +May 14 +June 17 +5.3 +Test for Conditional Homoscedasticity +Let us now consider a simple test of model specification of the tvDAR(1) model with +time-varying parameters. It is based on testing for the constancy of the variance function +in model 4.3 which is given by the following expression +σ(t/T) = +� +ω(t/T) + α(t/T)x2 +t−1,T , +t = 1, ..., T. +(5.8) +To test the null hypothesis H0 : σ(t/T) = σ0 ∀t, we consider the below test statistics +proposed by Chandler and Polonik (2017) for time-varying autoregressive processes +CPT = sup +α∈[0,1] +� +T +(γ(1 − γ)| ˆGT,γ(α) − αγ|, +(5.9) +where +• ˆGT,γ(α) = 1 +T +�[αT] +t=1 1(ˆϵ2 +t ≥ ˆq2 +γ), + +THIS VERSION: January 3, 2023 +38 +• ˆq2 +γ = min +� +q2 ≥ 0 : 1 +T +� +t∈[aT,bT] 1(ˆϵ2 +t > q2) ≤ γ +� +, +• ˆϵt = xt − ˆφ( t +T )xt−1. +The process ˆGT,γ(α) counts the number of squared residuals within the first (100×α)% +of the observations that are larger than the empirical quantile of the squared residuals +denoted by ˆq2 +γ. The series of squared residuals is constructed by computing ˆϵt = xt − +ˆφ( t +T )xt−1 for each period t where ˆφ( t +T ) in our case is the corresponding DAR(1) estimate +obtained in the previous section. +Chandler and Polonik (2017) shows that the test statistics CPT in equation (5.9) under +the null hypothesis converges asymptotically to the supremum of a Brownian bridge. This +asymptotic result is found to be still valid when the time-varying functions of the model +parameters are estimated nonparametrically (see Chandler and Polonik (2012)). +When computing the test statistics CPT , we consider different alternatives for the em- +pirical upper γ-quantile for comparison, particularly γ = (0.7, 0.8, 0.9) following Chandler +and Polonik (2017). Given the choice of γ, we then calculate the expression +� +T +(γ(1−γ)| ˆGT,γ(α)− +αγ| for different values of α and report the results in table 3. +Table 3: The calculated values of +� +T +(γ(1−γ)| ˆGT,γ(α) − αγ| for the given pairs of γ +and α. +α +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0.7 +0.922 +1.302 +2.466 +3.328 +4.251 +5.957 +6.759 +6.837 +3.539 +γ +0.8 +0.912 +1.478 +2.390 +3.233 +3.937 +5.471 +6.106 +6.327 +3.509 +0.9 +0.562 +1.031 +1.593 +2.155 +2.993 +3.923 +4.485 +5.047 +3.490 +The table shows that regardless of the choice of γ, the largest value is achieved when +α = 0.8. By the definition of supremum, the test statistics CPT should satisfy CPT ≥ +� +T +(γ(1−γ)| ˆGT,γ(α)−αγ| for all α ∈ [0, 1]. Hence, it is safe to say that we have CPT ≥ 5.047 +in the worst case scenario, i.e when γ = 0.9 is chosen.19 +Compared with the critical +19Alternatively, we could fix our choice of γ and find the exact value of α ∈ [0, 1] at which the expression +� +T +(γ(1−γ)| ˆGT,γ(α) − αγ| attains its maximum on this fixed interval. +This could slightly improve the + +THIS VERSION: January 3, 2023 +39 +values from the asymptotic distribution of the test statistics,20 this result leads us to the +conclusion that we can reject the null hypothesis of constant variance function even at +the 99% confidence level. In other words, we have a statistically significant evidence in +favor of the alternative that the variance function is varying over time, which consequently +justifies our strategy to estimate the model with time-varying parameters. +6 +Concluding Remarks +We show that the distributional and the dynamic properties of stablecoins have been +evolving over the sampling period. We implement local analysis to detect and describe +local explosive patterns, time-varying volatility and persistence. We model the dynamic of +the most important stablecoin which is Tether, and provide evidence that the tvDAR(1) +model with time varying coefficients provides locally a good fit and reliable short-term +predictions of Tether prices. Our modelling strategy enables us to have valid inference +even when the tvDAR(1) coefficient φt of Tether price is not locally different from 1. +The sample Lyapunov exponent computed from the parameter estimates of the model +provides a measure of stability. It confirms that at the end of the sampling period Tether +becomes relatively more stable and allows for comparing the stability of Tether with other +stablecoins. +Appendix A: Technical Results +This Appendix contains the proofs of Propositions 1, 2, 3 and 4. +A.1 +Proposition 1 +Because of the symmetry of the density of η, the Lyapunov exponent is an even function +of φ. Hence we can suppose that φ > 0 to find the expression of the Lyapunov exponent, +and then replace φ by |φ|. +For φ > 0 we have: +precision of the lower bound for the test statistics. For example, when γ = 0.9, the expression attains its +maximum value of 5.106 at α∗ = 0.8012. +20see https://homepages.ecs.vuw.ac.nz/ ray/Brownian/ for the distribution of the supremum of a Brow- +nian Bridge. + +THIS VERSION: January 3, 2023 +40 +λ(φ, α) = E ln |φ + η√α| = +� +ln |φ + η√α|ψ(α)dα. +Let us assume that η ∼ U[−1,1]. Then, its density is ψ(η) = 1 +21η∈[−1,1]. +We have: +λ(φ, α) = 1 +2 +� 1 +−1 ln |φ + η√α|dα. +We observe that: +φ + η√α > 0 +⇐⇒ η > −φ/√α, +φ + η√α < 0 +⇐⇒ η < −φ/√α. +Hence, +λ(φ, α) = 1 +2 +� 1 +−1 +1η>−φ/√α ln(φ + η√α)dη + 1 +2 +� 1 +−1 +1η<−φ/√α ln(−φ − η√α)dη +Let us now examine the two cases: +a) If φ/√α > 1 ⇐⇒ −φ/√α < −1, we get: +λ(φ, α) += 1 +2 +� 1 +−1 ln(φ + η√α)dη + 1 +20 = 1 +2 +1 +√α +� 1 +−1 ln(φ + η√α)d(√αη) += +1 +2√α +� φ+√α +φ−√α ln(u)du with change of variable u = φ + η√α += +1 +2√αu ln(u) − u|φ+√α +φ−√α += +1 +2√α [(φ + √α) ln(φ + √α) − (φ + √α) − (φ − √α) ln(φ − √α) + (φ − √α)] +b) If φ/√α < 1 ⇐⇒ −φ/√α > −1, we get: +λ(φ, α) = 1 +2 +� 1 +−φ/√α ln(φ + η√α)dη + 1 +2 +� −φ/√α +−1 +ln(−φ − η√α)dη += 1 +2 +1 +√α +� φ+√α +0 +ln(u)du + 1 +2 +� 1 +φ/√α ln(−φ + η√α)dη with the change of variable u = φ + η√α += +1 +2√α +� φ+√α +0 +ln(u)du + +1 +2√α +� −φ+√α +0 +ln(u)du += +1 +2√α [(φ + √α) ln(φ + √α) − (φ + √α) − [(−φ + √α) ln(−φ + √α)] + (−φ + √α)] +By putting the two expressions in a) and b) together we get for φ > 0: +λ(φ, α) = +1 +2√α +� +(φ + √α) ln(φ + √α) − (φ + √α) − |φ − √α| ln |φ − √α)| + |φ − √α| +� +The general expression of the Lyapunov Exponent without the sign constraint on φ is: +λ(φ, α) = +1 +2√α +� +(|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α)| + ||φ| − √α| +� +We observe a non-differentiability in φ = ±√α. Moreover, for φ = 0, we get a zero value +of the Lyapunov exponent: + +THIS VERSION: January 3, 2023 +41 +λ(φ, α) = +1 +2√α +�√α ln √α − √α ln √α + √α +� += 0 +A.2 +Proof of Proposition 2 +We need to show that when T → ∞, then +� +ln[ˆφT +η√ˆαT ]ψ0(η)dη → +� +ln(φ0+η√α0)ψ0(η)dη +in probability if (ˆφT , ˆαT ) → (φ0, α0) in probability, and if condition A.6 of Proposition 2: +∃δ > 0, such that +� +sup φ0 − δ < φ < φ0 + δ +α0 − δ < α < α0 + δ +| ln |φ + η√α||ψ0(η)dη < ∞ +(a.1) +holds. +Proof of convergence: +If (ˆφT , ˆαT ) → (φ0, α0) in probability, then they also converge in distribution. It follows +from the Skorokhod theorem that up to a change of probability space, we can assume that +the almost sure (a.s.) convergence also holds [Billingsley (1999)]. Therefore, if g is a con- +tinuous function of (φ, α), we have g(ˆφT , ˆαT ) → g(φ0, α0) a.s. and in distribution in that +new space. Then, g(ˆφT , ˆαT ) d→ g(φ0, α0) in the initial space and also in probability because +the limit is constant, we get the ”in probability” version of the continuous mapping theo- +rem. Therefore, we need only the condition ensuring that g(φ, α) = +� +ln |φ+η√α|ψ0(η)dη +is continuous. Condition (A.6) ensures the continuity of integral function g, which follows +from the dominated convergence theorem. +A.3 +Proof of Proposition 3 +We first prove a general lemma, which is next applied to the DAR model and Lyapunov +estimator λ2,T . +Lemma +Let us consider a sequence GT (θ) of stochastic functions of θ, θ ∈ Θ, and a sequence +of estimators ˆθT . We assume that: +i) Θ is compact and θ0 is in the interior of Θ. +ii) ˆθT tends in probability to θ0. +iii) GT (θ) tends in probability to a limit G(θ), ∀θ ∈ Θ. +iv) Sufficient Lipschitz condition for stochastic equicontinuity: + +THIS VERSION: January 3, 2023 +42 +There exists a stochastic sequence BT with BT = Op(1) and an increasing function +h : [0, ∞) → [0, ∞) continuous at zero, with h(0) = 0 and such that for all ˜θ, θ ∈ Θ, +|GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)). +Then, ˆGT (ˆθT ) tends in probability to G(θ0). +Proof: +We have : +|GT (ˆθT ) − G(θ0)| += |GT (ˆθT ) − GT (θ0) + GT (θ0) − G(θ0)| +≤ |GT (ˆθT ) − G(θ0)| + |GT (θ0) − G(θ0)| +≤ BT h[d(ˆθT , θ0)] + |GT (θ0) − G(θ0)|. +We know that if XT +P→ 0, YT +P→ 0 => XT + YT +P→ 0, i.e. the sum of op(1) is op(1). +Under condition iii) |GT (θ0) − G(θ0)| = op(1). It remains to be shown that BT h[d(ˆθT , θ0)] +is op(1). +We have: +[BT < M and h[d(ˆθT , θ0)] < ϵ/M] => [BT h[d(ˆθT , θ0)]] < ϵ] +⇐⇒ +� +(BT < M) ∩ [h[d(ˆθT , θ0)] < ϵ/M] +� +⊂ [BT h[d(ˆθT , θ0)] < ϵ] +Consider the complement: (A ∩ B)c = Ac ∪ Bc. We get: +((BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M) ⊃ [BT h[d(ˆθT , θ0)] > √ϵ] +It follows that: +P[BT h[d(ˆθT , θ0)] > ϵ] +≤ P[(BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M)] +≤ P[BT > M] + P[h[d(ˆθT , θ0)] > h−1(ϵ/M)], +because P[A ∪ B] ≤ P(A) + P(B). +Then, for any ϵ, we can choose a value of M and a number of observations T sufficiently +large to get P[BT h[d(ˆθT , θ0)] > ϵ] arbitrarily small. Therefore, BT h[d(ˆθT , θ0)] tends to +zero in probability. +QED +Then, the lemma can be applied with GT (θ) = 1 +T +�T +t=2 g(xt, xt−1; θ) and g(xt, xt−1; θ) = +ln |φ+ xt−φxt−1 +√ +ω+αx2 +t−1 +√α|. Under assumptions A.1, A.4, A.7, conditions i), ii), iii) are satisfied. +For example: +GT (θ) P→ E0g(xt, xt−1; θ), +by the weak law of large numbers applied to the transformation g(xt, xt−1; θ) of the ergodic +stationary process (xt). Assumption A.8 corresponds to condition iv) of the lemma. + +THIS VERSION: January 3, 2023 +43 +A.4 +Proof of Proposition 4 +a) Proof of convergence +We need to show that ˆφ2 +T + ˆαT → φ2 +0 + α0 in probability if (ˆφ, ˆα) → (φ0, α0) in +probability when T → ∞. +Thi is a consequence of the ”in probability” version of the continuous mapping theorem +given in Appendix A.2 +b) Proof of Normality +The Taylor series expansion pre-multiplied by +√ +T implies: +√ +T +� +(ˆφ2 +T + ˆαT ) − (φ2 +0 + α0) +� += +� 2φ0 +1 +�′ √ +T +� ˆφT − φ0 +ˆαT − α0 +� ++ op(1) += A′√ +T +� ˆφT − φ0 +ˆαT − α0 +� ++ op(1) +where A′ = [2φ0 1]. We get the asymptotic normal distribution of ˆξT : +√ +T(ˆξT − ξ0) ∼ N(0, Vξ), +where Vξ = A′Ω∗A. The matrix Ω∗ is: Ω∗ = diag(Σ−1, V (ˆα)) where Σ = E0(y2/(ω0+α0y2) +given in Section 4.3.1. +Matrix V (ˆα) = (E0 +1 +(ω0+α0y2)2 )/ ˜V0(y2) and ˜V0(y2) = ˜E0(y4) − +( ˜E0y2)2. In this formula, ˜E0 denotes the expectation of variables y4 +1 +(ω0+α0y2)2 /E0 +1 +(ω0+α0y2)2 +and y2 +1 +(ω0+α0y2)2 /E0 +1 +(ω0+α0y2)2 [see, section 4.3.1 and Ling (2004) for the variance estima- +tor formula]. +Appendix B: Simulation Results +The purpose of this section is to illustrate the derived results in Appendix A using simu- +lation experiments. We distinguish the case where the distribution of the innovation η is +known and the case it is not. +First, we use the result in Proposition 1 and plot the Lyapunov exponent λ = E(ln(|φ+ +√αη|)) for different values of the parameter φ and α. To do so, we assume η ∼ U[−1, 1]. +Figure B1 shows that the Lyapunov exponent λ remains lower than zero as as φ varies in +{−1, −0.8, −0.6, . . . , 0.6, 0.8, 1} and α varies in {0, 0.1, 0.2, 0.3, . . . , 0.8, 0.9, 1}. +Second, we assume η ∼ N(0, 1), set the true parameters to φ0 = 0.7, α0 = 0.5, +ω0 = 0.01. Note that the parameters are chosen to be close to their estimated value from +the entire data in our application. The estimated densities are based on 4, 000 simulations + +THIS VERSION: January 3, 2023 +44 +and obtained via kernel density estimation. Figure B2 plots the estimated density for +the Lyapunov exponent ˆλ2,T and the stability measure ˆξT when the three parameters are +estimated from a sample of size T and plugged in. The results on panel (a) of the figure +show that the mostly frequent estimated value is below zero for T = 50 or T = 100, +implying valid inference. In addition, panel (b) of Figure B2 shows that the density of the +estimated alternative stability measure ˆξT = ˆφ2 +T + ˆαT has its mode around the true value +of ξ, which is ξ0 = φ2 +0 + α0 = 0.99. +As a by-product, we present Figure B3, which shows the estimated density for ˆφT , ˆαT +and ˆωT for T = 50 and T = 100. The three panels in the figure provide evidence that the +three parameters are fairly accurately estimated. More specifically, the estimated values +have modes close to their true unknown parameters. The accuracy improves as the sample +size increases from T = 50 to T = 100. +Figure B1: Lyapunov exponent λ = E(ln(|φ + √αη|)) in terms of φ and α when η ∼ +U[−1, 1] + +0 +-2 +-4 ~ +-6 +1 +0.8 +0.6 +0.4 +0 +0.2 +-0.5 +0 +-11 +0.5THIS VERSION: January 3, 2023 +45 +-3.5 +-3 +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +0 +0.5 +1 +1.5 +2 +2.5 +Density +T=50 +T=100 +(a) Density of the Lyapunov exponent ˆλ2,T +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +Density +T=50 +T=100 +(b) Density of the alternative stability measure ˆξT +Figure B2: Densities for stability measures based on estimated parameters + +THIS VERSION: January 3, 2023 +46 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Density +T=50 +T=100 +0 +(a) Density of ˆφT +-0.005 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Density +T=50 +T=100 +0 +(b) Density of ˆωT +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.5 +1 +1.5 +2 +2.5 +3 +Density +T=50 +T=100 +0 +(c) Density of ˆαT +Figure B3: Densities of estimators for φ, α and ω + +THIS VERSION: January 3, 2023 +47 +Appendix C: More Empirical Results +Given that the proposed stability measure can be used as a mechanical tool to detect +periods of instability in stablecoins, we use the results of Proposition 4 in Appendix A to +construct an interval for ξ employing the same rolling window approach as before. Figure +C1 presents the results and contains, in its first panel, the estimated coefficient DAR model +using the rolling windows approach, in its second panel, the conditional heteroskedasticity, +and in the third panel, the Lyapunov exponent over time. In addition to the episodes of +high persistence mentioned above, we observed, around September 2020, an important +instability that is not due to high persistence in Tether price, but more frequent changes +in the conditional heteroskedasticity, which can be seen in the second panel. This period +can also be linked to higher local volatility in the observed data in Figure 4. There is no +specific event we can associate with this movement, as is sometimes the case in crypto +markets. However, the proposed model allows capturing those changes. +As explained above, the measure of stability ξ plotted in Figure C1 is more conservative +than the Lyapunov exponent λ. Because we can have ξ ≥ 1 while λ < 0 so that valid +inference is still possible, the rejection of ξ < 1 should be interpreted as the need for +investors, regulators or stablecoin issuers to be cautious when predicting Tether future +price around the tested periods. + +THIS VERSION: January 3, 2023 +48 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-1 +-0.5 +0 +0.5 +1 +1.5 +=1 +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +Conditional Volatility +Jul 2017 +Jan 2018 +Jul 2018 +Jan 2019 +Jul 2019 +Jan 2020 +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +-4 +-2 +0 +2 +4 +6 +Figure C1: tvDAR(1) parameter φ(t/T) and Lyapunov exponent ξ(t/T) + +THIS VERSION: January 3, 2023 +49 +References +Allen, F., Gu, X., and J. Jagtiani (2022): “Fintech, Cryptocurrencies, and CBDC: Finan- +cial Structural Transformation in China”, Journal of International Money and Finance, +Elsevier, vol. 124(C). +Andrews, D. (1987): “Consistency in Nonlinear Econometric Models: A Generic Uniform +Law of Large Numbers”, Econometrica, 55, 1465-1471. +Barry, C. B., and R. L., Winkler (1976) : “Nonstationarity and Portfolio Choice”, The +Journal of Financial and Quantitative Analysis, Vol. 11, No. 2, pp. 217-235. +Bandi, F., and P., Phillips (2009) : “Nonstationary Continuous-Time Processes”, in Hand- +book of Financial Econometrics, Y., Ait Sahalia, and L., Hansen eds., 140-199, Elsevier. +Baum¨ohl, E. and T. Vyrost (2020) : “Stablecoins as a crypto safe haven? +Not all of +them!”, ZBW-Leibniz Information Centre for Economics, Kiel, Hamburg +Bianchi, D., Rossini, L and M., Iacopini (2022) “Stablecoins and Cryptocurrency Returns: +What is the Role of Tether”, Working Paper, University of Milan? +Billingsley, P. (1999): “Convergence of Probability Measures”, New York, Wiley. +Blanchard, O. and M. Watson (1982): “Bubbles, Rational Expectations, and Financial +Markets”, P. Wachtel (ed.) Crisis in the Economic and Financial Structure, Lexington +Books, Lexington, Mass. +Borkovec, M. (2000): “Extremal Behavior of the Autoregressive Process with ARCH(1) +Errors”, Stochastic Processes and their Applications, 85, 189-207. +Borkovec, M. and C. Kluppenberg (2001): “The Tail of the Stationary Distribution of +the Autoregressive Process with ARCH(1) Errors”, Annals of Applied Probability, 11, +1220-1241. +Bullman, D., Klemm, J., and A.,Pinna (2019): “In Search for Stability in Crypto-assets: +Are Stablecoins the Solution?”, European Central Bank Occasional Paper Series No 230 +Catalini, C., and A.,de Gortari (2021): “On the Economic Design of Stablecoins”, Avail- +able at SSRN: https://ssrn.com/abstract=3899499 or http://dx.doi.org/10.2139/ssrn.3899499. + +THIS VERSION: January 3, 2023 +50 +Chen, M., Li, D. and S. Ling (2014): “Non-Stationarity and Quasi-Maximum Likelihood +Estimation on a Double Autoregressive Model”, Journal of Time Series Analysis, 35: 189– +202. +Chen, M., Qin, C., and X., Zhang (2022): “Cryptocurrency price discrepancies under +uncertainty: Evidence from COVID-19 and lockdown nexus,” Journal of International +Money and Finance, Elsevier, vol. 124(C). +Chandler, G., and W., Polonik (2012): “Mode Identification of Volatility in Time-Varying +Autoregression”, Journal of the American Statistical Association, 107(499), 1217-1229. +Chandler, G., and W., Polonik (2017): “ Residual Empirical Processes and Weighted Sums +for Time-Varying Processes with Applications to Testing for Homoscedasticity”, Journal +of Time Series Analysis, vol. 38, 72-98. +Dahlhaus, R. (2000): “A Likelihood Approximation for Locally Stationary Processes”, +Annals of Statistics, 28, 1782-1794. +Dahlhaus, R., S. Richter and W. Wu (2019): “Towards a General Theory for Nonlinear +Locally Stationary Processes”, Bernoulli, 25, 1013-1044. +Day, W. (1976): “A Reform of the European Currency Snake”, IMF Econ Rev 23, 580?597. +Dechert, W.D. and R. Gencay (1992): “Lyapunov Exponents as a Nonparametric Diag- +nostic for Stability Analysis”, Journal of Applied Econometrics, VOL. 7, S41-S60 +Fan, J., M. Farmen and I. Gijbels (1998): “Local Maximum Likelihood Estimation and +Inference”, Journal of the Royal Statistical Society, series B, 60, Part 3, 591-608. +Froot, K., and M. Obstfeld (1991): “Intrinsic Bubbles: The Case of Stock Prices”, Amer- +ican Economic Review, 81, pp. 1189-1214. +Gourieroux, C.: “ARCH Models and Financial Applications”, New York: Springer-Verlag, +1997. +Gourieroux C. and J. Jasiak (2019): “Robust Analysis of the Martingale Hypothesis”, +Econometrics and Statistics, Vol 9, 17-41. +Gourieroux, C., and J.M., Zakoian (2017): “Local Explosion Modelling by Noncausal +Processes”, Journal of the Royal Statistical Society (JRSS), Series B, 79, 737-756. + +THIS VERSION: January 3, 2023 +51 +Griffin, J. M. , and A.,Shams (2020): “Is Bitcoin Really Untethered?”, The Journal of +Finance, vol. 75, issue 4 +Hong, H., and J. C., Stein (2002): “A Unified Theory of Underreaction, Momentum +Trading, and Overreaction in Asset Markets”, The Journal of Finance, 54, issue 6, 2143- +2184 +Huisman, R., Koedijik, K.G., and Pownall, R.A.J., (1998): “VaR-x: Fat Tails in Financial +Risk Management”, Papers 98-54, Southern California - School of Business Administra- +tion. +Kortian, T. (1995): “Modern Approaches to Asset Price Formation: A Survey of Re- +cent Theoretical Literature”, RBA Research Discussion Papers rdp9501, Reserve Bank of +Australia. +Lebaron, B. (1994): “Chaos and Nonlinear Forecastability in Economics and Finance”, +Philosophical Transactions of the Royal Society of London. Series A: Physical and Engi- +neering Sciences, 348, 397-404. +Li, Q. (1999): “Consistent Model Specification Tests for Time Series Econometric Models”, +Journal of Econometrics, 92, 101-147. +Li, W. K. (1992): “On the Asymptotic Standard Errors of Residual Autocorrelations in +Nonlinear Time Series Modeling”, Biometrika, 79, 435-437. +Li, W. K. and Mak, T. K. (1994): “On the Squared Residual Autocorrelations in Non- +linear Time Series with Conditional Heteroskedasticity”, Journal of Time Series Analysis, +15, 627-636. +Li, D., Guo, S., and K., Zhu (2019): “A Double AR Model without Intercept: An Al- +ternative to Modeling Nonstationarity and Heteroscedasticity”, Econometric Reviews, 38, +issue 3, 319-331. +Li, D., Ling, S. and R., Zhang (2016): “On a Threshold Double Autoregressive Model”, +Journal of Business and Economic Statistics, 34, 68-80. +Li, Y. and Mayer, S., (2022) “Money Creation in Decentralized Finance: A Dynamic +Model of Stablecoin and Crypto Shadow Banking”, Fisher College of Business Working +Paper No. 2020-03-030, Charles A. Dice Center Working Paper No. 2020-30 + +THIS VERSION: January 3, 2023 +52 +Ling, S. (2004): “Estimation and Testing Stationarity for Double Autoregressive Models”, +JRSS Series B, 66, 63-78 +Ling, S. (2007): “A Double AR(p) Model: Structure and Estimation”, Statistica Sinica”, +Vol. 17, No. 1., 161-175 +Ling, S. and D. Li (2008): “Asymptotic Inference for a Nonstationary Double AR(1) +Model”, Biometrika , 95, 1, pp. 257–263 +Liu, F, Li, D. and X. Kang (2018) “Sample Path Properties of an Explosive Double +Autoregressive Model”, Econometric Reviews, 37, 484-490. +Lyons, R. K., and G., Viswanath-Natraj (2020): “What Keeps Stablecions Stable?”, +NBER Working Papers 27136, National Bureau of Economic Research, Inc. +Nelson, D.B. (1990): “Stationarity and Persistence in the GARCH(1,1) Model”, Econo- +metric Theory, 6, 318-334. +Newey, W. (1991): “Uniform Convergence in Probability and Stochastic Continuity”, +Econometrica, Vol 59, 1161-1167. +Potcher, B. and J. Prucha (1989): “Uniform Law of Large Numbers for Dependent and +Heteregeneous Processes,” Econometrica, 57, 675-683. +President’s Working Group (2021): “President’s Working Group on Financial Markets Re- +leases Report and Recommendations on Stablecoins”, https://home.treasury.gov/news/press- +releases/jy0454. +Sprott, J.C. (2003): “Chaos and Time-Series Analysis”, Oxford University Press, Oxford +Sprott, J.C. (2014): “Numerical Calculation of Largest Lyapunov Exponent”, working +paper, University of Wisconsin. +Wang, G., Ma, X., and H., Wu. (2020): “Are Stablecoins truly diversifiers, hedges, or Safe +Havens against traditional cryptocurrencies as their names?”, Research in International +Business and Finance, 54, p. 101-225. +Zakoian, J.M. (1994): “Threshold heteroskedastic models”, Journal of Economic Dynamics +and Control, 18, 931-955. + diff --git a/1tAyT4oBgHgl3EQfofi7/content/tmp_files/load_file.txt b/1tAyT4oBgHgl3EQfofi7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c837dc165d956afb465ef71923babec3b4b49543 --- /dev/null +++ b/1tAyT4oBgHgl3EQfofi7/content/tmp_files/load_file.txt @@ -0,0 +1,1257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf,len=1256 +page_content='Time-Varying Coefficient DAR Model and Stability Measures for Stablecoin Prices: An Application to Tether Antoine Djogbenou,∗ Emre Inan,† Joann Jasiak‡ This Version: January 3, 2023 Abstract This paper examines the dynamics of Tether, the stablecoin with the largest market capitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We show that the distributional and dynamic proper- ties of Tether/USD rates have been evolving from 2017 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We use local analysis methods to detect and describe the local patterns, such as short-lived trends, time-varying volatility and persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To accommodate these pat- terns, we consider a time varying parameter Double Autoregressive tvDAR(1) model under the assumption of local stationarity of Tether/USD rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We es- timate the tvDAR model non-parametrically and test hypotheses on the func- tional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the application to Tether, the model provides a good fit and reliable out-of-sample forecasts at short horizons, while being robust to time-varying persistence and volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, the model yields a simple plug-in measure of stability for Tether and other stablecoins for assessing and comparing their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Keywords: Stablecoins, Tether, Prices, DAR Model, Persistence, Time- Varying Parameters, Conditional Heteroskedasticity, Local Stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' JEL number: C58, C13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ∗York University, Canada, e-mail: daa@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ca †York University, Canada, e-mail: emreynan@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ca ‡York University, Canada, e-mail: jasiakj@yorku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The authors thank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gourieroux and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Kim and the participants of CMStatistics 2022 and Canadian Economic Association (CEA) 2022 meetings for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This project was supported by the Digital Currency Research Clusters Initiative, the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Social Sciences and Humanities Research Council of Canada (SSHRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='00509v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='EM] 2 Jan 2023 THIS VERSION: January 3, 2023 1 1 Introduction The total market capitalization of cryptocurrencies is currently over 1 trillion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' dollar, with the top three cryptocurrencies in terms of market capitalization being Bitcoin (BTC), Ethereum (ETH), and Tether (USDT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While Bitcoin and Euthereum are characterized by high price volatility, Tether is a stablecoin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' a cryptocurrency designed to maintain a stable price compared to other cryptocurrencies such as Bitcoin and Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is the first and by far the largest stablecoin in the market with the highest daily volume of over $100 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In order to achieve price stability, the value of Tether is pegged 1-to-1 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' There also exist other stablecoins with values to other currency or gold and managed by either a single authority (usually the service provider) or a network of participants (the whole protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Allen, Gu, and Jagtiani (2022) recently discussed how stable cryptocurrencies provide alternative financial instruments for market participants and how appropriately regulated crypto markets could allow increased public confidence and lead to growth and innova- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The November 2021 report by the US President’s Working Group on Financial Markets (PWG), the Federal Deposit Insurance Corporation (FDIC), and the Office of the Comptroller of the Currency (OCC) highlight various risks that need to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These include user protection and run risk, payment system risk, systemic risk and con- centration of economic power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' They provided various recommendations, including the requirement for stablecoin issuers to be insured depository institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' See President’s Working Group (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Furthermore, Li and Mayer (2021) show that collateralized stablecoins like Tether could create systemic risk if the issuer does not have enough reserve to maintain its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More recently, Chen, Qin, and Zhang (2022, page 5) pointed out the important role of Tether in the trading volume of Bitcoin compared to US dollars since 2017 and noted the limited reserve of Tether according to anecdotal evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Despite the increased interest in stablecoins and the recommendations of more scrutiny by regulators, these crypto assets’ stability is still ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, TerraUSD, an algorithmic stablecoin, collapsed in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This situation posits the need for predictability of stablecoin prices and easy tools for proactive assessment of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To address those issues, this paper made the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' First, we analyze THIS VERSION: January 3, 2023 2 the local Tether price from historical data and pin down important features in its dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These features include the local pattern of the mean and the conditional pattern of Tether price as well as the role of specific events in this dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Second, we develop a time- varying model for Tether price that incorporates these specificities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Third, we propose, based on the model, measures that can be used to assess the stability of stablecoins and mitigate risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More specifically, we examine the dynamics of Tether/USD rates and documents the time varying distributional properties of this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We apply local analysis methods to reveal the time varying mean, volatility and persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In particular, we observe periods when Tether rates deviate from the peg, which are often combined with increased volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' During those episodes, local persistence measures increase, suggesting unit root dynamics of Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Based on these findings, we consider an extension of the Double-Autoregressive (DAR) model, called the dynamic time-varying parameter Double-Autoregressive (tvDAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The DAR model [Ling (2004)] accommodates the conditional heteroscedasticity and nests the ARCH and the autoregressive of order one AR(1) models, including the unit root model with the autoregressive coefficient equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More specifically, the DAR is a nonlinear Markov 1 process, which becomes a stationary martingale when the autoregressive coef- ficient is equal to 1 [Gourieroux, Jasiak (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The DAR model, unlike the traditional autoregressive AR(1)-ARCH process, provides valid inference and consistent parameter estimators for the autoregressive coefficient values including 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The proposed extension to a deterministic time-varying parameters model relies on the assumption of local strict sta- tionarity of the process, following the approach of Dahlhaus (2000) and Dahlhaus, Richter, Wu (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, during the episodes of unit root dynamics, the process satisfies locally the stationary martingale condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The time varying tvDAR model provides a good fit to the Tether/USD rates and gives reliable one step ahead out-of-sample predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To obtain the empirical results, we employed a rectangular kernel and an Epanechnikov ker- nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The first is an asymmetric kernel, which permits the incorporation of past information in a pre-specified window and could be used for out-of-sample prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The second is a symmetric kernel that uses information around any time period and is more suitable for inference on the parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, the tvDAR model provides a simple plug-in measure of stability for sta- THIS VERSION: January 3, 2023 3 blecoins, based on the Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This measure is commonly used to assess the stability of deterministic dynamical systems and to test for chaos [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Sprott (2014)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The Lyapunov exponent for the AR(1)-ARCH model has been determined by Borkovec and Kluppenberg (2001), and shown to be the condition of strict stationarity of that process [see also Borkovec (2000)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It has been also considered by Nelson (1990) in the context of the IGARCH model and by Cline and Pu (2004) in a non-parametric framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The Lyapunov exponent was also used as a stability measure in application to the Vector Autoregressive VAR model by Dechert and Gencay (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Those authors have introduced an alternative stability measure based on the noise-to-signal ratio for linear dynamic models [see also LeBaron (1994) for introduction to chaos].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In this paper, the sample Lyapunov exponent is computed from the model parameter estimates and proposed as a measure of stability for stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' A more conservative measure, based on the condition of second-order stationarity is also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Both measures can be computed locally and used to assess the stability of a stablecoin over time, or to compare the stability of different stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The time-varying coefficient approach based on the assumption of local stationarity dis- tinguishes our approach from the literature that relies on the assumption of global strong stationarity of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, Baum¨ohl and Vyrost (2022) use high frequency data to compute a spectral density-based quantile dependence measure under a strict stationarity condition, which does not seem to be satisfied by Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Bianchi, Rossini, and Iacopini (2022) estimate a Bayesian VAR with stochastic volatility and Student-t dis- tributed shocks (BVAR-SV-t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the conditional volatility equation is constrained to unit root dynamics, which is inconsistent with the empirical evidence provided in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Section 2 describes the stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Section 3 dis- cusses the local dynamic analysis of the price of Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Section 4 discusses the modelling approach, estimation procedures and stability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Section 5 presents the empiri- cal results based on the estimation and inference on the DAR(1) and tvDAR(1) models, including the sample stability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Appendix A contains the technical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Simulation and additional empirical results on stability measures are relegated to Appendices B and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 4 2 Stablecoins This section defines stablecoins and discusses their classification, issuance, and redemption mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, we discuss how the market prices of stablecoins are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Definition and Classification of Stablecoins Stablecoins are a type of cryptocurrency designed to maintain a stable price and reduced volatility, compared to other cryptocurrencies such as Bitcoin and Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Conven- tionally, stablecoin companies peg the value of their coins to that of a physical asset such as a fiat currency or gold with the assumption that the market price of their coins will eventually stabilize, establishing equivalency with the reference asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The strategies used to achieve price stability of stablecoins are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' There is currently no standard in place that private enterprises should comply with to qualify as a legitimate stablecoin company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This leaves stablecoin enterprises with un- limited design options to choose from to differentiate their business models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Currently, business models of stablecoin companies differ in their economic design, the quality of backing they maintain, stability assumptions they rely on, and legal protection they pro- vide for coin holders (Catalini and de Gortari, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While the underlining business models may be diverse and complex, there is interest in the elements of such models to understand their economic implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, one element of interest is the mechanism stabelcoins rely on to stabilize price and another is how the responsibilities are distributed over stablecoin protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' There exist two alternative mechanisms used by stablecoin companies to achieve price stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' They either hold collaterals in their reserves to back the value of their coins or they adjust the supply of coins through software codes to restore the peg with the reference asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When the market value of a cryptocurrency is backed by collaterals, the cryptocurrency is referred to as a collateralized stablecoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Conventionally, collateralized stablecoins are split into two sub-categories including off-chain collateralized stablecoins and on-chain collateralized stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Off-chain collateralized stablecoins are backed by a set of collaterals that have an economic value outside of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The reserves of this type of stablecoins usually consist of a fiat currency such as the US dollar for Tether (USDT) or a commodity such as gold for PAX Gold (PAXG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Stablecoins are labelled as THIS VERSION: January 3, 2023 5 on-chain collateralized if the underlying collaterals are composed of crypto assets that are created in a digital form and recorded on a distributed ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, Dai (DAI), the largest on-chain stablecoin project, supports 18 collateral assets including not only cryptocurriencies such as Ethereum (ETH) and Chainlink (LINK) but also stablecoins such as Tether (USDT), USD Coin (USDC), TrueUSD (TUSD) and PAX dollar (USDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Some projects opt for developing software codes to minimize price fluctuations instead of collateralizing their coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This type of cryptocurrencies is called an algorithmic stable- coin as they try to stabilize their price around the peg by contracting or expanding the coin supply with the help of computer algorithms embedded in their design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' TerraUSD (UST) was until May 2022 the only example of an algortihmic stablecoin that has a market capitalization over a billion US dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In terms of distribution of responsibilities, stablecoins can be categorized as centralized or decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Centralized stablecoins rely on a single legal entity to maintain the price stability, to manage and protect the collaterals, and to fulfill its obligations to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, Tether Limited is the legal entity that has the authority as well as the respon- sibility over every Tether in the circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Unlike centralized stablecoins, decentralized stablecoins distribute these responsibilities within their network through smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This allows network participants to take an active role in determining the rules of the stablecoin protocol such as the set of eligible collaterals and the minimum collateral re- quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The decentralized stablecoin DAI grants users who hold its governance token Maker (MKR) the right to vote on the changes to its protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li and Mayer (2021) noted that the introduction of stablecoins is comparable to “the unregulated creation of safe assets to meet agents’ transactional demands” known as shadow banking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Unlike stablecoin issuers, shadow banks must play the role of credit guarantees in the case of insolvency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, as we will see later, the observed prices of stablecoins tend to deviate from the peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, these crypto assets face multiple risks including the risk of liquidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For stablecoins designed to be on par with a fiat currency, the use of reserve allows the stablecoins issuers to sell or buy the currency to achieve its price stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This mechanism helps stablecoin companies to underpin the market value of their coins and protect against the highly volatile nature of the cryptocurrency markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It resembles fixed exchange rate regimes currently implemented in Panama, Qatar, and Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the THIS VERSION: January 3, 2023 6 fixed exchange rate regime, the central bank also uses its foreign reserves to buy or sell its domestic currency to maintain the fixed parity with the currency peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When the reserve system fails, the domestic currency can be devaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Lyons and Viswanath-Natraj (2020) documented that contrary to central banks with some macroeconomic mandate, including keeping inflation around its target, stablecoin issuers do not have any policy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, stablecoin companies cannot use the interest rate or devaluation policy to control the exchange rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For our analysis, we focus on Tether, which is by far the primarily traded stablecoin in terms of market capitalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' As we will show later, Tether price tends to be noticeably affected by events in the crypto world, leading to deviations from the peg despite using reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Given the aforementioned significant risks for stablecoin holders, there is a need to develop appropriate tools to assess their predictability and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This paper develops a model based on the properties of Tether price and uses it to propose tests for its sta- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Before discussing the specificity of the cryptocurrency of interest and the modeling strategy, we provide further explanation on the issuance and redemption of stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Issuance and Redemption Issuance and redemption are the two fundamental market activities that determine the equilibrium quantity of a stablecoin in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The equilibrium quantity goes up when new coins are issued, and it goes down when existing coins are redeemed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While the equilibrium quantity changes with issuance and redemption, the price of a stablecoin is held constant during these transactions by the service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This constant price policy is the result of the pegging strategy explained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Issuance and redemption of stablecoins are presumably initiated by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 How these transactions are executed depends on whether stablecoin has a centralized or decentralized structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Issuance takes place following the transfer of funds by user to the stablecoin enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Depending on whether stablecoin is centralized or decentralized, these funds are deposited either into banking accounts of a custodian or into a cryptographical vault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, an individual or a business who wants to buy Tether should transfer the funds, specifically the US dollar, to Tether Limited’s accounts at Cathay Bank and Hwatai 1See Griffin and Shams (2020) for further discussion on whether Tether issuances are supply-driven or demand-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 7 Bank in Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The collection of these funds constitutes the reserves of the stablecoin enterprise, and they are meant to be kept as a collateral to back the value of every coin in the circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Once the funds are successfully deposited, stablecoins are issued through smart contracts and credited into the user’s wallet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For centralized stablecoins, it is the issuer or the agent that authorizes the issuance of the coins whereas it is done automatically by the blockchain technology for decentralized stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Redemption of stablecoins is also initiated by user but the difference is that the trans- actions take place in the reserve order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To redeem stablecoins, users place an order on the blockchain to exchange their stablecoins for the collateral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Upon the order, the stablecoin enterprise becomes obliged to withdraw the stablecoins from circulation and give the user the corresponding amount of collateral in return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The stablecoins that the user redeems are destroyed subsequently from the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Stablecoin projects pledge in their whitepa- pers that their coins are 100% redeemable and redemptions can be performed any time users want at the predetermined price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence, one can argue that redeemability becomes the liability of stablecoin enterprises and plays a key role in the sustainability of their projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is the issuer that is liable to users for redemptions in centralized stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Decentralized stablecoins, on the other hand, have no single legal entity that shoulders the responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is the whole network that is responsible for undertaking redemptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 Market Price of Stablecoins Stablecoin prices are usually fluctuating around the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While their value in terms of the reference asset is fixed during issuance and redemptions, they are often traded at a premium or a discount on the exchange platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 This splits the market for stablecoins between the primary market and the secondary market, which can be considered analogous to the market for traditional securities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The primary market for stablecoins is where stablecoins are created (issuance) or destroyed (redemption) at the fixed exchange rate predetermined by the stablecoin initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The secondary market is where the users trade stablecoins within and across cryptocurrency exchanges such as Binance, Coinbase Exchange, Kraken and Bitfinex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The price of stablecoins in the secondary market is determined as a result of the market activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' According to Griffin 2See Lyons and Viswanath-Natraj (2020) for the detailed analysis of premium and discount on stablecoin prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 8 and Shams (2020), the secondary market activities account for most of the aggregate Tether flow from 2014 to 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence, the price of Tether and other stablecoins in the secondary market can be considered as the effective rate at which individuals or businesses can buy and sell stablecoins on a day-to-day basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' At first glance, the design elements stands out as the primary mechanism through which stablecoin projects try to achieve the price stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the price stabilization in the stablecoin market could be multifaceted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, Lyons and Viswanath-Natraj (2020) argue that the price gap between the primary market and the secondary market, which corresponds to the deviation from the peg, can be mitigated also by arbitrage activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' As long as investors have access to the primary market, the price deviations in the secondary market creates an opportunity for them to make profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When the price of a stablecoin in the secondary market is above the peg, the arbitrager can buy the coin at the target exchange rate from the primary market and sell it in the secondary market to make profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This increases the supply of the stablecoin in the secondary market, so it puts downward pressure on its price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Similarly, when the price of a stablecoin in the secondary market is below the parity, an arbitrager can buy the coin from the secondary market and redeem it at the peg ratio in the primary market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In this case, the demand from the arbitrager puts upward pressure on the secondary market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence, one could expect that the price of stablecoins across cryptocurrency exchanges would stabilize around the peg through arbitrage activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, introduction of Tether to the Ethereum blockchain in April 2019, which is associated with increased direct access of investors to the primary market, is found to have a stabilizing effect on the price of Tether in the secondary market (Lyons and Viswanath-Natraj, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Bullman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2019) provides the list of alternative tools that each type of stablecoins can adopt to maintain the peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The list consists of fees, redemption limits, and penalty fees to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Fees and redemption limits could be used by collateralized stablecoins to limit the users’ transactions and prevent sudden liquidations while penalty fees can help maintain the minimum level of collateralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, Tether Limited imposes the minimum amount of 100,000 USD required for a fiat withdrawal or deposit and charges the greater of $1,000 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1% fee per fiat withdrawal and per fiat deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 The stabilization strategies of Tether have not always been fully successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The next 3https://tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='to/fees/ THIS VERSION: January 3, 2023 9 section presents empirical evidence based on the dynamic analysis of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 3 Tether Dynamics This section examines the patterns in Tether dynamics in the sample of T = 1361 daily closing prices recorded between November 9, 2017, and July 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure 1 displays the evolution of daily closing rates of the Tether against US Dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observe the episodes of explosive dynamics mixed with more stable periods as well as the convergence of the process at the end of the trajectory to a smooth process taking values close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The convergence of Tether towards the peg and its reduced range after 2021 are associated with increased volume displayed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='08 Daily Price Figure 1: Tether/USD daily closing rates During the sampling period, the lowest and highest price were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9666 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0779, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Although 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0334 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0779 deviations from the one US dollar parity may look small, they can provide important arbitrage opportunities if the investor is holding a large position in the crypto asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While the mean over this period is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='022 and close to one, as expected, the volatility around the mean is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The evolution of the price shows an alternate of relatively large and small deviation in the stablecoin price due to changes in its demand and lags in intervention by Tether to maintain price stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 10 The fluctuation of the Tether price around the one-dollar peg can be connected with the European snake in the tunnel currency system created in April 1972 by an agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To increase the convergence among the different currencies in the European Economic Community (EEC), the agreement objective was to create a single currency band within which all the EEC currencies could fluctuate and not deviate too much from a peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The peg was defined using first gold and, later on, the US dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More details on the system can be found in Day (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To achieve stability around the peg, central banks had to use their reserve to intervene by buying or selling local currencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The system was difficult to sustain as several currencies left the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Although stablecoin issuers do not have a macroeconomic policy function as central banks, the difficulties in maintaining the snake currency system also speak to the challenge of maintaining stablecoin prices around its peg using the reserve system discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To understand the price movements of Tether, we first discuss factors that affect its demand during the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure 2: Time varying volume of Tether The daily volume series in Figure 2 exhibits larger fluctuations during the year 2021 of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Although the overall trend was positive, the daily volume varied Volume in million USD (USDT) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 Jui 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022THIS VERSION: January 3, 2023 11 roughly between $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 billion and $279 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The highest daily volume of $279 billion was achieved on May 19, 2021, when the Chinese government cracked down on its domestic market for cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Later, the daily volume plunged to as low as $33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7 billion in July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The high levels of daily volume in 2020 and 2021 could be explained by an increasing interest from investors as the market for cryptocurrencies grew substantially during the initial stages of the Covid-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Tether’s daily volume increased drastically from less than $10 billion in 2018 to as high as $279 billion in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' While the daily volume is observed to be increasing almost steadily in 2018 and 2019, it exhibits rather a volatile pattern in 2020 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, in early 2021, the daily volume of Tether more than doubled in a matter of a few months and reached a peak of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 billion USD on the March 13th, a day after the infamous “Crypto Black Thursday”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the pattern in Figure 2 indicates that the daily volume of Tether decreased between May and July of 2021 and returned to its pre-pandemic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The convergence to reduced range and small variation around the constant value of 1 occurs first in Tether in May 2018 and is interrupted by the end of September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the environnement of Tether, the convergence is observed simultaneously for other mostly traded traded stablecoins such as USD Coin, Binance USD, True USD, and Pax Dollar starting from July 2020 and in Dai starting from December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' During this period, these stablecoins displayed a period of improved stability towards the end of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Also, the Bitcoin and Etheureum prices in US Dollars have increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This period of stability overlaps with the period of bullish run in the cryptocurrency market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The cryptocurrency market indices such as the S&P Cryptocurrency Broad Digital Market (BDM) Index recorded more than fivefold increase between September 2020 and May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 The strong demand for cryptocurrencies also benefited the stablecoin companies as the total market capitalization of the top 10 stablecoins went up from approximately $20 billion in September 2020 to slightly over $100 billion in July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 The stability is interrupted again for all stablecoins when the cryptocurrency market shrunk by over $300 billion on April 17, 2021 in less than 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 While the waves 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='spglobal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/spdji/en/indices/digital-assets/sp-cryptocurrency-broad-digital-market- index/#overview 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/statistics/1255835/stablecoin-market-capitalization/ 6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/sites/jonathanponciano/2021/04/18/crypto-flash-crash-wiped-out-300- THIS VERSION: January 3, 2023 12 of sell-off caused the price of Bitcoin to plummet by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5%, the price of all stablecoins increased simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This can be evidence in favor of the previous studies which suggest that stablecoins could provide hedging opportunities for cryptocurrency investors against Bitcoin’s volatility, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Wang and Wu (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, one could also argue that the risk mitigating properties of stablecoins, which are closely linked to the comovements between the price of stablecoins and that of Bitcoin, could be changing locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For exam- ple, stablecoins showed resilience against even a much stronger market crash in mid-May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' On May 19, 2021, the Chinese government announced that the banks in China are banned from providing cryptocurrency services to their clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7 The market reacted to this news almost immediately as Bitcoin shed 30% of its value over the course of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' During the crash, all stablecoins except for Terra USD managed to keep their price stable around the one-dollar peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Local Analysis of Tether Price This subsection analyzes the local dynamics of Tether price series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We examine its local means, variances, and autocorrelations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (a) Local mean and variance This section studies the evolution of Tether/USD rates and identifies local patterns in this series by considering time varying descriptive statistics computed by rolling over a window of 50 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' billion-in-less-than-24-hours-spurring-massive-bitcoin-liquidations/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='sh=7d60735b2c89 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/technology/2021/may/19/bitcoin-falls-30-after-china-crackdown THIS VERSION: January 3, 2023 13 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 Local Mean =1 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0 1 2 3 Local Variance 10 4 Figure 3: Local mean and variance for the price of Tether The locally estimated marginal mean µt and variance σ2 t are displayed in panels a) and b) of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 The figure’s top panel reports the local mean of the price series and shows its evolution over the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observe that: the local mean of Tether varies across sub- periods, and it displays local trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Especially, in the first half of the sampling period, a strong local trend is observed, which is interrupted by a return of the series to values close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, in August 2019, the local mean increases, which is akin to the pattern of financial bubbles observed in stock prices (rational stochastic bubbles as in Blanchard and Watson (1982) or intrinsic bubbles as in Froot and Obstfeld (1991)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' See Kortian (1995) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These patterns, however, disappear towards the end of the sampling period and the local mean becomes more stable and close to the target value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, there are periods where the target value of 1 falls within the confidence intervals of local means: April 19, 2018 to May 5, 2018, May 9, 2018 to June 16, 2018, 8The lower and upper bounds of the confidence interval at the 95% level are calculated under the iid assumption as ˆµt ∓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 � ˆσt n for each window of n days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 14 October 10, 2018 to October 17, 2018, January 2, 2019 to January 3, 2019, and April 2, 2020 to May 1, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The local variance of the price series is plotted in the bottom panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It varies over time, and its variation is much higher in the first half of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, in 2018, Tether had periods of high volatility from January to March as well as periods of low volatility such as from April to November.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' On the other hand, Tether is less volatile during the second half of the sampling period as the local variance takes smaller values except for a short period of increased volatility between mid-March and early May of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Although the rolling window approach helps detect local trends, it needs to be inter- preted with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For a window size of n days, the first n-1 observations in the dataset are eliminated due to rolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, using a longer rolling window (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', 100 days) may over-smooth the changes in the mean and variance as compared to a shorter window (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', 50 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore, we use the window of 50 days for further computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9 The distributional changes in Tether also concern the range and quantiles of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Overall, we observe that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The local mean is changing over time and is close to 1 between April and June 2018, and after January 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The variance is time varying and diminishes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2, we identify a series of events that are closely related to Tether, and provide a detailed explanation of the reason why those events could be the driving force behind the changes we observe in the local statistics of Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Event Analysis The dynamics of Tether are strongly influenced by events, which can be used to distinguish the episodes of distinct patterns in the local mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure 4 shows the evolution of the local mean and the local variance of Tether along with the series of events that can be important for the dynamics of Tether, which can 9Also, note that n=50 is large enough to estimate parameters within each window consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Fur- thermore, n=50 divided by the sample size of T=1361 is the bandwidth bT = n/T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0367 in our local analysis, which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the literature, an optimal choice should satisfy Tb3 T = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In our case, we have Tb3 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0675, which is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' See Dahlhaus, Richter, Wu (2019, page 1039) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 15 help explain the trend reversals in its local statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Total of 11 such events are identified including 7 events for the local mean and 4 events for the local variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In 2018, the local mean shows a downward trend for the most part of the year, except for a brief period of recovery between early May and Late September.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This should come as no surprise because 2018 was a very tumultuous year for Tether Limited and its business partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Tether Limited was being scrutinized by the media and scholars for the quality of its reserves and its close ties to the cryptocurrency exchange Bitfinex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More specifically, Tether was publicly accused for not holding enough reserves to back all of its coins in circulation and for manipulating the price of Bitcoin by pumping unbacked supply of Tether into the market through Bitfinex to buy Bitcoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Amid these controversies, the local mean of Tether is found to be decreasing for the most part of the year, which could be linked directly or indirectly to a shift in investor sentiment towards Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In 2018, there is also a short period of a slight upward trend in the local mean roughly between early May and late September.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In early May, the owners of Tether Limited made their first significant attempt to show their willingness to address the investors‘ concerns about the accountability of their business and that of Bitfinex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' On May 7, 2018, the cryptocurrency exchange Bitfinex officially announced that Peter Warrack, who worked previously at RBC Royal Bank for 20 years as an anti-money laundering specialist, joins their team as the Chief Compliance Officer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Upon this news, the local mean of Tether enjoys a period of recovery and hovers above the one-dollar peg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Nevertheless, the local mean of Tether starts to decrease once again in September and reaches its lowest level in November.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The downfall of Tether during this period could be triggered by the introduction of USD Coin (USDC) on September 26, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' USD Coin, which is also designed to be on par with the US dollar, relies on the business principles similar to that of Tether but it claims to offer its users an improved transparency in its business activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 16 Figure 4: Important events for the local mean and the local variance of Tether Note: This note provides a description of the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Peter Warrack: Peter Warrack was hired by Bitfinex as the Chief Compliance Officer on May 7, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' USDC launched: USD Coin was launched on September 26, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Partial Backing: Bloomberg suggested on December 12, 2018 that Tether could be fully backed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Partial Backing 2: On March 14, 2019, Tether made changes to its backing policy on its official website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Tether wins appellate: Bitfinex won a motion in the New York Supreme Court to delay sub- mission of its business documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' WHO Covid: WHO made an announcement on Twitter on December 31, 2019 to acknowledge the cases of pneumonia in Wuhan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Coinbase Outage: Bitcoin shed over 10% of its value in a matter of minutes on May 9, 2020, which was followed by an outage in the cryptocurrency exchange Coinbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Wire Deposits: Bitfinex “temporarily paused” EUR, USD, JPY, and GBP wire deposits on October 11, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' TRON: Tether went live on the Tron network on April 17, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Black Thursday: Black Thursday: Bitcoin’s price reduced by around 50% in less than a day on March 12, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Chain Swap: On June 22, 2020, Tether announced on Twitter that they would implement a chain swap for a sizable amount of USDT from Tron TRC20 to ERC20 protocol on June 29th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Local Mean (USDT) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 Warrack PIAOS OHM 6 uedde 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='015 no ieie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='985 Jui 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 ×10-4 Local Variance(USDT) TRON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Deposits 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 Jui 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022THIS VERSION: January 3, 2023 17 Tether makes up for the loses quickly as the local mean increases remarkably from its lowest level in November 2018 to its highest level in January 2019 just under two months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The surge in the mean price of Tether could be a sign of positive reaction from the investors as the bank statements obtained by Bloomberg showed that on the contrary to the allegations, Tether Limited could be holding enough reserves to back its coins in the circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='10 However, this positive attitude towards Tether does not seem to last long as the local mean diminishes once again starting in January 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The downward pressure on the local mean was so strong during this period that it persisted up until August 2019 despite Tether’s effort to be more transparent about the composition of its reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' According to coindesk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com,11 Tether changed the terms on its website in Mid-February implying that while every Tether is always 100% backed by their reserves, the reserves are not necessarily composed of only fiat currency as they claimed before but also includes cash equivalents, other assets, and receivables from loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This update, however, did not stop the New York State Attorney General (NYSAG) sue Bitfinex and Tether Limited on April 24, 2019 on the basis of an ongoing fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='12 As the downward trend in the local mean dies out in August 2019, another episode of increasing local trend is observed subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The trend becomes more noticeable around September 24, 2019 when iFinex Inc, the parent company of Bitfinex and Tether Limited, won a motion in the court against NYSAG meaning that the company does not have to hand over the documents related to its business activities until further notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='13 Following this small victory in the court, the local mean of Tether diverges from the peg and keeps increasing until the end of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' On December 31, 2019, the World Health Organization (WHO) announced via its official Twitter account that they were informed of cases of pneumonia of unknown cause in Wuhan City, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='14 With this announcement, WHO acknowledged the problem of an epidemic in China, which would soon turn into a global health and economic crisis of 10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/news/articles/2018-12-18/crypto-mystery-clues-suggest-tether-has-the- billions-it-promised 11https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='coindesk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/markets/2019/03/14/tether-says-its-usdt-stablecoin-may-not-be-backed- by-fiat-alone/ 12https://ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='gov/press-release/2019/attorney-general-james-announces-court-order-against-crypto- currency-company 13https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/sites/michaeldelcastillo/2019/09/24/bitfinex-and-tether-win-appeal-from- new-york-supreme-court-in-900-million-case/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='sh=7c8bf41132bc 14see https://twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/who/status/1213795226072109058?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='lang=en for the original tweet from WHO THIS VERSION: January 3, 2023 18 Figure 5: Autocorrelation functions of Tether over different periods COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Subsequently, the local mean of Tether starts to decline cancelling out the gains from the last quarter of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 Persistence in Tether Series Let us now examine the serial correlation in Tether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The autocorrelation function (ACF) of Tether is computed from the entire sampling period 2017-2021 as well as from each calendar year separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure 5 presents the computed ACF functions: the ACF over the entire period (panel (a)), and in years 2017-2021 in panels (b) to (e), consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The ACF calculated from the entire sample exhibits a long range persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the subperiod analysis reveals that the persistence in the ACF of Tether is strong up to and including 2019,15 whereas the series has a short memory in 2020 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When combined with the results from the local statistics, it can be inferred that the period of long-range persistence in Tether coincides with the period of level shifts and high volatility as documented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Likewise, when the variation is small and the local mean stabilizes around the one-dollar peg as in 2020 and 2021, Tether displays a short memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 15In 2017, there are only 53 observations, which could be the reason for the weak evidence for the persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Panel (a): 2017-2021 Panel (b): 2017 Panel (c): 2018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 20 10 15 20 10 15 10 Lag Lag Lag Panel (d): 2019 Panel (e): 2020 Panel (f): 2021 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 29 10 15 20 5 10 15 10 15 Lag Lag LagTHIS VERSION: January 3, 2023 19 In brief, the empirical results show that the analysis of Tether based on global statistics would provide unreliable results, especially concerning the serial correlation of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, different values and range of serial correlation are obtained in year 2021, as compared to years 2018-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Let us now focus on the autocorrelation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More specifically, the autocorrela- tion at lag one of the series can be estimated from the autoregressive coefficient of an autoregressive of order 1 (AR(1)) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We first consider the autoregressive coefficient estimated from the AR(1) model fitted to the whole sample of demeaned Tether prices xt = yt − µ xt = ρxt−1 + σeet, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1) where et is assumed to be a white noise with mean 0 and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The AR(1) process is stationary when |ρ| < 1 and nonstationary and explosive when ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1) is estimated globally by the OLS, or equivalently by maximizing the Gaussian Maximum Likelihood as follows: ˆθT = Argmaxθ T � t=1 l(xt|xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = Argmaxθ T � t=2 −1 2 � log � 2πσ2 e � + (xt − ρxt−1)2 σ2e � where θ = (ρ, σ2 e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimated parameter values are ˆρT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='674 and ˆσ2 e,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Next, model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1) is fitted locally and estimated by rolling with a window of length 50 and displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The top panel of Figure 6 shows the autoregressive coefficient/correlation at lag 1 estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observe that the autoregressive coefficient is close to 1 during the explosive episodes in 2018 and 2019, which violates the stationary condition of the AR(1) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The unit root dynamics of Tether resembles the stock prices and exchange rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It suggests that Tether is then locally efficient in financial terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The autocorrelation at lag 1 is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 at the end of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In comparison with the results presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1, estimating the autocorrelation at lag one based on the AR(1) model gives us greater flexibility to assess the change in the persistence of Tether as we are able to examine its evolution at a daily frequency rather than on a yearly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For THIS VERSION: January 3, 2023 20 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='014 Figure 6: AR(1) parameter estimates and conditional volatility for xt example, the estimated values of the AR(1) coefficient ρ suggest that the autocorrelation at lag one ranges between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='20 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 in 2018 whereas in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1, the ACF at lag 1 was estimated to be slightly over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='76 for the entire year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The bottom panel shows the conditional volatility ˆσe(t) = � 1 50 �t τ=t−49(xτ − ˆρ(τ) xτ−1)2 of the price of Tether under the AR(1) assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The figure shows that this price ex- hibits periods of low volatility and high volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Especially from October 2020 onwards, the conditional volatility decreases remarkably and becomes very close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This result is consistent with the period of stability we observed in the price series of Tether during the cryptocurrency bull market explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' If the volatility was computed over the full sample we would have a constant estimate which does not capture the changes in volatility over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To accommodate this feature, we consider the time-varying volatil- ity model which also allows us to have valid inference when the estimated correlation parameter is close to one unlike the AR(1) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 Properties of rolling estimators Let us now examine the properties of the rolling estimators of time varying parameters written as deterministic functions of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' First, we estimated by rolling the time varying marginal mean and variance functions, hoping to approximate m(t) and σ2(t) in a simple model yt = m(t) + σ(t)ut, under the simplifying assumption of Normally distributed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' process ut with mean 0 and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, in finite sample ˆm(t) = 1 50 t � τ=t−49 yτ ∼ N �m(t) + · · · + m(t − 49) 50 , σ2(t) + · · · + σ2(t − 49) 250 � We see that ˆm(t) is biased of m(t) towards an integrated mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' By the same argument it can be shown that ˆσ2(t) is biased of both σ2(t) and the integrated variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Let us now consider the time varying parameters θ(t) = (ρ(t), σ2 e(t)) of a time varying parameter AR(1) model: xt = ρ(t)xt−1 + σe(t)et, where xt = yt −m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, the normality-based MLE estimator ˆθ∗(t) obtained by rolling can be written as the following kernel MLE estimator (see Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1998) for the method of local kernel-weighted likelihood estimation using local polynomial fitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ˆθ∗ T (t) = Argmaxθ 1 50 t � τ=t−49 l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = Argmaxθ 1 50 T � τ=1 1t−49≤τ≤t l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = Argmaxθ T � τ=1 � 1 501−49≤τ−t≤0 l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) � = Argmaxθ T � τ=1 1 50 1−49/50≤(τ−t)/50≤0 l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = Argmaxθ T � τ=1 1 50 K �τ − t 50 � l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ), t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T THIS VERSION: January 3, 2023 22 with the kernel K(u) = 1[−1,0](u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is easy to see that the dimension of the parameter of interest [θ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', θ(T)] depends on the number of observations T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To circumvent this difficulty, we can replace the functional parameter [θ∗(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', θ∗(T)] with t ∈ N by an alternative functional parameter θ(c), c ∈ (0, 1) on [0,1], such that θ∗(t) = θ(t/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The rolling MLE is such that ˆθ∗ T (t) = ˆθT (t/T) = Argmaxθ T � τ=1 T 50 K �τ/T − t/T 50/T � l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ), t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This formula can be extended to any value of argument c ∈ [0, 1]: ˆθT (c) = Argmaxθ T � τ=1 T 50 K �τ/T − c 50/T � l(xτ|xτ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ), c ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Dahlhaus (2000) and Dahlhaus, Richter, Wu (2019) show that under regularity conditions the functional parameters θ(c) of a locally stationary process can be consistently estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Instead of considering the time varying parameters in calendar time, we have now defined the functional parameters in a deformed time t → t/T that depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The functional parameter is now independent of the observations, while the effect of T is introduced by considering a triangular array approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This leads to a sequence of models indexed by T: xt,T = ρ(t/T)xt−1,T + σe(t/T)et,T , where xt,T = yt,T − m(t/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This approach motivates our modelling approach presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4 DAR(1) Model for Tether Price To account for time-varying conditional mean and volatility, we introduce the Double Autoregressive (tvDAR) process of order 1 with time varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The first part of this section recalls the constant parameter DAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Next, the estimation of both type of models is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The last part of this section presents the stability measures and their estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Model with Constant Parameters We consider the DAR process of order 1 for the demeaned Tether price series: xt = φxt−1 + ηt � ω + αx2 t−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2) where ω > 0, α > 0, and ηt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T is an independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=') sequence with mean 0 and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The parameter φ captures the conditional mean dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Parameter α represents the past dependence in the conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The model is semi-parametric and conditionally heteroskedastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='16 Borkovec and Kluppenberg (2001), Ling (2004) show that there exists a unique strictly stationary and ergodic solution to model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2) when the following assumptions hold: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1: ηt has a symmetric and continuous density with mean 0 and variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2: The parameter space is Θ = {θ = (φ, ω, α) : E(ln|φ+ηt √α|) < 0 with |φ| ≤ ˜φ, ω ≤ ω ≤ ˜ω, α ≤ α ≤ ˜α where ˜φ, ω, ˜ω, α, ˜α are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 is not a stringent assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is satisfied in particular if ηt, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T are normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 is the existence and negativity of the Lyapunov exponent ensuring the existence and uniqueness of a stationary solution [Borkovec and Kluppenberg (2001)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The region of φ, α that satisfy the negativity condi- tion is displayed in Figure 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 64 Ling (2004) and Figure 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 191 Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It includes cases when φ ≥ 1 as well as E(x2 t ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The processes xt that satisfy As- sumption 2 are strictly stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Some of these processes are also weakly (second-order) stationary and satisfy additionally the condition φ2 + α < 1 ensuring that E(x2 t ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Thus, the marginal variance of those processes is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When φ = 1, and E(ln |1 + ηt √α|) < 0, the process xt is a strictly stationary martin- gale process with volatility induced “mean-reversion” [Gourieroux, Jasiak (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2) is non-stationary when the Lyapunov exponent is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In particular, it is nonstationary at the boundary points (φ, α) = (±1, 0) and nests the standard unit root models at these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When φ = 0, the process is an ARCH(1) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, process (xt) is strictly stationary when x0 is drawn from a stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 16More on the models with conditional heteroscedasticity and their applications in finance can be found in Gourieroux (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Zakoian (1994) also proposed maximum likelihood and least squares estimators for conditionally heteroscedastic model with threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 24 Under assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2, the parameter space Θ is compact and there exists a unique strictly stationary solution of the model for any θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, we assume that: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3: The model is well-specified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' the process satisfies equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2) for the true value of parameter θ0 = (φ0, w0, α0) and the true density ψ0 of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The true parameter value θ0 is an interior point in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4: The observed process is the unique, strictly stationary solution asso- ciated with (θ0, ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These two conditions are introduced for the identification of the model and parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Model with Time-Varying Parameters The DAR(1) model can be extended to a time-varying parameter model by using the triangular array approach for locally stationary processes [Dahlhaus (2000), Dahlhaus, Richter, Wu (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The time varying tvDAR(1) model is written for locally demeaned observations xt,T , t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T indexed by t and T (triangular array) and defined by: xt,T = φ(t/T)xt−1,T + ηt,T � ω(t/T) + α(t/T)x2 t−1,T , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3) where for each time T, (ηt,T ) is a strong (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='d) white noise with mean zero, unit variance and a symmetric distribution invariant in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] are deterministic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We assume that these functions are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1: The functions φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ), ω(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ), α(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ), are positive, deterministic and twice differentiable on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, the trajectories of the process have to be little responsive to small changes of the parameters, which is ensured by a Lipschitz condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More precisely, let us consider process xt(c) defined by: xt(c) = φ(c)xt−1(c) + ηt(c) � ω(c) + α(c)xt−1(c)2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4) We assume that the following condition holds: Assumption a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2: THIS VERSION: January 3, 2023 25 Let the Lq norm for q > 0 be denoted by ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='||q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, i) For each c ∈ (0, 1), process {xt(c)} is stationary and ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ii) c → xt(c) is continuous for any t and ||supcxt(c)||q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' iii) There exists α, 1 ≥ α > 0 and CB > 0, such that ||xt(c) − xt(c′)||q < CB|c − c′|α uniformly in t and c, c′ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Under Assumptions a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2, if T is large and t/T in a small interval (c−ϵ, c), then the parameters are almost constant over that interval and locally model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4) is close to model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3) with φ = φ(c), ω = ω(c), α = α(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This explains the local stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When all the observations xt,T , t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T are considered, the variation of the pa- rameters prevents the DAR process from being globally stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, it is locally stationary, if Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 is locally satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' E[ln|φ(c) + ηt(c)α(c)|] < 0 for any c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This is the condition on the negativity of the local Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 Estimation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Estimation of the Model with Constant Parameters The parameter estimates of model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2) are obtained by maximizing the quasi-maximum likelihood (QML) objective function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' the log-likelihood function for normally dis- tributed ηt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' LT (θ) = −1 2 T � t=2 ln � ω + αx2 t−1 � − 1 2 T � t=2 (xt − φxt−1)2 � ω + αx2 t−1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5) where θ = [φ, ω, α]′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The QML estimators of Model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2): ˆθT = Argmaxθ∈ΘLT (θ) are consistent under Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4, and the vector of QMLE estimators ˆθT = [ˆφT ˆωT ˆαT ]′ → θ0 in probability, where θ0 = [φ0 ω0 α0]′ [Ling (2004,2007)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, if the following assumption: Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5: E(η4 t ) < ∞ is satisfied, Li and Ling (2008) and Chen, Li and Ling (2014) show that the Quasi Maxi- mum Likelihood estimators (QMLE) of θ are also asymptotically normal when |φ| ≥ 1 17 as well as E(x2 t ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 17The ML/OLS estimators of φ from a linear autoregressive AR(1) model with constant parameters are not asymptotically normal when φ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 26 √ T(ˆθT − θ0) → N(0, diag(Σ−1, κΩ−1)) where this convergence is in distribution, Σ = E0[x2 t−1/(ω0 + α0x2 t−1)] Ω = E0 � 1 (ω0 + α0x2 t−1)2 � 1 x2 t−1 x2 t−1 x4 t−1 �� , diag(Σ−1, κΩ−1) denotes the block-diagonal matrix with Σ−1 as the upper left block and κΩ−1 as the bottom right block and κ is the kurtosis less 1 of the distribution of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In particular, κ = 2 when ηt is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These asymptotic results are valid for any true distribution of η, not necessarily a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The consistent estimators of Σ and Ω are ˆΣT = 1 T − 1 T � t=2 [x2 t−1/(ˆωT + ˆαT x2 t−1)], ˆΩT = 1 T − 1 T � t=2 1 (ˆωT + ˆαT x2 t−1)2 � 1 x2 t−1 x2 t−1 x4 t−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The model residuals are defined as: ˆηt,T = (xt − ˆφT xt−1)/ � ˆωT + ˆαT x2 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The model residuals ˆηt,T , t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T allow us to estimate non-parametrically the error density to verify ex-post the symmetry assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The parameter κ is estimated by ˆκT = 1 T−1 �T t=2 ˆη2 t,T − 1 = 1 T−1 �T t=2 (xt−ˆφxt−1) 4 (ˆω+ˆαx2 t−1) 2 − 1 allowing us to accommodate the heavy tailed distribution of the stablecoin prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Estimation of the Model with Time-Varying Parameters Let us consider the locally stationary tvDAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The dynamic model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3) of triangular arrays xt,T , t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T is non-parametric and depends on the functional param- eters φ(c), ω(c) > 0, α(c) > 0, c ∈ [0, 1] and on the density function of the noise ηt,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimation of φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ), ω(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ), α(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=') can be done by the local-in-time QML estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We con- sider a kernel K defined on [−1/2, 1/2] and bandwith bT , bT > 0 (following the notation THIS VERSION: January 3, 2023 27 used in Dahlhaus, Richter, Wu (2019), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 1035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The local negative log-conditional quasi likelihood is LT,b(c, φ, ω, α) = 1 TbT T � t=2 K �t/T − c bT � � �−1 2 ln � ω + αx2 t−1,T � − 1 2 (xt,T − φxt−1,T )2 � ω + αx2 t−1,T � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6) Then, the local negative QML estimator of θ(c) = [φ(c), ω(c), α(c)] is: ˆθT,b(c) = ArgmaxθLT,b(c, φ, ω, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7) Under suitable regularity conditions given in [Dahlhaus, Richter, Wu (2019)], this functional QML estimator ˆθT,bT (c), c ∈ [0, 1] is consistent of θ(c), c ∈ [0, 1] and its limiting distribution is normal: � TbT (ˆθb(c) − θ0(c)) → N(0, � K(y)2dy J(c)−1I(c)J(c)−1), where → denotes the weak convergence of processes indexed by c, J(c) is the Hessian matrix and I(c) is the outer product of scores, both evaluated at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Under the symmetry assumption on ηt and for a strictly stationary and ergodic xt, the information matrix I(c) simplifies to a block diagonal matrix [Ling (2004), Remark 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The regularity conditions concern the functional parameter, the distribution of noise ηt, the kernel K and the bandwidth bT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' They can be found in Dahlhaus, Richter, Wu (2019), since the DAR model is a nonlinear autoregressive model discussed in Dahlhaus, Richter, Wu (2019), Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In particular, the bandwidth has to satisfy the conditions bT → 0, TbT → ∞, Tb3 T → 0 when T tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 Stability measures The Lyapunov exponent measures the average logarithmic rate of separation or conver- gence of initially close trajectories in chaotic systems, and the sensitivity to initial con- ditions, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' A negative value of the Lyapunov exponent indicates the stability of the dynamical system, while a positive value indicates chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The more negative the Lyapunov exponent, the more stable the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore, it is used for testing for chaos THIS VERSION: January 3, 2023 28 [Sprott (2003)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In this section, the Lyapunov exponent is proposed as a measure of stabil- ity of Tether and other stable coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the framework of the DAR model, the Lyapunov exponent is: λ = E(ln |φ + η√α|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The more negative the Lyapunov exponent, the less explosive the process18 and more likely its marginal variance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The behavior of E(ln |φ+ηt √α|) as a function of φ, α can be examined analytically for selected densities of η, and/or simulated and illustrated graphically [see, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2018) for graphical illustration].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 shows the analytical formula of λ for a uniformly distributed sequence {ηt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More precisely: Proposition 1: If η ∼ U[−1,1], the Lyapunov exponent is given by: λ(φ, α) = 1 2√α � (|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α| + ||φ| − √α| � Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This example clarifies that the Lyapunov exponent is a continuous function of φ, α, although with points of non-differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, it is easy to show that that E(ln |φ + ηt √α|) is always an even function of φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' it takes the same value for φ and −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To see that, consider a symmetric density function ψ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, E(ln | − φ + ηt √α|) = � (ln | − φ + ηt √α|)ψ(η)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Because ψ(η) is symmetric, we can change the variable η → −η: E(ln|−φ+ηt √α|) = � (ln |−φ−ηt √α|)ψ(−η)dη = � (ln |φ+ηt √α|)ψ(η)dη = E(ln |φ+ηt √α|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This proves that the Lyapunov exponent is even in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The Lyapunov exponent can be computed by plug-in from the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Let us first consider the constant parameter DAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The following estimators can be considered: a) Suppose that the true density function ψ = ψ0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, the estimator ˆλ1,T of the Lyapunov exponent is: ˆλ1,T = � ln |ˆφT + η � ˆαT | ψ0(η) dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 18A strictly stationary process can have infinite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 29 Proposition 2: When the density ψ(η) = ψ0(η) is known, then under assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 and the following condition: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6) ∃δ > 0, such that � sup φ0 − δ < φ < φ0 + δ α0 − δ < α < α0 + δ | ln |φ + η√α||ψ0(η)dη < ∞ the estimator ˆλ1,T converges in probability to the true value λ0 = � ln |φ0 +η√α0|ψ0(η)dη of the Lyapunov exponent when T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 b) When the density ψ(η) is unknown, the model is semi-parametric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The Lyapunov exponent can be estimated by the estimator ˆλ2,T such that: ˆλ2,T = 1 T T � t=1 ln |ˆφT + ˆηt,T � ˆαT |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore this estimator is equal to : ˆλ2,T = 1 T T � t=1 ln ������ ˆφT + xt − ˆφT xt−1 � ˆωT + ˆαT x2 t−1 � ˆαT ������ = 1 T T � t=1 g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ˆθT ) where g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = ln ����φ + xt−φxt−1 √ ω+αx2 t−1 √α ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We also introduce the notation GT (θ) = 1 T �T t=1 g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proposition 3: Let us introduce the additional conditions: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7) Eθ0g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) < ∞, ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8) Sufficient Lipschitz condition for stochastic equicontinuity: There exists a stochas- tic sequence BT with BT = Op(1) and an increasing function h from [0, ∞) to [0, ∞), continuous at 0 with h(0) = 0, such that for all ˜θ, θ ∈ Θ, |GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, under assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 and conditions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 the estimator ˆλ2,T = GT (ˆθT ) → G(θ0) = λ0 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The asymptotic distributions of estimators ˆλ1,T , ˆλ2,T cannot be obtained asymptot- ically from the Taylor series expansion because function φ, α → � ln |φ + η√α|ψ(η)dη THIS VERSION: January 3, 2023 30 does not satisfy the necessary differentiability assumption, as pointed out in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the distribution of ˆλ1,T , ˆλ2,T can be determined by simulations and used for hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' c) An alternative stability measure is ξ = φ2+α, which depicts the region of parameter space ξ < 1 where the marginal variance of xt remains finite, so that the process is both strictly and weakly stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In that sense ξ is a more conservative measure of stability than the Lyapunov exponent because there is a region of parameter values φ, α where the condition ξ < 1 no longer holds, while the condition λ < 0 remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimator ˆξT of ξ is: ˆξT = ˆφ2 T + ˆαT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proposition 4: Under Assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5, the estimator ˆξT converges in probability to ξ0 when T → ∞ and it is asymptotically Normally distributed: √ T(ˆξT − ξ0) A∼ N(0, Vξ), where the formula of the asymptotic variance Vξ is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The asymptotic variance provides the asymptotically valid standard errors that can be used to test the null hypothesis ξ < ξ0 using a Wald test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The interpretation of measure ξ is similar to that of λ: the smaller ξ, the more stable xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Model with time varying parameters The Lyapunov exponent λ2(c) can be estimated locally by computing ˆλ2,T (c) from the plugged in parameter estimates and residual values of the tvDAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, the Lyapunov exponent can be estimated from a kernel-weighted formula ˆλ2,T (c) = 1 TbT T � t=1 K �t/T − c bT � (ln|ˆφ(t/T) + ˆηt,T � ˆα(t/T)|), with the Epanechnikov kernel K(c) = 3 2(1−(2c)2) for c ∈ [−1/2, 1/2] and K(c) = 0 other- wise, which satisfies the regularity condition for localizing kernel [see Dahlhaus, Richter, Wu (2019, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 31 A similar approach can be used to estimate locally the measure ξ(c) = φ2(c) + α(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The local plug-in estimator ˆλ2,T (c) is illustrated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 5 Empirical Results This section presents the parameter estimates for the constant parameter DAR model and the time varying parametr tvDAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The time varying parameters DAR model is estimated first by rolling, which is equivalent to the use of an asymmetric rectangular kernel, as shown in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Next, the model is estimated by using a kernel which assigns higher weights to the observations close to the estimation date, providing consistent and asymptotically normally distributed estimates, which are used for testing hypotheses on the constancy of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimators of stability measures are also computed and illustrated graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimation of the DAR(1) process with constant parameters is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' First, we demean the price series of Tether by subtracting the total mean of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='0022 and then fit the model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 to the entire series of the demeaned prices, which gives the following result Table 1: Estimation of the DAR(1) model using the entire sample DAR(1) parameters φ ω α Estimates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='699 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='102e−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='484 Standard deviation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='034) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='174e−06) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='205) where the standard deviations of the parameters are obtained using the asymptotic dis- tribution described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In the following sections, the model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 is estimated by using two types of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The first one is the asymmetric rectangular kernel, equivalent to the rolling estimation over the window of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This window ensures good properties of the estimators, while preventing the over-smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 32 The second approach relies on the Epanechnikov kernel and produces consistent and asymptotically normally distributed parameter estimates used for hypotheses testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Rectangular kernel The tvDAR(1) is estimated from the demeaned Tether price by rolling, which is a common practice in applied literature, and a window of length 50 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This is equivalent to using an asymmetric rectangular kernel, K(u) = 1(−1,0)(u), bT = 50/T which is computationally simple, but does not satisfy the smoothness conditions ensuring locally the validity of asymptotic distribution of the QMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The rolling estimate and the confidence interval of the parameter of interest φ(t/T) is displayed in the first paned of Figure 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 =1 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='014 Conditional Volatility Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 6 5 4 3 2 1 0 Figure 7: tvDAR(1) parameter φ(t/T), conditional volatility and Lyapunov exponent λ2(t/T) THIS VERSION: January 3, 2023 33 The second panel shows the estimated conditional volatility � ˆω(t/T) + ˆα(t/T)x2 t for Tether price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The third panel in Figure 7 presents the local estimates ˆλ2,T (t/T) of the Lyapunov exponent computed by plugging in the local parameter estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observe that there are periods when the autoregressive coefficient is not signifi- cantly different from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For instance, this is the case between October and December 2018 and between February and May 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These results from the tvDAR (1) model confirm our initial observation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 that the price series of Tether shows strong persis- tence in 2018 and 2019 whilst allowing us to pinpoint its exact timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimated conditional volatility based on the estimated parameters has a pattern consistent with the local variance estimator in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The results in the third panel suggests that Assump- tion 2 holds and the Lyapunov exponent remains negative even for the highest recorded persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The sample Lyapunov exponent varies across time becoming on average more negative before the end of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This indicates that Tether achieves higher stability over that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Full Sample Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='08 y=1 Predicted E(yt+1|yt) y October 2018 to October 2019 Jul 2018 Oct 2018 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Jan 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='04 y=1 Predicted E(yt+1|yt) y Figure 8: Tether prices compared to one-step ahead out-of-sample forecasts THIS VERSION: January 3, 2023 34 The tvDAR(1) model estimated with the asymmetric rectangular kernel can be used for forecasting at short horizons, under the assumption that the parameter functions remain constant and equal to the last estimated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure 8 presents the observed Tether price and the estimate ˆyt+1 of the one-day-ahead conditional mean E(yt+1|yt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=') of the price of Tether using a rolling window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To get ˆyt+1, we add the local mean of Tether price to the estimated ˆφ using data over 50 days up to date t times xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The figure shows a close match between Tether price and its best prediction based on the previous day price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, the computed mean square prediction error is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7428 × 10−5 which is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Under the assumption of locally constant parameters, the 95 % asymptotically valid prediction intervals in Figure 9 are given by � ˆyt+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 √ T � x2 t ˆΣ−1, ˆyt+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 √ T � x2 t ˆΣ−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Full Sample Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='05 y=1 October 2018 to October 2019 Jul 2018 Oct 2018 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Jan 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='05 y=1 Figure 9: One-step ahead out-of-sample predicted Tether prices and prediction intervals Next, we conduct further investigations to analyze the goodness of fit of the tvDAR(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For this analysis, we keep the estimation window of 50 and perform the Ljung-Box test of THIS VERSION: January 3, 2023 35 white noise on ˆηt and ˆη2 t , while rolling the sample used for the test [see Li (1992), and Li and Mak (1994)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Because we consider all subsamples of 50 consecutive dates, it is likely that at some dates the serial correlation is not fully captured by the tvDAR(1) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Our results show that for most periods, residuals ˆηt and ˆη2 t are serially uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More precisely, we reject the null of no serial correlation for ˆηt only for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='52% of the subsamples, while we reject the null of no serial correlation for ˆη2 t only for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='08% of the subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The results suggest that the model captures most of the nonlinear serial correlation in Tether prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Epanechnikov kernel We now use a symmetric Epanechnikov kernel producing consistent and asymptotically normally distributed estimates for hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The first panel of Figure 10 plots the estimated time-varying autoregressive coefficient and its confidence band, while the second panel presents the time-varying estimates for of the Lyapunov exponent using the ˆλ2,T (t/T) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For the bandwidth, we choose bT = 50/T using the same window as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The results confirm that when the estimation is conducted locally over each period, the Lyapunov exponent displayed in the second panel of Figure 10 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence the critical validity condition holds for all the dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, the Lyapunov exponent is on average more negative at the end of the sampling period, confirming that Tether has achieved higher stability at the end of the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Furthermore, the estimated dynamic of estimated parameter φ for Tether price in the first panel of of Figure 10 remains similar to that of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' These findings confirm that Tether price has causal dynamics, as the autoregressive parameter and its confidence interval are mostly between −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, there are few dates when this estimate is not statistically different from 1, which suggests strong persistence in Tether prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 36 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 =1 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 7 6 5 4 3 2 1 0 Figure 10: Kernel-based parameter φ estimates and Lyapunov exponent λ(t/T) To detect the periods of strong persistence, we test the null hypothesis H0 : φ = 1 using the series of ˆφ(t/T) and its 95% confidence intervals for each t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We conduct the hypothesis test using the series of ˆφ(t/T) obtained from the estimation with the Epanechnikov kernel and report the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The results suggest that although Tether price is predictable most of the time, there are intervals of time periods where this tends not to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' During these episodes presented in Table 2, we find high persistence in Tether price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, as explained above, the proposed tvDAR approach remains valid and accommodates strong persistence in the price series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In light of the results in Figure 4, we further inspect the identified periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observed that the identified episodes in 2018 and 2019 overlap with the periods of high volatility observed in Figure 4, which ended in February 2020, but started around the introduction in September 2018 of USD Coin, another stablecoin designed to maintain price equivalence to the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, the episodes in 2020 match with a small rise in Tether price volatility by the end of July 2020, while the episodes in 2021 THIS VERSION: January 3, 2023 37 can be associated with the period of increased volatility at the end of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Table 2: Episodes of high persistence in Tether characterized by φ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Year From To 2018 September 24 October 29 December 3 October 5 2019 January 11 January 30 February 9 February 11 February 16 February 21 2020 July 25 August 8 August 16 August 19 2021 May 14 June 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 Test for Conditional Homoscedasticity Let us now consider a simple test of model specification of the tvDAR(1) model with time-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It is based on testing for the constancy of the variance function in model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 which is given by the following expression σ(t/T) = � ω(t/T) + α(t/T)x2 t−1,T , t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8) To test the null hypothesis H0 : σ(t/T) = σ0 ∀t, we consider the below test statistics proposed by Chandler and Polonik (2017) for time-varying autoregressive processes CPT = sup α∈[0,1] � T (γ(1 − γ)| ˆGT,γ(α) − αγ|, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9) where ˆGT,γ(α) = 1 T �[αT] t=1 1(ˆϵ2 t ≥ ˆq2 γ), THIS VERSION: January 3, 2023 38 ˆq2 γ = min � q2 ≥ 0 : 1 T � t∈[aT,bT] 1(ˆϵ2 t > q2) ≤ γ � , ˆϵt = xt − ˆφ( t T )xt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The process ˆGT,γ(α) counts the number of squared residuals within the first (100×α)% of the observations that are larger than the empirical quantile of the squared residuals denoted by ˆq2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The series of squared residuals is constructed by computing ˆϵt = xt − ˆφ( t T )xt−1 for each period t where ˆφ( t T ) in our case is the corresponding DAR(1) estimate obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Chandler and Polonik (2017) shows that the test statistics CPT in equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9) under the null hypothesis converges asymptotically to the supremum of a Brownian bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This asymptotic result is found to be still valid when the time-varying functions of the model parameters are estimated nonparametrically (see Chandler and Polonik (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' When computing the test statistics CPT , we consider different alternatives for the em- pirical upper γ-quantile for comparison, particularly γ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9) following Chandler and Polonik (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Given the choice of γ, we then calculate the expression � T (γ(1−γ)| ˆGT,γ(α)− αγ| for different values of α and report the results in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Table 3: The calculated values of � T (γ(1−γ)| ˆGT,γ(α) − αγ| for the given pairs of γ and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='922 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='302 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='466 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='328 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='251 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='957 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='759 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='837 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='539 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='912 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='478 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='390 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='233 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='937 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='471 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='327 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='562 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='031 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='593 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='155 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='993 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='923 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='485 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='047 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='490 The table shows that regardless of the choice of γ, the largest value is achieved when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' By the definition of supremum, the test statistics CPT should satisfy CPT ≥ � T (γ(1−γ)| ˆGT,γ(α)−αγ| for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence, it is safe to say that we have CPT ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='047 in the worst case scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e when γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9 is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='19 Compared with the critical 19Alternatively, we could fix our choice of γ and find the exact value of α ∈ [0, 1] at which the expression � T (γ(1−γ)| ˆGT,γ(α) − αγ| attains its maximum on this fixed interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This could slightly improve the THIS VERSION: January 3, 2023 39 values from the asymptotic distribution of the test statistics,20 this result leads us to the conclusion that we can reject the null hypothesis of constant variance function even at the 99% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In other words, we have a statistically significant evidence in favor of the alternative that the variance function is varying over time, which consequently justifies our strategy to estimate the model with time-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 6 Concluding Remarks We show that the distributional and the dynamic properties of stablecoins have been evolving over the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We implement local analysis to detect and describe local explosive patterns, time-varying volatility and persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We model the dynamic of the most important stablecoin which is Tether, and provide evidence that the tvDAR(1) model with time varying coefficients provides locally a good fit and reliable short-term predictions of Tether prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Our modelling strategy enables us to have valid inference even when the tvDAR(1) coefficient φt of Tether price is not locally different from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The sample Lyapunov exponent computed from the parameter estimates of the model provides a measure of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It confirms that at the end of the sampling period Tether becomes relatively more stable and allows for comparing the stability of Tether with other stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Appendix A: Technical Results This Appendix contains the proofs of Propositions 1, 2, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 Proposition 1 Because of the symmetry of the density of η, the Lyapunov exponent is an even function of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence we can suppose that φ > 0 to find the expression of the Lyapunov exponent, and then replace φ by |φ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For φ > 0 we have: precision of the lower bound for the test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example, when γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9, the expression attains its maximum value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='106 at α∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 20see https://homepages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='vuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='nz/ ray/Brownian/ for the distribution of the supremum of a Brow- nian Bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 40 λ(φ, α) = E ln |φ + η√α| = � ln |φ + η√α|ψ(α)dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Let us assume that η ∼ U[−1,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, its density is ψ(η) = 1 21η∈[−1,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We have: λ(φ, α) = 1 2 � 1 −1 ln |φ + η√α|dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We observe that: φ + η√α > 0 ⇐⇒ η > −φ/√α, φ + η√α < 0 ⇐⇒ η < −φ/√α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' λ(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α) = 1 2 � 1 −1 1η>−φ/√α ln(φ + η√α)dη + 1 2 � 1 −1 1η<−φ/√α ln(−φ − η√α)dη Let us now examine the two cases: a) If φ/√α > 1 ⇐⇒ −φ/√α < −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' we get: λ(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α) = 1 2 � 1 −1 ln(φ + η√α)dη + 1 20 = 1 2 1 √α � 1 −1 ln(φ + η√α)d(√αη) = 1 2√α � φ+√α φ−√α ln(u)du with change of variable u = φ + η√α = 1 2√αu ln(u) − u|φ+√α φ−√α = 1 2√α [(φ + √α) ln(φ + √α) − (φ + √α) − (φ − √α) ln(φ − √α) + (φ − √α)] b) If φ/√α < 1 ⇐⇒ −φ/√α > −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' we get: λ(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α) = 1 2 � 1 −φ/√α ln(φ + η√α)dη + 1 2 � −φ/√α −1 ln(−φ − η√α)dη = 1 2 1 √α � φ+√α 0 ln(u)du + 1 2 � 1 φ/√α ln(−φ + η√α)dη with the change of variable u = φ + η√α = 1 2√α � φ+√α 0 ln(u)du + 1 2√α � −φ+√α 0 ln(u)du = 1 2√α [(φ + √α) ln(φ + √α) − (φ + √α) − [(−φ + √α) ln(−φ + √α)] + (−φ + √α)] By putting the two expressions in a) and b) together we get for φ > 0: λ(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α) = 1 2√α � (φ + √α) ln(φ + √α) − (φ + √α) − |φ − √α| ln |φ − √α)| + |φ − √α| � The general expression of the Lyapunov Exponent without the sign constraint on φ is: λ(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' α) = 1 2√α � (|φ| + √α) ln(|φ| + √α) − (|φ| + √α) − ||φ| − √α| ln ||φ| − √α)| + ||φ| − √α| � We observe a non-differentiability in φ = ±√α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Moreover, for φ = 0, we get a zero value of the Lyapunov exponent: THIS VERSION: January 3, 2023 41 λ(φ, α) = 1 2√α �√α ln √α − √α ln √α + √α � = 0 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 Proof of Proposition 2 We need to show that when T → ∞, then � ln[ˆφT +η√ˆαT ]ψ0(η)dη → � ln(φ0+η√α0)ψ0(η)dη in probability if (ˆφT , ˆαT ) → (φ0, α0) in probability, and if condition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 of Proposition 2: ∃δ > 0, such that � sup φ0 − δ < φ < φ0 + δ α0 − δ < α < α0 + δ | ln |φ + η√α||ψ0(η)dη < ∞ (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proof of convergence: If (ˆφT , ˆαT ) → (φ0, α0) in probability, then they also converge in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It follows from the Skorokhod theorem that up to a change of probability space, we can assume that the almost sure (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=') convergence also holds [Billingsley (1999)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore, if g is a con- tinuous function of (φ, α), we have g(ˆφT , ˆαT ) → g(φ0, α0) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and in distribution in that new space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, g(ˆφT , ˆαT ) d→ g(φ0, α0) in the initial space and also in probability because the limit is constant, we get the ”in probability” version of the continuous mapping theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore, we need only the condition ensuring that g(φ, α) = � ln |φ+η√α|ψ0(η)dη is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6) ensures the continuity of integral function g, which follows from the dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3 Proof of Proposition 3 We first prove a general lemma, which is next applied to the DAR model and Lyapunov estimator λ2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Lemma Let us consider a sequence GT (θ) of stochastic functions of θ, θ ∈ Θ, and a sequence of estimators ˆθT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We assume that: i) Θ is compact and θ0 is in the interior of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' ii) ˆθT tends in probability to θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' iii) GT (θ) tends in probability to a limit G(θ), ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' iv) Sufficient Lipschitz condition for stochastic equicontinuity: THIS VERSION: January 3, 2023 42 There exists a stochastic sequence BT with BT = Op(1) and an increasing function h : [0, ∞) → [0, ∞) continuous at zero, with h(0) = 0 and such that for all ˜θ, θ ∈ Θ, |GT (˜θ) − GT (θ)| ≤ BT h(d(˜θ, θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, ˆGT (ˆθT ) tends in probability to G(θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Proof: We have : |GT (ˆθT ) − G(θ0)| = |GT (ˆθT ) − GT (θ0) + GT (θ0) − G(θ0)| ≤ |GT (ˆθT ) − G(θ0)| + |GT (θ0) − G(θ0)| ≤ BT h[d(ˆθT , θ0)] + |GT (θ0) − G(θ0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We know that if XT P→ 0, YT P→ 0 => XT + YT P→ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' the sum of op(1) is op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Under condition iii) |GT (θ0) − G(θ0)| = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' It remains to be shown that BT h[d(ˆθT , θ0)] is op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We have: [BT < M and h[d(ˆθT , θ0)] < ϵ/M] => [BT h[d(ˆθT , θ0)]] < ϵ] ⇐⇒ � (BT < M) ∩ [h[d(ˆθT , θ0)] < ϵ/M] � ⊂ [BT h[d(ˆθT , θ0)] < ϵ] Consider the complement: (A ∩ B)c = Ac ∪ Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We get: ((BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M) ⊃ [BT h[d(ˆθT , θ0)] > √ϵ] It follows that: P[BT h[d(ˆθT , θ0)] > ϵ] ≤ P[(BT > M) ∪ (h[d(ˆθT , θ0)] > ϵ/M)] ≤ P[BT > M] + P[h[d(ˆθT , θ0)] > h−1(ϵ/M)], because P[A ∪ B] ≤ P(A) + P(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Then, for any ϵ, we can choose a value of M and a number of observations T sufficiently large to get P[BT h[d(ˆθT , θ0)] > ϵ] arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Therefore, BT h[d(ˆθT , θ0)] tends to zero in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' QED Then, the lemma can be applied with GT (θ) = 1 T �T t=2 g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) and g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) = ln |φ+ xt−φxt−1 √ ω+αx2 t−1 √α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Under assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7, conditions i), ii), iii) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' For example: GT (θ) P→ E0g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ), by the weak law of large numbers applied to the transformation g(xt, xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' θ) of the ergodic stationary process (xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 corresponds to condition iv) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 43 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 Proof of Proposition 4 a) Proof of convergence We need to show that ˆφ2 T + ˆαT → φ2 0 + α0 in probability if (ˆφ, ˆα) → (φ0, α0) in probability when T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Thi is a consequence of the ”in probability” version of the continuous mapping theorem given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 b) Proof of Normality The Taylor series expansion pre-multiplied by √ T implies: √ T � (ˆφ2 T + ˆαT ) − (φ2 0 + α0) � = � 2φ0 1 �′ √ T � ˆφT − φ0 ˆαT − α0 � + op(1) = A′√ T � ˆφT − φ0 ˆαT − α0 � + op(1) where A′ = [2φ0 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We get the asymptotic normal distribution of ˆξT : √ T(ˆξT − ξ0) ∼ N(0, Vξ), where Vξ = A′Ω∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The matrix Ω∗ is: Ω∗ = diag(Σ−1, V (ˆα)) where Σ = E0(y2/(ω0+α0y2) given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Matrix V (ˆα) = (E0 1 (ω0+α0y2)2 )/ ˜V0(y2) and ˜V0(y2) = ˜E0(y4) − ( ˜E0y2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In this formula, ˜E0 denotes the expectation of variables y4 1 (ω0+α0y2)2 /E0 1 (ω0+α0y2)2 and y2 1 (ω0+α0y2)2 /E0 1 (ω0+α0y2)2 [see, section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1 and Ling (2004) for the variance estima- tor formula].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Appendix B: Simulation Results The purpose of this section is to illustrate the derived results in Appendix A using simu- lation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' We distinguish the case where the distribution of the innovation η is known and the case it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' First, we use the result in Proposition 1 and plot the Lyapunov exponent λ = E(ln(|φ+ √αη|)) for different values of the parameter φ and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' To do so, we assume η ∼ U[−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure B1 shows that the Lyapunov exponent λ remains lower than zero as as φ varies in {−1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8, 1} and α varies in {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='9, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Second, we assume η ∼ N(0, 1), set the true parameters to φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='7, α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5, ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Note that the parameters are chosen to be close to their estimated value from the entire data in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The estimated densities are based on 4, 000 simulations THIS VERSION: January 3, 2023 44 and obtained via kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure B2 plots the estimated density for the Lyapunov exponent ˆλ2,T and the stability measure ˆξT when the three parameters are estimated from a sample of size T and plugged in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The results on panel (a) of the figure show that the mostly frequent estimated value is below zero for T = 50 or T = 100, implying valid inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition, panel (b) of Figure B2 shows that the density of the estimated alternative stability measure ˆξT = ˆφ2 T + ˆαT has its mode around the true value of ξ, which is ξ0 = φ2 0 + α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' As a by-product, we present Figure B3, which shows the estimated density for ˆφT , ˆαT and ˆωT for T = 50 and T = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The three panels in the figure provide evidence that the three parameters are fairly accurately estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' More specifically, the estimated values have modes close to their true unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' The accuracy improves as the sample size increases from T = 50 to T = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure B1: Lyapunov exponent λ = E(ln(|φ + √αη|)) in terms of φ and α when η ∼ U[−1, 1] 0 2 4 ~ 6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5THIS VERSION: January 3, 2023 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 Density T=50 T=100 (a) Density of the Lyapunov exponent ˆλ2,T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 2 Density T=50 T=100 (b) Density of the alternative stability measure ˆξT Figure B2: Densities for stability measures based on estimated parameters THIS VERSION: January 3, 2023 46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 Density T=50 T=100 0 (a) Density of ˆφT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='005 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='035 0 20 40 60 80 100 120 140 160 180 Density T=50 T=100 0 (b) Density of ˆωT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 3 Density T=50 T=100 0 (c) Density of ˆαT Figure B3: Densities of estimators for φ, α and ω THIS VERSION: January 3, 2023 47 Appendix C: More Empirical Results Given that the proposed stability measure can be used as a mechanical tool to detect periods of instability in stablecoins, we use the results of Proposition 4 in Appendix A to construct an interval for ξ employing the same rolling window approach as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Figure C1 presents the results and contains, in its first panel, the estimated coefficient DAR model using the rolling windows approach, in its second panel, the conditional heteroskedasticity, and in the third panel, the Lyapunov exponent over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' In addition to the episodes of high persistence mentioned above, we observed, around September 2020, an important instability that is not due to high persistence in Tether price, but more frequent changes in the conditional heteroskedasticity, which can be seen in the second panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' This period can also be linked to higher local volatility in the observed data in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' There is no specific event we can associate with this movement, as is sometimes the case in crypto markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' However, the proposed model allows capturing those changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' As explained above, the measure of stability ξ plotted in Figure C1 is more conservative than the Lyapunov exponent λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Because we can have ξ ≥ 1 while λ < 0 so that valid inference is still possible, the rejection of ξ < 1 should be interpreted as the need for investors, regulators or stablecoin issuers to be cautious when predicting Tether future price around the tested periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 48 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='5 =1 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='014 Conditional Volatility Jul 2017 Jan 2018 Jul 2018 Jan 2019 Jul 2019 Jan 2020 Jul 2020 Jan 2021 Jul 2021 Jan 2022 4 2 0 2 4 6 Figure C1: tvDAR(1) parameter φ(t/T) and Lyapunov exponent ξ(t/T) THIS VERSION: January 3, 2023 49 References Allen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Jagtiani (2022): “Fintech, Cryptocurrencies, and CBDC: Finan- cial Structural Transformation in China”, Journal of International Money and Finance, Elsevier, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 124(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Andrews, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1987): “Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers”, Econometrica, 55, 1465-1471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Barry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Winkler (1976) : “Nonstationarity and Portfolio Choice”, The Journal of Financial and Quantitative Analysis, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 11, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 217-235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Bandi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Phillips (2009) : “Nonstationary Continuous-Time Processes”, in Hand- book of Financial Econometrics, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Ait Sahalia, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Hansen eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', 140-199, Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Baum¨ohl, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Vyrost (2020) : “Stablecoins as a crypto safe haven?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Not all of them!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', ZBW-Leibniz Information Centre for Economics, Kiel, Hamburg Bianchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Rossini, L and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Iacopini (2022) “Stablecoins and Cryptocurrency Returns: What is the Role of Tether”, Working Paper, University of Milan?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Billingsley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1999): “Convergence of Probability Measures”, New York, Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Blanchard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Watson (1982): “Bubbles, Rational Expectations, and Financial Markets”, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Wachtel (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=') Crisis in the Economic and Financial Structure, Lexington Books, Lexington, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Borkovec, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2000): “Extremal Behavior of the Autoregressive Process with ARCH(1) Errors”, Stochastic Processes and their Applications, 85, 189-207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Borkovec, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Kluppenberg (2001): “The Tail of the Stationary Distribution of the Autoregressive Process with ARCH(1) Errors”, Annals of Applied Probability, 11, 1220-1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Bullman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Klemm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=',Pinna (2019): “In Search for Stability in Crypto-assets: Are Stablecoins the Solution?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', European Central Bank Occasional Paper Series No 230 Catalini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=',de Gortari (2021): “On the Economic Design of Stablecoins”, Avail- able at SSRN: https://ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='com/abstract=3899499 or http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='3899499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 50 Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Ling (2014): “Non-Stationarity and Quasi-Maximum Likelihood Estimation on a Double Autoregressive Model”, Journal of Time Series Analysis, 35: 189– 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Qin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Zhang (2022): “Cryptocurrency price discrepancies under uncertainty: Evidence from COVID-19 and lockdown nexus,” Journal of International Money and Finance, Elsevier, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 124(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Chandler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Polonik (2012): “Mode Identification of Volatility in Time-Varying Autoregression”, Journal of the American Statistical Association, 107(499), 1217-1229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Chandler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Polonik (2017): “ Residual Empirical Processes and Weighted Sums for Time-Varying Processes with Applications to Testing for Homoscedasticity”, Journal of Time Series Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 38, 72-98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Dahlhaus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2000): “A Likelihood Approximation for Locally Stationary Processes”, Annals of Statistics, 28, 1782-1794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Dahlhaus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Richter and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Wu (2019): “Towards a General Theory for Nonlinear Locally Stationary Processes”, Bernoulli, 25, 1013-1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Day, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1976): “A Reform of the European Currency Snake”, IMF Econ Rev 23, 580?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Dechert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gencay (1992): “Lyapunov Exponents as a Nonparametric Diag- nostic for Stability Analysis”, Journal of Applied Econometrics, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 7, S41-S60 Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Farmen and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gijbels (1998): “Local Maximum Likelihood Estimation and Inference”, Journal of the Royal Statistical Society, series B, 60, Part 3, 591-608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Froot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Obstfeld (1991): “Intrinsic Bubbles: The Case of Stock Prices”, Amer- ican Economic Review, 81, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 1189-1214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gourieroux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=': “ARCH Models and Financial Applications”, New York: Springer-Verlag, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gourieroux C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Jasiak (2019): “Robust Analysis of the Martingale Hypothesis”, Econometrics and Statistics, Vol 9, 17-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Gourieroux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Zakoian (2017): “Local Explosion Modelling by Noncausal Processes”, Journal of the Royal Statistical Society (JRSS), Series B, 79, 737-756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' THIS VERSION: January 3, 2023 51 Griffin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' , and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=',Shams (2020): “Is Bitcoin Really Untethered?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', The Journal of Finance, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 75, issue 4 Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Stein (2002): “A Unified Theory of Underreaction, Momentum Trading, and Overreaction in Asset Markets”, The Journal of Finance, 54, issue 6, 2143- 2184 Huisman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Koedijik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and Pownall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', (1998): “VaR-x: Fat Tails in Financial Risk Management”, Papers 98-54, Southern California - School of Business Administra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Kortian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1995): “Modern Approaches to Asset Price Formation: A Survey of Re- cent Theoretical Literature”, RBA Research Discussion Papers rdp9501, Reserve Bank of Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Lebaron, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1994): “Chaos and Nonlinear Forecastability in Economics and Finance”, Philosophical Transactions of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Series A: Physical and Engi- neering Sciences, 348, 397-404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1999): “Consistent Model Specification Tests for Time Series Econometric Models”, Journal of Econometrics, 92, 101-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1992): “On the Asymptotic Standard Errors of Residual Autocorrelations in Nonlinear Time Series Modeling”, Biometrika, 79, 435-437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and Mak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1994): “On the Squared Residual Autocorrelations in Non- linear Time Series with Conditional Heteroskedasticity”, Journal of Time Series Analysis, 15, 627-636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Zhu (2019): “A Double AR Model without Intercept: An Al- ternative to Modeling Nonstationarity and Heteroscedasticity”, Econometric Reviews, 38, issue 3, 319-331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Ling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Zhang (2016): “On a Threshold Double Autoregressive Model”, Journal of Business and Economic Statistics, 34, 68-80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and Mayer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', (2022) “Money Creation in Decentralized Finance: A Dynamic Model of Stablecoin and Crypto Shadow Banking”, Fisher College of Business Working Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2020-03-030, Charles A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Dice Center Working Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 2020-30 THIS VERSION: January 3, 2023 52 Ling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2004): “Estimation and Testing Stationarity for Double Autoregressive Models”, JRSS Series B, 66, 63-78 Ling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2007): “A Double AR(p) Model: Structure and Estimation”, Statistica Sinica”, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 17, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', 161-175 Ling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Li (2008): “Asymptotic Inference for a Nonstationary Double AR(1) Model”, Biometrika , 95, 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 257–263 Liu, F, Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Kang (2018) “Sample Path Properties of an Explosive Double Autoregressive Model”, Econometric Reviews, 37, 484-490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Lyons, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Viswanath-Natraj (2020): “What Keeps Stablecions Stable?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', NBER Working Papers 27136, National Bureau of Economic Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Nelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1990): “Stationarity and Persistence in the GARCH(1,1) Model”, Econo- metric Theory, 6, 318-334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Newey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1991): “Uniform Convergence in Probability and Stochastic Continuity”, Econometrica, Vol 59, 1161-1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Potcher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Prucha (1989): “Uniform Law of Large Numbers for Dependent and Heteregeneous Processes,” Econometrica, 57, 675-683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' President’s Working Group (2021): “President’s Working Group on Financial Markets Re- leases Report and Recommendations on Stablecoins”, https://home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='gov/news/press- releases/jy0454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Sprott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2003): “Chaos and Time-Series Analysis”, Oxford University Press, Oxford Sprott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2014): “Numerical Calculation of Largest Lyapunov Exponent”, working paper, University of Wisconsin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (2020): “Are Stablecoins truly diversifiers, hedges, or Safe Havens against traditional cryptocurrencies as their names?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=', Research in International Business and Finance, 54, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' 101-225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' Zakoian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} +page_content=' (1994): “Threshold heteroskedastic models”, Journal of Economic Dynamics and Control, 18, 931-955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQfofi7/content/2301.00509v1.pdf'} diff --git a/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/2301.02141v1.pdf.txt b/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/2301.02141v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bba71b1c63865b259db3722848d75606d5fee45f --- /dev/null +++ b/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/2301.02141v1.pdf.txt @@ -0,0 +1,955 @@ +arXiv:2301.02141v1 [math.NT] 5 Jan 2023 +A refinement of Lang’s formula for the sum of powers of integers +Jos´e Luis Cereceda +Collado Villalba, 28400 (Madrid), Spain +jl.cereceda@movistar.es +Abstract +In 2011, W. Lang derived a novel, explicit formula for the sum of powers of integers Sk(n) = +1k + 2k + · · · + nk involving simultaneously the Stirling numbers of the first and second kind. In +this note, we first recall and then slightly refine Lang’s formula for Sk(n). As it turns out, the +modified Lang’s formula constitutes a special case of a general relationship discovered by Merca +between the power sums, the elementary symmetric functions, and the complete homogeneous +symmetric functions. +1 +Introduction +For integers n ≥ 1 and k ≥ 0, let Sk(n) denote the sum of k-th powers of the first n positive integers +1k + 2k + · · · + nk. In a 2011 technical note [8], W. Lang derived the following explicit formula for +Sk(n) (in our notation): +Sk(n) = +min (k,n−1) +� +m=0 +(−1)m(n − m) +� +n + 1 +n + 1 − m +��n + k − m +n +� +, +(1) +see [8, Equation (10)], where +�k +j +� +and +�k +j +� +are the (unsigned) Stirling numbers of the first and second +kind, respectively. +For completeness and for its intrinsic interest, in Section 2 of the present note we outline the +proof of the formula (1) as given by Lang. Then, in Section 3, we slightly refine the formula (1). The +refinement made essentially amounts to the removal of n from the factor (n − m). In Section 4, we +show that the modified Lang’s formula arises as a direct consequence of the Newton-Girard identities +involving the power sums Sk(n) and the elementary symmetric functions with natural arguments. +Moreover, in Section 5, we point out that, actually, the modified Lang’s formula constitutes a +special case of a general relationship discovered by Merca (see [10, Lemma 2.1]) between the power +sums, the elementary symmetric functions, and the complete homogeneous symmetric functions. +2 +Proof of Lang’s formula +Following Lang’s own derivation [8], next we give a simplified proof sketch of the formula (1). We +start with the ordinary generating function of Sk(n), i.e. +Gn(x) = +∞ +� +k=0 +(1k + 2k + · · · + nk)xk = +n +� +j=1 +1 +1 − jx. +This generating function can be rewritten in the form +Gn(x) = +Pn(x) +�n +j=1(1 − jx), +(2) +1 + +where Pn(x) is the following polynomial in x of degree n − 1 with coefficients Pn,r: +Pn(x) = +n +� +j=1 +n +� +l=1 +l̸=j +(1 − lx) = +n−1 +� +r=0 +Pn,rxr. +(3) +Hence, noting that +1 +�n +j=1(1−jx) = �∞ +m=0 +�n+m +n +� +xm, from (2) and (3) it follows that +Sk(n) = +min (k,n−1) +� +m=0 +Pn,m +�n + k − m +n +� +. +(4) +Now, as pointed out by Lang [8], the elementary symmetric functions σm(1, 2, . . . , n) enter the +scene because we have that +n +� +j=1 +(1 − jx) = +n +� +m=0 +(−1)mσm(1, 2, . . . , n)xm, +(5) +with σ0 = 1. In view of (3) and (5), it is clear that, by symmetry, Pn(x) must be of the form +Pn(x) = +n−1 +� +m=0 +Cn,m(−1)mσm(1, 2, . . . , n)xm, +for certain positive integer coefficients Cn,m. Indeed, it can be seen that +Pn,0 = n, +Pn,1 = (n − 1)(−1)(1 + 2 + · · · + n) = (n − 1)(−1)σ1(1, 2, . . . , n), +Pn,2 = (n − 2)(1 · 2 + 1 · 3 + · · · + (n − 1)n) = (n − 2)σ2(1, 2, . . . , n), +and, in general, +Pn,m = n +�n−1 +m +� +�n +m +� (−1)mσm(1, 2, . . . , n) = (n − m)(−1)mσm(1, 2, . . . , n), +so that Cn,m = n − m, for m = 0, 1, . . . , n − 1. +Therefore, recalling (4), and invoking the well-known relationship σm(1, 2, . . . , n) = +� +n+1 +n+1−m +� +(see, e.g., [7, Equation (2.6)]), we get (1). +3 +A refinement of Lang’s formula +Having considered Lang’s original formula for the sum of powers of integers, we show that this +formula can be simplified somewhat. To see this, we write (1) in the equivalent form +Sk(n) = n +min (k,n) +� +m=0 +(−1)m +� +n + 1 +n + 1 − m +��n + k − m +n +� ++ +min (k,n) +� +m=1 +(−1)m−1 m +� +n + 1 +n + 1 − m +��n + k − m +n +� +, +2 + +where the second summation on the right-hand side is zero when k = 0 or, in other words, it applies +for the case that k ≥ 1. Regarding the first summation, it turns out that +min (k,n) +� +m=0 +(−1)m +� +n + 1 +n + 1 − m +��n + k − m +n +� += δk,0, +(6) +where δk,0 is the Kronecker’s delta. This is so because + +� +i≥0 +(−1)i +� n + 1 +n + 1 − i +� +xi + + + +� +j≥0 +�n + j +n +� +xj + + = 1. +Consequently, Lang’s original formula (1) can be reduced to +Sk(n) = n δk,0 + +min (k,n) +� +m=1 +(−1)m−1 m +� +n + 1 +n + 1 − m +��n + k − m +n +� +, +(7) +which holds for any integers n ≥ 1 and k ≥ 0, and where, as noted above, the summation on the +right-hand side is zero when k = 0. Moreover, for the general case where k ≥ 1, the formula (7) +can in turn be expressed without loss of generality as +Sk(n) = +k +� +m=1 +(−1)m−1 m +� +n + 1 +n + 1 − m +��n + k − m +n +� +, +k ≥ 1, +(8) +assuming the natural convention that +� +n+1 +n+1−m +� += σm(1, 2, . . . , n) = 0 whenever m > n. +4 +Connection with the Newton-Girard identities +As we shall presently see, the modified Lang’s formula for Sk(n) in eq. (8) can be readily ob- +tained from the Newton-Girard identities (cf. Exercise 2 of [3]). +Let {x1, x2, . . . , xn} denote a +(possibly infinite) set of variables and let σm(x1, x2, . . . , xn) denote the corresponding elementary +symmetric function. Generally speaking, the Newton-Girard identities are, within the ring of sym- +metric functions, the connection formulas between the generating sets {σm(x1, x2, . . . , xn)}k +m=1 and +{pm(x1, x2, . . . , xn)}k +m=1, where k stands for any fixed positive integer and the pm’s stand for the +power sums pm(x1, x2, . . . , xn) = xm +1 + xm +2 + · · · + xm +n . +For our purposes here, we focus on the case where xi = i, ∀i. Also, to abbreviate the notation, +in what follows we write σm(1, 2, . . . , n) in the shortened form σm(n). Then, for any given positive +integer m, the Newton-Girard identities can be formulated as follows (see, e.g., [5, Equation (5)] +and [14, Theorem 1.2]) +m−1 +� +j=1 +σm−j(n)Sj(n) + Sm(n) + mσm(n) = 0, +m ≥ 1, +(9) +where σj(n) = (−1)jσj(n), and where the summation on the left-hand side is zero when m = 1. +Thus, letting successively m = 1, 2, 3, . . . , k in (9) yields the following system of k equations in the +3 + +unknowns S1(n), S2(n), . . . , Sk(n): +S1(n) = −σ1(n), +σ1(n)S1(n) + S2(n) = −2σ2(n), +σ2(n)S1(n) + σ1(n)S2(n) + S3(n) = −3σ3(n), +... +σk−1(n)S1(n) + σk−2(n)S2(n) + · · · + σ1(n)Sk−1(n) + Sk(n) = −kσk(n), +which can be expressed in matrix form as + + + + + + + + + +1 +0 +0 +· · · +0 +σ1(n) +1 +0 +... +0 +σ2(n) +σ1(n) +1 +... +0 +... +... +... +... +0 +σk−1(n) +σk−2(n) +· · · +σ1(n) +1 + + + + + + + + + + + + + + + + + + +S1(n) +S2(n) +S3(n) +... +Sk(n) + + + + + + + + + += + + + + + + + + + +−σ1(n) +−2σ2(n) +−3σ3(n) +... +−kσk(n) + + + + + + + + + +. +On the other hand, it is easily seen that the orthogonality relation in eq. (6) is equivalent to the +matrix identity + + + + + + + + + +1 +0 +0 +· · · +0 +σ1(n) +1 +0 +... +0 +σ2(n) +σ1(n) +1 +... +0 +... +... +... +... +0 +σk−1(n) +σk−2(n) +· · · +σ1(n) +1 + + + + + + + + + +−1 += + + + + + + + + + +1 +0 +0 +· · · +0 +h1(n) +1 +0 +... +0 +h2(n) +h1(n) +1 +... +0 +... +... +... +... +0 +hk−1(n) +hk−2(n) +· · · +h1(n) +1 + + + + + + + + + +, +where hk(n) = +�n+k +n +� +and h0(n) = 1. Hence, it follows that + + + + + + + + + +S1(n) +S2(n) +S3(n) +... +Sk(n) + + + + + + + + + += + + + + + + + + + +1 +0 +0 +· · · +0 +h1(n) +1 +0 +... +0 +h2(n) +h1(n) +1 +... +0 +... +... +... +... +0 +hk−1(n) +hk−2(n) +· · · +h1(n) +1 + + + + + + + + + + + + + + + + + + +−σ1(n) +−2σ2(n) +−3σ3(n) +... +−kσk(n) + + + + + + + + + +. +Finally, solving for Sk(n), we get (8). +We conclude this section with the following two remarks. +Remark 1. The Newton-Girard identities (9) can equally be written as the recurrence relation +Sm(n) = (−1)m−1mσm(n) − +m−1 +� +j=1 +(−1)jσj(n)Sm−j(n), +m ≥ 1, +giving Sm(n) in terms of σ1(n), σ2(n), . . . , σm(n) and the earlier power sums Sj(n), j = 1, 2, . . . , m− +1. This recurrence may be compared with the following one appearing in [3, Remark 3]: +Sm(n) = m! +�n + m +m + 1 +� +− +m−1 +� +j=1 +σj(m − 1)Sm−j(n), +m ≥ 1. +4 + +Remark 2. It should be mentioned that the formula for Sk(n) in eq. (8) was (re)discovered by +Merca in [9, Theorem 1] by manipulating the formal power series for the Stirling numbers. +5 +Generalized Lang’s formula +The proof given in the preceding section of the formula (8) naturally generalizes to arbitrary +elementary symmetric functions σm(x1, x2, . . . , xn), complete homogenous symmetric functions +hm(x1, x2, . . . , xn), and associated power sums pm(x1, x2, . . . , xn). Indeed, as shown by Merca (see +[10, Lemma 2.1]), the power sum pk(x1, x2, . . . , xn) can be expressed in terms of the σm(x1, x2, . . . , xn) +and hk−m(x1, x2, . . . , xn) as +pk(x1, x2, . . . , xn) = +k +� +m=1 +(−1)m−1mσm(x1, x2, . . . , xn)hk−m(x1, x2, . . . , xn), +(10) +which becomes the formula (8) when xi = i, ∀i. Next, we describe some other applications of the +formula (10). +Consider first the case in which xi = 1, ∀i. Then, recalling that σm(1, 1, . . . , 1) = +� n +m +� +and +hm(1, 1, . . . , 1) = +�n+m−1 +m +� +, from (10) we obtain the identity +k +� +m=1 +(−1)m−1 m +�n +m +��n + k − m − 1 +k − m +� += n, +which holds for any integers k, n ≥ 1. +On the other hand, for integers 1 ≤ r ≤ n, it turns +out that the r-Stirling numbers of the first kind are the elementary symmetric functions of the +numbers r, r + 1, . . . , n, that is, +� +n+1 +n+1−m +� +r = σm(r, r + 1, . . . , n); and the r-Stirling numbers of +the second kind are the complete symmetric functions of the numbers r, r + 1, . . . , n, that is, +�n+m +n +� +r = hm(r, r + 1, . . . , n) (see [2, Section 5]). Therefore, from (10), we find that +rk + (r + 1)k + · · · + nk = +k +� +m=1 +(−1)m−1 m +� +n + 1 +n + 1 − m +� +r +�n + k − m +n +� +r +, +where +� +n+1 +n+1−m +� +r = σm(r, r + 1, . . . , n) = 0 whenever m > n + 1 − r. In particular, for r = 1, this +equation reduces to (8). A further generalization of (8) in terms of the r-Whitney numbers of both +kinds and the Bernoulli polynomials can be found in [11]. +As another application of eq. (10), we can evaluate the sum of even powers of the first n positive +integers, S2k(n) = 12k + 22k + · · · + n2k, by using the fact that (see [12]) +u(n + 1, n + 1 − m) = (−1)mσm(12, 22, . . . , n2), +and +U(n + m, n) = hm(12, 22, . . . , n2), +where u(n, k) [respectively, U(n, k)] are the central factorial numbers of the first [respectively, +second] kind with even indices. Therefore, we have [12, Theorem 1.1] +12k + 22k + · · · + n2k = − +k +� +m=1 +m u(n + 1, n + 1 − m)U(n + k − m, n). +5 + +Likewise, noting that (see [12]) +v(n, n − m) = (−1)mσm(12, 32, . . . , (2n − 1)2), +and +V (n − 1 + m, n − 1) = hm(12, 32, . . . , (2n − 1)2), +where v(n, k) [respectively, V (n, k)] are the central factorial numbers of the first [respectively, +second] kind with odd indices, we can evaluate the sum of even powers of the first n odd integers, +12k + 32k + · · · + (2n − 1)2k, as follows +12k + 32k + · · · + (2n − 1)2k = − +k +� +m=1 +m v(n, n − m)V (n − 1 + k − m, n − 1). +Incidentally, it is to be noted that the above power sum can alternatively be expressed in the form +(see [6, 4]) +12k + 32k + · · · + (2n − 1)2k = n +k +� +m=1 +dk,mN m, +where N = (2n − 1)(2n + 1), and where the dk,m are certain (non-zero) rational coefficients. +Our last application concerns the so-called Legendre-Stirling (LS) numbers of the first and +second kind, which, following [1], we denote by Ps(j) +n +and PS(j) +n , respectively. Furthermore, we +assume that n and j are non-negative integers fulfilling 0 ≤ j ≤ n. Table 1 (2) displays the first +few LS numbers of the first (second) kind. The LS numbers of the first kind are the elementary +symmetric functions of the numbers 2, 6, . . . , n(n + 1), i.e +Ps(n+1−k) +n+1 += (−1)kσk(2, 6, . . . , n(n + 1)), +whereas the LS numbers of the second kind are the complete homogeneous symmetric functions of +the numbers 2, 6, . . . , n(n + 1), i.e +PS(n) +n+k = hk(2, 6, . . . , n(n + 1)). +Equivalently, we can write the above two expressions as +Ps(n+1−k) +n+1 += (−1)k2kσk(T1, T2, . . . , Tn), +n\ j +0 +1 +2 +3 +4 +5 +6 +7 +0 +1 +1 +0 +1 +2 +0 +−2 +1 +3 +0 +12 +−8 +1 +4 +0 +−144 +108 +−20 +1 +5 +0 +2880 +−2304 +508 +−40 +1 +6 +0 +−86400 +72000 +−17544 +1708 +−70 +1 +7 +0 +3628800 +−3110400 +808848 +−89280 +4648 +−112 +1 +Table 1: The LS numbers of the first kind, Ps(j) +n , up to n = 7. +6 + +n\ j +0 +1 +2 +3 +4 +5 +6 +7 +0 +1 +1 +0 +1 +2 +0 +2 +1 +3 +0 +4 +8 +1 +4 +0 +8 +52 +20 +1 +5 +0 +16 +320 +292 +40 +1 +6 +0 +32 +1936 +3824 +1092 +70 +1 +7 +0 +64 +11648 +47824 +25664 +3192 +112 +1 +Table 2: The LS numbers of the second kind, PS(j) +n , up to n = 7. +and +PS(n) +n+k = 2khk(T1, T2, . . . , Tn), +where Tn = 1 +2n(n + 1) is the n-th triangular number. Therefore, we conclude from (10) that +T k +1 + T k +2 + · · · + T k +n = − 1 +2k +k +� +m=1 +m Ps(n+1−m) +n+1 +PS(n) +n+k−m. +(11) +In particular, for k = 1, we have +T1 + T2 + · · · + Tn = +�n + 2 +3 +� += −1 +2Ps(n) +n+1. +In addition, we note that the sum of k-th powers of the first n triangular numbers can also be +expressed by +T k +1 + T k +2 + · · · + T k +n = 1 +2k +k +� +j=0 +�k +j +� +Sk+j(n) = 1 +2k +k +� +j=0 +�k +j +�Bk+j+1(n + 1) − Bk+j+1(1) +k + j + 1 +, +(12) +where the Bk(n) are the Bernoulli polynomials. Moreover, Merca showed that, see [13, Corollary +1.1] (in our notation) +− +k +� +m=1 +m Ps(n+1−m) +n+1 +PS(n) +n+k−m = +(−1)k +(k + 1) +�2k+2 +k+1 +� + +k +� +j=0 +�k +j +�Bk+j+1(n + 1) +k + j + 1 +. +(13) +Hence, combining (11) and (13), and taking into account (12), we obtain the identity +k +� +j=0 +(−1)j +�k +j +� Bk+j+1 +k + j + 1 = +1 +(k + 1) +�2k+2 +k+1 +�, +k ≥ 1, +where the Bk are the Bernoulli numbers. +7 + +6 +Conclusion +In this note, we have brought to light an outstanding (though largely unnoticed) contribution of +W. Lang to the subject of sums of powers of integers, namely, his formula for Sk(n) stated in +eq. (1). We have shown that Lang’s original formula (1) can be slightly refined so that the integer +variable n can be effectively removed from the factor (n − m), as can be seen by looking at formula +(7). Furthermore, we have shown that the modified Lang’s formula for Sk(n) in eq. (8) follows +straightforwardly from the Newton-Girard identities formulated in eq. (9). Finally, to broaden the +scope of the present note, we have pointed out several extensions of the formula (8) achieved by +Merca [10, 11, 12, 13]. +References +[1] G. E. Andrews, W. Gawronski, and L. L. Littlejohn, The Legendre-Stirling numbers, Discrete +Math., 311(14):1255–1272 (2011). +[2] A. Z. Broder, The r-Stirling numbers. Discrete Math., 49(3):241–259 (1984). +[3] J. L. Cereceda, Sums of powers of integers and Stirling numbers, Resonance, 27(5):769–784 +(2022). +[4] J. L. Cereceda, Explicit polynomial for sums of powers of odd integers, Int. Math. Forum, +9(30):1441–1446 (2014). +[5] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci +sequences, Fibonacci Quart., 37(2):135–140 (1999). +[6] S. Guo and Y. Shen, On sums of powers of odd integers, J. Math. Res. Appl., 33(6):666–672 +(2013). +[7] D. E. Knuth, Two notes on notation, Amer. Math. Monthly, 99(5):403–422 (1992). +[8] W. Lang, A196837: Ordinary generating functions for sums of powers of the first n positive +integers, online note (2011), available at http://oeis.org/A196837/a196837.pdf +[9] M. Merca, An alternative to Faulhaber’s formula, Amer. Math. Monthly, 122(6):599–601 +(2015). +[10] M. Merca, New convolutions for complete and elementary symmetric functions, Integral Trans- +forms Spec. Funct., 27(12):965–973 (2016). +[11] M. Merca, A new connection between r-Whitney numbers and Bernoulli polynomials, Integral +Transforms Spec. Funct., 25(12):937–942 (2014). +[12] M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period. +Math. Hungar., 73(2):259–264 (2016). +[13] M. Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J. Num- +ber Theory, 151:223–229 (2015). +[14] M. +Moss´e, +Newton’s +identities, +online +note +(2019), +available +at +https://web.stanford.edu/~marykw/classes/CS250_W19/Netwons_Identities.pdf +8 + diff --git a/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/load_file.txt b/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0763541029d6d168f54407221d528f5b0b74809 --- /dev/null +++ b/2tA0T4oBgHgl3EQfM_-6/content/tmp_files/load_file.txt @@ -0,0 +1,361 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf,len=360 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='02141v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='NT] 5 Jan 2023 A refinement of Lang’s formula for the sum of powers of integers Jos´e Luis Cereceda Collado Villalba, 28400 (Madrid), Spain jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='cereceda@movistar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='es Abstract In 2011, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Lang derived a novel, explicit formula for the sum of powers of integers Sk(n) = 1k + 2k + · · · + nk involving simultaneously the Stirling numbers of the first and second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In this note, we first recall and then slightly refine Lang’s formula for Sk(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' As it turns out, the modified Lang’s formula constitutes a special case of a general relationship discovered by Merca between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 1 Introduction For integers n ≥ 1 and k ≥ 0, let Sk(n) denote the sum of k-th powers of the first n positive integers 1k + 2k + · · · + nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In a 2011 technical note [8], W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Lang derived the following explicit formula for Sk(n) (in our notation): Sk(n) = min (k,n−1) � m=0 (−1)m(n − m) � n + 1 n + 1 − m ��n + k − m n � , (1) see [8, Equation (10)], where �k j � and �k j � are the (unsigned) Stirling numbers of the first and second kind, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' For completeness and for its intrinsic interest, in Section 2 of the present note we outline the proof of the formula (1) as given by Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Then, in Section 3, we slightly refine the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' The refinement made essentially amounts to the removal of n from the factor (n − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In Section 4, we show that the modified Lang’s formula arises as a direct consequence of the Newton-Girard identities involving the power sums Sk(n) and the elementary symmetric functions with natural arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Moreover, in Section 5, we point out that, actually, the modified Lang’s formula constitutes a special case of a general relationship discovered by Merca (see [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='1]) between the power sums, the elementary symmetric functions, and the complete homogeneous symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 2 Proof of Lang’s formula Following Lang’s own derivation [8], next we give a simplified proof sketch of the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' We start with the ordinary generating function of Sk(n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Gn(x) = ∞ � k=0 (1k + 2k + · · · + nk)xk = n � j=1 1 1 − jx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' This generating function can be rewritten in the form Gn(x) = Pn(x) �n j=1(1 − jx), (2) 1 where Pn(x) is the following polynomial in x of degree n − 1 with coefficients Pn,r: Pn(x) = n � j=1 n � l=1 l̸=j (1 − lx) = n−1 � r=0 Pn,rxr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (3) Hence, noting that 1 �n j=1(1−jx) = �∞ m=0 �n+m n � xm, from (2) and (3) it follows that Sk(n) = min (k,n−1) � m=0 Pn,m �n + k − m n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (4) Now, as pointed out by Lang [8], the elementary symmetric functions σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) enter the scene because we have that n � j=1 (1 − jx) = n � m=0 (−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n)xm, (5) with σ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In view of (3) and (5), it is clear that, by symmetry, Pn(x) must be of the form Pn(x) = n−1 � m=0 Cn,m(−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n)xm, for certain positive integer coefficients Cn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Indeed, it can be seen that Pn,0 = n, Pn,1 = (n − 1)(−1)(1 + 2 + · · · + n) = (n − 1)(−1)σ1(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n), Pn,2 = (n − 2)(1 · 2 + 1 · 3 + · · · + (n − 1)n) = (n − 2)σ2(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n), and, in general, Pn,m = n �n−1 m � �n m � (−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) = (n − m)(−1)mσm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n), so that Cn,m = n − m, for m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Therefore, recalling (4), and invoking the well-known relationship σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) = � n+1 n+1−m � (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', [7, Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='6)]), we get (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 3 A refinement of Lang’s formula Having considered Lang’s original formula for the sum of powers of integers, we show that this formula can be simplified somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' To see this, we write (1) in the equivalent form Sk(n) = n min (k,n) � m=0 (−1)m � n + 1 n + 1 − m ��n + k − m n � + min (k,n) � m=1 (−1)m−1 m � n + 1 n + 1 − m ��n + k − m n � , 2 where the second summation on the right-hand side is zero when k = 0 or, in other words, it applies for the case that k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Regarding the first summation, it turns out that min (k,n) � m=0 (−1)m � n + 1 n + 1 − m ��n + k − m n � = δk,0, (6) where δk,0 is the Kronecker’s delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' This is so because \uf8eb \uf8ed� i≥0 (−1)i � n + 1 n + 1 − i � xi \uf8f6 \uf8f8 \uf8eb \uf8ed� j≥0 �n + j n � xj \uf8f6 \uf8f8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Consequently, Lang’s original formula (1) can be reduced to Sk(n) = n δk,0 + min (k,n) � m=1 (−1)m−1 m � n + 1 n + 1 − m ��n + k − m n � , (7) which holds for any integers n ≥ 1 and k ≥ 0, and where, as noted above, the summation on the right-hand side is zero when k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Moreover, for the general case where k ≥ 1, the formula (7) can in turn be expressed without loss of generality as Sk(n) = k � m=1 (−1)m−1 m � n + 1 n + 1 − m ��n + k − m n � , k ≥ 1, (8) assuming the natural convention that � n+1 n+1−m � = σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) = 0 whenever m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 4 Connection with the Newton-Girard identities As we shall presently see, the modified Lang’s formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (8) can be readily ob- tained from the Newton-Girard identities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Exercise 2 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Let {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn} denote a (possibly infinite) set of variables and let σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) denote the corresponding elementary symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Generally speaking, the Newton-Girard identities are, within the ring of sym- metric functions, the connection formulas between the generating sets {σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn)}k m=1 and {pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn)}k m=1, where k stands for any fixed positive integer and the pm’s stand for the power sums pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) = xm 1 + xm 2 + · · · + xm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' For our purposes here, we focus on the case where xi = i, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Also, to abbreviate the notation, in what follows we write σm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) in the shortened form σm(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Then, for any given positive integer m, the Newton-Girard identities can be formulated as follows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', [5, Equation (5)] and [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='2]) m−1 � j=1 σm−j(n)Sj(n) + Sm(n) + mσm(n) = 0, m ≥ 1, (9) where σj(n) = (−1)jσj(n), and where the summation on the left-hand side is zero when m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Thus, letting successively m = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , k in (9) yields the following system of k equations in the 3 unknowns S1(n), S2(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , Sk(n): S1(n) = −σ1(n), σ1(n)S1(n) + S2(n) = −2σ2(n), σ2(n)S1(n) + σ1(n)S2(n) + S3(n) = −3σ3(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' σk−1(n)S1(n) + σk−2(n)S2(n) + · · · + σ1(n)Sk−1(n) + Sk(n) = −kσk(n), which can be expressed in matrix form as \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 · · 0 σ1(n) 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 σ2(n) σ1(n) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 σk−1(n) σk−2(n) · · σ1(n) 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed S1(n) S2(n) S3(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Sk(n) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed −σ1(n) −2σ2(n) −3σ3(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' −kσk(n) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' On the other hand, it is easily seen that the orthogonality relation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (6) is equivalent to the matrix identity \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 · · 0 σ1(n) 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 σ2(n) σ1(n) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 σk−1(n) σk−2(n) · · σ1(n) 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 −1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 · · 0 h1(n) 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 h2(n) h1(n) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 hk−1(n) hk−2(n) · · h1(n) 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , where hk(n) = �n+k n � and h0(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Hence, it follows that \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed S1(n) S2(n) S3(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Sk(n) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 · · 0 h1(n) 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 h2(n) h1(n) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 0 hk−1(n) hk−2(n) · · h1(n) 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed −σ1(n) −2σ2(n) −3σ3(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' −kσk(n) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Finally, solving for Sk(n), we get (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' We conclude this section with the following two remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' The Newton-Girard identities (9) can equally be written as the recurrence relation Sm(n) = (−1)m−1mσm(n) − m−1 � j=1 (−1)jσj(n)Sm−j(n), m ≥ 1, giving Sm(n) in terms of σ1(n), σ2(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , σm(n) and the earlier power sums Sj(n), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , m− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' This recurrence may be compared with the following one appearing in [3, Remark 3]: Sm(n) = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' �n + m m + 1 � − m−1 � j=1 σj(m − 1)Sm−j(n), m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 4 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' It should be mentioned that the formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (8) was (re)discovered by Merca in [9, Theorem 1] by manipulating the formal power series for the Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 5 Generalized Lang’s formula The proof given in the preceding section of the formula (8) naturally generalizes to arbitrary elementary symmetric functions σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn), complete homogenous symmetric functions hm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn), and associated power sums pm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Indeed, as shown by Merca (see [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='1]), the power sum pk(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) can be expressed in terms of the σm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) and hk−m(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) as pk(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn) = k � m=1 (−1)m−1mσm(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn)hk−m(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , xn), (10) which becomes the formula (8) when xi = i, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Next, we describe some other applications of the formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Consider first the case in which xi = 1, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Then, recalling that σm(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , 1) = � n m � and hm(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , 1) = �n+m−1 m � , from (10) we obtain the identity k � m=1 (−1)m−1 m �n m ��n + k − m − 1 k − m � = n, which holds for any integers k, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' On the other hand, for integers 1 ≤ r ≤ n, it turns out that the r-Stirling numbers of the first kind are the elementary symmetric functions of the numbers r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n, that is, � n+1 n+1−m � r = σm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' and the r-Stirling numbers of the second kind are the complete symmetric functions of the numbers r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n, that is, �n+m n � r = hm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) (see [2, Section 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Therefore, from (10), we find that rk + (r + 1)k + · · · + nk = k � m=1 (−1)m−1 m � n + 1 n + 1 − m � r �n + k − m n � r , where � n+1 n+1−m � r = σm(r, r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n) = 0 whenever m > n + 1 − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In particular, for r = 1, this equation reduces to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' A further generalization of (8) in terms of the r-Whitney numbers of both kinds and the Bernoulli polynomials can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' As another application of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (10), we can evaluate the sum of even powers of the first n positive integers, S2k(n) = 12k + 22k + · · · + n2k, by using the fact that (see [12]) u(n + 1, n + 1 − m) = (−1)mσm(12, 22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n2), and U(n + m, n) = hm(12, 22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n2), where u(n, k) [respectively, U(n, k)] are the central factorial numbers of the first [respectively, second] kind with even indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Therefore, we have [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='1] 12k + 22k + · · · + n2k = − k � m=1 m u(n + 1, n + 1 − m)U(n + k − m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 5 Likewise, noting that (see [12]) v(n, n − m) = (−1)mσm(12, 32, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , (2n − 1)2), and V (n − 1 + m, n − 1) = hm(12, 32, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , (2n − 1)2), where v(n, k) [respectively, V (n, k)] are the central factorial numbers of the first [respectively, second] kind with odd indices, we can evaluate the sum of even powers of the first n odd integers, 12k + 32k + · · · + (2n − 1)2k, as follows 12k + 32k + · · · + (2n − 1)2k = − k � m=1 m v(n, n − m)V (n − 1 + k − m, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Incidentally, it is to be noted that the above power sum can alternatively be expressed in the form (see [6, 4]) 12k + 32k + · · · + (2n − 1)2k = n k � m=1 dk,mN m, where N = (2n − 1)(2n + 1), and where the dk,m are certain (non-zero) rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Our last application concerns the so-called Legendre-Stirling (LS) numbers of the first and second kind, which, following [1], we denote by Ps(j) n and PS(j) n , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Furthermore, we assume that n and j are non-negative integers fulfilling 0 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Table 1 (2) displays the first few LS numbers of the first (second) kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' The LS numbers of the first kind are the elementary symmetric functions of the numbers 2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n(n + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='e Ps(n+1−k) n+1 = (−1)kσk(2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n(n + 1)), whereas the LS numbers of the second kind are the complete homogeneous symmetric functions of the numbers 2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n(n + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='e PS(n) n+k = hk(2, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , n(n + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Equivalently, we can write the above two expressions as Ps(n+1−k) n+1 = (−1)k2kσk(T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , Tn), n\\ j 0 1 2 3 4 5 6 7 0 1 1 0 1 2 0 −2 1 3 0 12 −8 1 4 0 −144 108 −20 1 5 0 2880 −2304 508 −40 1 6 0 −86400 72000 −17544 1708 −70 1 7 0 3628800 −3110400 808848 −89280 4648 −112 1 Table 1: The LS numbers of the first kind, Ps(j) n , up to n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 6 n\\ j 0 1 2 3 4 5 6 7 0 1 1 0 1 2 0 2 1 3 0 4 8 1 4 0 8 52 20 1 5 0 16 320 292 40 1 6 0 32 1936 3824 1092 70 1 7 0 64 11648 47824 25664 3192 112 1 Table 2: The LS numbers of the second kind, PS(j) n , up to n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' and PS(n) n+k = 2khk(T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' , Tn), where Tn = 1 2n(n + 1) is the n-th triangular number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Therefore, we conclude from (10) that T k 1 + T k 2 + · · · + T k n = − 1 2k k � m=1 m Ps(n+1−m) n+1 PS(n) n+k−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (11) In particular, for k = 1, we have T1 + T2 + · · · + Tn = �n + 2 3 � = −1 2Ps(n) n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' In addition, we note that the sum of k-th powers of the first n triangular numbers can also be expressed by T k 1 + T k 2 + · · · + T k n = 1 2k k � j=0 �k j � Sk+j(n) = 1 2k k � j=0 �k j �Bk+j+1(n + 1) − Bk+j+1(1) k + j + 1 , (12) where the Bk(n) are the Bernoulli polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Moreover, Merca showed that, see [13, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='1] (in our notation) − k � m=1 m Ps(n+1−m) n+1 PS(n) n+k−m = (−1)k (k + 1) �2k+2 k+1 � + k � j=0 �k j �Bk+j+1(n + 1) k + j + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (13) Hence, combining (11) and (13), and taking into account (12), we obtain the identity k � j=0 (−1)j �k j � Bk+j+1 k + j + 1 = 1 (k + 1) �2k+2 k+1 �, k ≥ 1, where the Bk are the Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' 7 6 Conclusion In this note, we have brought to light an outstanding (though largely unnoticed) contribution of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Lang to the subject of sums of powers of integers, namely, his formula for Sk(n) stated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' We have shown that Lang’s original formula (1) can be slightly refined so that the integer variable n can be effectively removed from the factor (n − m), as can be seen by looking at formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Furthermore, we have shown that the modified Lang’s formula for Sk(n) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (8) follows straightforwardly from the Newton-Girard identities formulated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Finally, to broaden the scope of the present note, we have pointed out several extensions of the formula (8) achieved by Merca [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Andrews, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Gawronski, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Littlejohn, The Legendre-Stirling numbers, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 311(14):1255–1272 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Broder, The r-Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 49(3):241–259 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Cereceda, Sums of powers of integers and Stirling numbers, Resonance, 27(5):769–784 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Cereceda, Explicit polynomial for sums of powers of odd integers, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Forum, 9(30):1441–1446 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 37(2):135–140 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Guo and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Shen, On sums of powers of odd integers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 33(6):666–672 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Knuth, Two notes on notation, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Monthly, 99(5):403–422 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Lang, A196837: Ordinary generating functions for sums of powers of the first n positive integers, online note (2011), available at http://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='org/A196837/a196837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='pdf [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Merca, An alternative to Faulhaber’s formula, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Monthly, 122(6):599–601 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Merca, New convolutions for complete and elementary symmetric functions, Integral Trans- forms Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 27(12):965–973 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Merca, A new connection between r-Whitney numbers and Bernoulli polynomials, Integral Transforms Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 25(12):937–942 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Merca, Connections between central factorial numbers and Bernoulli polynomials, Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=', 73(2):259–264 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Merca, A connection between Jacobi-Stirling numbers and Bernoulli polynomials, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Num- ber Theory, 151:223–229 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content=' Moss´e, Newton’s identities, online note (2019), available at https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='edu/~marykw/classes/CS250_W19/Netwons_Identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} +page_content='pdf 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tA0T4oBgHgl3EQfM_-6/content/2301.02141v1.pdf'} diff --git a/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/2301.02608v1.pdf.txt b/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/2301.02608v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..38f5f5d45ea00610cbcfa5f7094e555f8548e612 --- /dev/null +++ b/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/2301.02608v1.pdf.txt @@ -0,0 +1,1639 @@ +A CAD System for Colorectal Cancer from WSI: A +Clinically Validated Interpretable ML-based Prototype +Pedro C. Netoa,b,1, Diana Montezumac,f,d,1, Sara P. Oliveiraa,b,1, Domingos +Oliveirac, Jo˜ao Fragae, Ana Monteiroc, Jo˜ao Monteiroc, Liliana Ribeiroc, +Sofia Gon¸calvesc, Stefan Reinhardg, Inti Zlobecg, Isabel M. Pintoc, Jaime S. +Cardosoa,b +aInstitute for Systems and Computer Engineering, Technology and Science (INESC +TEC), R. Dr. Roberto Frias, Porto, 4200-465, Porto, Portugal +bFaculty of Engineering, University of Porto (FEUP), R. Dr. Roberto +Frias, Porto, 4200-465, Porto, Portugal +cIMP Diagnostics, Praca do Bom Sucesso, 61, sala +809, Porto, 4150-146, Porto, Portugal +dCancer Biology and Epigenetics Group, IPO-Porto, R. Dr. Ant´onio Bernardino de +Almeida 865, Porto, 4200-072, Porto, Portugal +eDepartment of Pathology, IPO-Porto, R. Dr. Ant´onio Bernardino de Almeida +865, Porto, 4200-072, Porto, Portugal +fSchool of Medicine and Biomedical Sciences, University of Porto (ICBAS), R. Jorge de +Viterbo Ferreira 228, Porto, 4050-313, Porto, Portugal +gInstitute of Pathology, University of Bern, Uni Bern, Murtenstrasse +31, Bern, 3008, Bern, Switzerland +Abstract +The integration of Artificial Intelligence (AI) and Digital Pathology has +been increasing over the past years. Nowadays, applications of deep learn- +ing (DL) methods to diagnose cancer from whole-slide images (WSI) are, +more than ever, a reality within different research groups. Nonetheless, the +development of these systems was limited by a myriad of constraints re- +garding the lack of training samples, the scaling difficulties, the opaqueness +of DL methods, and, more importantly, the lack of clinical validation. As +such, we propose a system designed specifically for the diagnosis of colorectal +samples. The construction of such a system consisted of four stages: (1) a +careful data collection and annotation process, which resulted in one of the +largest WSI colorectal samples datasets; (2) the design of an interpretable +1These authors contributed equally. +Preprint submitted to . +January 9, 2023 +arXiv:2301.02608v1 [eess.IV] 6 Jan 2023 + +mixed-supervision scheme to leverage the domain knowledge introduced by +pathologists through spatial annotations; (3) the development of an effective +sampling approach based on the expected severeness of each tile, which de- +creased the computation cost by a factor of almost 6x; (4) the creation of +a prototype that integrates the full set of features of the model to be eval- +uated in clinical practice. During these stages, the proposed method was +evaluated in four separate test sets, two of them are external and completely +independent. On the largest of those sets, the proposed approach achieved +an accuracy of 93.44%. DL for colorectal samples is a few steps closer to stop +being research exclusive and to become fully integrated in clinical practice. +Keywords: +Clinical Prototype, Colorectal Cancer, Interpretable Artificial +Intelligence, Deep Learning, Whole-Slide Images +1. Introduction +Colorectal cancer (CRC) incidence and mortality are increasing, and it +is estimated that they will keep growing at least until 2040 [1], according to +estimations of the International Agency for Research on Cancer. Nowadays, +it is the third most incident (10.7% of all cancer diagnoses) and the second +most deadly type of cancer [1]. Due to the effect of lifestyle, genetics, envi- +ronmental factors and an increase in life expectancy, the current increase in +world wealth and the adoption of western lifestyles further advocates for an +increase in the capabilities to perform more CRC evaluations for potential +diagnosis [2, 3]. Despite the pessimist predictions for an increase in the in- +cidence, CRC is preventable and curable when detected in its earlier stages. +Thus, effective screening through medical examination, imaging techniques +and colonoscopy are of utmost importance [4, 5]. +Despite the CRC detection capabilities shown by imaging techniques, +the diagnosis of cancer is always based on the pathologist’s evaluation of +biopsies/surgical specimen samples. The stratification of neoplasia develop- +ment stages consists of non-neoplastic (NNeo), low-grade dysplasia (LGD), +high-grade dysplasia (HGD, including intramucosal carcinomas), and inva- +sive carcinomas, from the initial to the latest stage of cancer progression, +respectively. In spite of the inherent subjectivity of the dysplasia grading +system [6], recent guidelines from the European Society of Gastrointestinal +Endoscopy (ESGE), as well as those from the US multi-society task force on +CRC, consistently recommend shorter surveillance intervals for patients with +2 + +polyps with high-grade dysplasia, regardless of their dimension [5, 7]. Hence, +grading dysplasia is still routinely performed by pathologists worldwide when +assessing colorectal tissue samples. +Private datasets of digitised slides are becoming widely available, in the +form of whole-slide images (WSI), with an increase in the adoption of digital +workflows [8, 9, 10]. Despite the burden of the additional scanning step, WSI +eases the revision of old cases, data sharing and quick peer-review [11, 12]. +It has also created several research opportunities within the computer vision +domain, especially due to the complexity of the problem and the high dimen- +sions of WSI [13, 14, 15, 16]. As such, robust and high-performance systems +can be valuable assets to the digital workflow of a laboratory, especially if +they are transparent and interpretable [11, 12]. However, some limitations +still affect the applicability of such solutions in practice [17]. +The majority of the works on CRC diagnosis direct their focus towards the +classification of cropped regions of interest, or small tiles, instead of tackling +the challenging task of diagnosing the entire WSI [18, 19, 17, 20]. Notwith- +standing, some authors already presented methods to assess the grading of +the complete slide of colorectal samples. In 2020, Iizuka et al. [21] used a re- +current neural network (RNN) to aggregate the predictions of individual tiles +processed by an Inception-v3 network into non-neoplastic, adenoma (AD) +and adenocarcinoma (ADC). Due to the large dimensions of WSI related to +their pyramidal format (with several magnification levels) [22], usually over +50,000 × 50,000 pixels, it is usual to use a scheme consisting of a grid of +tiles. This scheme permits the acceleration of the processing steps since the +tiles are small enough to fit in the memory of the graphics processing units +(GPU), popular units for the training of deep learning (DL). Wei et al. [23] +studied the usage of an ensemble of five distinct ResNet networks, in order to +distinguish the types of CRC adenomas H&E stained slides. Song et al. [24] +experimented with a modified DeepLab-v2 network for tile classification, and +proposed pixel probability thresholding to detect CRC adenomas. Both Xu et +al. [25] and Wang et al. [26, 27] looked into the performance of the Inception- +v3 architecture to detect CRC, with the latter also retrieving a cluster-based +slide classification and a map of predictions. The MuSTMIL [28] method, +classifieds five colon-tissue findings: normal glands, hyperplastic polyps, low- +grade dysplasias, high-grade dysplasias and carcinomas. This classification +originates from a multitask architecture that leverages several levels of mag- +nification of a slide. Ho et al. [29] extended the experiments with multitask +learning, but instead of leveraging the magnification, its model aims to jointly +3 + +segment glands, detect tumour areas and sort the slides into low-risk (benign, +inflammation or reactive changes) and high-risk (adenocarcinoma or dyspla- +sia) categories. The architecture of this model is considerably more complex, +with regard to the number of parameters, and is known as Faster-RCNN with +a ResNet-101 backbone network for the segmentation task. Further to this +task, a gradient-boosted decision tree completes the pipeline that results in +the final grade. +As stated in the previous paragraph, recent state-of-the-art computer- +aided diagnosis systems are based on deep learning approaches. These sys- +tems rely on large volumes of data to learn how to perform a particular task. +Increasing the complexity of the task often demands an increase in the data +available. Collecting this data is expensive and tedious due to the annotation +complexity and the need for expert knowledge. Despite recent publications +that present approaches using large volumes of data to train CAD systems, +the majority do not publicly release the data used. To reverse this trend, +the novel and completely anonymised dataset introduced in this document +will have the majority of the available slides publicly released. This dataset +contains, approximately, 10500 high-quality slides. The available slides origi- +nate one of the largest colorectal samples (CRS) datasets to be made publicly +available. This high volume of data, in addition to the massive resolution +of the images, creates a significant bottleneck of deep learning approaches +that extract patches from the whole slide. Hence, we introduce an efficient +sampling approach that is performed once without sacrificing prediction per- +formance. The proposed sampling leverages knowledge learnt from the data +to create a proxy that reduces the rate of important information discarded, +when compared to random sampling. Due to the cost of annotating the en- +tire dataset, it is only annotated for a portion at the pixel level, while the +remaining portion is labelled for a portion at the slide level. To leverage these +two levels of supervision, we propose a mixed supervision training scheme. +One other increasingly relevant issue with current research is the lack +of external validation. It is not uncommon to observe models that perform +extremely well on a test set collected from the same data distribution used +to train, but fail on samples from other laboratories or scanners. Hence, +we validate our proposed model in two different external datasets that vary +in quality, country of origin and laboratory. While the results on a similar +dataset show the performance of the model if implemented in the institution +that collects that data, the test on external samples indicates its capabilities +to be deployed in other scenarios. +4 + +In order to bring this CAD system into production, and to infer its ca- +pabilities within clinical practice, we developed a prototype that has been +used by pathologists in clinical practice. We further collected information on +the misdiagnoses and the pathologists’ feedback. Since it is expected that +the model indicates some rationale behind its predictions, we developed a +visual approach to explain the decision and guide pathologists’ focus toward +more aggressive areas. This mechanism increased the acceptance of the al- +gorithm in clinical practice, and its usefulness. When collected from routine +the slides can be digitised with duplicated tissue areas, known as fragments, +which might be of lower quality. +Hence, the workflow for the automatic +diagnostic also included an automatic fragment detection and counting sys- +tem [30]. Moreover, it was possible to utilize this prototype to do the second +round of labelling based on the test set prediction given by the model. This +second round led to the correction of certain labels (that the model pre- +dicted correctly and were mislabeled by the expert pathologist) and insights +regarding the areas where the model has to be improved. +To summarise, in this paper we propose a novel dataset with more than +thirteen million tiles, a sampling approach to reduce the difficulty of using +large datasets, a deep learning model that is trained with mixed supervision, +evaluated on two external datasets, and incorporated in a prototype that +provides a simple integration in clinical practice and visual explanations of +the model’s predictions. This way, we come a step closer to making CAD +tools a reality for colorectal diagnosis. +2. Methods +In this section, after defining the problem at hand, we introduce the pro- +posed dataset used to train, validate and test the model, the external datasets +to evaluate the generalisation capabilities of the model and the pre-processing +pipeline. Afterwards, we describe in detail the methodology followed to cre- +ate the deep learning model and to design the experiments. Finally, we also +detail the clinical assessment and evaluation of the model. +2.1. Problem definition +Digitised colorectal cancer histological samples have large dimensions, +which are far larger than the dimensions of traditional images used in medi- +cal or general computer vision problems. Labelling such images is expensive +and highly dependent on the availability of expert knowledge. It limits the +5 + +availability of whole slide images, and, in scenarios where these are available, +meaningful annotations are usually lacking. On the other hand, it is easier +to label the dataset at the slide level. The inclusion of detailed spatial an- +notations on approximately 10% of the dataset has been shown to positively +impact the performance of deep learning algorithms [17, 31]. +Figure 1: Problem definition as a fully supervised task (on top), and as a weakly-supervised +task (bottom). +To fully leverage the potential of spatial and slide labels, we propose a +deep learning pipeline, based on previous approaches [17, 31], using mixed +supervision. Each slide, S is composed of a set of tiles Ts,n, where s represents +the index of the slide and n ∈ {1, · · · , ns} the tile number. Furthermore, +there is an inherent order in the grading used to classify the input into one of +the C(k) classes, which represents a variation in severity. For fully supervised +learning, only strongly annotated slides are useful, and for those, the label of +each tile Cs,n is known. The remaining slides are deprived of these detailed +labels, hence, they can only be leveraged by training algorithms with weakly +supervision. To be used by these algorithms, the weakly annotated slides +have only a single label for the entire bag (set) of tiles, as seen in Figure 1. +6 + +Following the order of the labels and the clinical knowledge, we assume that +the predicted slide label Cs is the most severe case of the tile labels: +Cs = maxn{Cs,n}. +In other words, if there is at least one tile classified as containing high- +grade dysplasia, then the entire slide that contains the tile is classified ac- +cordingly. On the other end of the spectrum, if the worst tile is classified as +non-neoplastic, then it is assumed that there is no dysplasia in the entire set +of tiles. This is a generalisation of multiple-instance learning (MIL) to an +ordinal classification problem, as proposed by Oliveira et al. [17]. +2.2. Datasets +The spectrum of large-scale CRC/CRS datasets is slowly increasing due to +the contributions of several researchers. Two datasets that have been recently +introduced in the literature are the CRS1K [17] and CRS4K [31] datasets. +Since the latter is an extension of the former with roughly four times more +slides, it will be the baseline dataset for the remaining of this document. +Moreover, we further extend these with the CRS10K dataset, which contains +9.26x and 2.36x more slides than CRS1K and CRS4K, respectively. Similarly, +the number of tiles is multiplied by a factor of 12.2 and 2.58 (Table 1). This +volume of slides is translated into an increase in the flexibility to design +experiments and infer the robustness of the model. Thus, the inclusion of a +test set separated from the validation set is now facilitated. +The set is composed of colorectal biopsies and polypectomies (excluding +surgical specimens). Following the same annotation process as the previ- +ous datasets, CRS10K slides are labelled according to three main categories: +non-neoplastic (NNeo), low-grade lesions (LG), and high-grade lesions (HG). +The first, contains normal colorectal mucosa, hyperplasia and non-specific +inflammation. LG lesions categorise conventional adenomas with low-grade +dysplasia. Finally, HG lesions are composed of adenomas with high-grade +dysplasia (including intra-mucosal carcinomas) and invasive adenocarcino- +mas. In order to avoid diversions from the main goal, slides with suspicion +of known history of inflammatory bowel disease/infection, serrated lesions or +other polyp types were not included in the dataset. +The slides, retrieved from an archive of previous cases, were digitised with +Leica GT450 WSI scanners, at 40× magnification. The cases were initially +seen and classified (labelled) by one of three pathologists. The pathologist +revised and classified the slides, and then compared them with the initial +report diagnosis (which served as a second-grader). If there was a match +7 + +between both, no further steps were taken. +In discordant cases, a third +pathologist served as a tie-breaker. Roughly 9% of the dataset (967 slides and +over a million tiles) were manually annotated by a pathologist and rechecked +by the other, in turn, using the Sedeen Viewer software [32]. For complex +cases, or when the agreement for a joint decision could not be reached, a +third pathologist reevaluated the annotation. +The CRS10K dataset was divided into train, validation and test sets. +The first includes all the strongly annotated slides and other slides randomly +selected. +Whereas the second is composed of only non-annotated slides. +Finally, the test set was selected from the new data added to extend the +previous datasets. Thus, it is completely separated from the training and +validation sets of previous works. The test set, will be publicly available, so +that future research can directly compare their results and use that set as a +benchmark. +Table 1: Dataset summary, with the number of slides (annotated samples are detailed in +parentheses) and tiles distributed by class for all the datasets used in this study. +NNeo +LG +HG +Total +# slides +300 (6) +552 (35) +281 (59) +1133 (100) +CRS1K dataset [17] +# annotated tiles +49,640 +77,946 +83,649 +211,235 +# non-annotated tiles +- +- +- +1,111,361 +# slides +663 (12) +2394 (207) +1376 (181) +4433 (400) +CRS4K dataset [31] +# annotated tiles +145,898 +196,116 +163,603 +505,617 +# non-annotated tiles +- +- +- +5,265,362 +# slides +1740 (12) +5387 (534) +3369 (421) +10,496 (967) +CRS10K dataset +# annotated tiles +338,979 +371,587 +341,268 +1,051,834 +# non-annotated tiles +- +- +- +13,571,871 +CRS Prototype +# slides +28 +44 +28 +100 +# non-annotated tiles +- +- +- +244,160 +PAIP [33] +# slides +- +- +100 +100 +# non-annotated tiles +- +- +- +97,392 +TCGA [34] +# slides +1 +1 +230 +232 +# non-annotated tiles +- +- +- +1,568,584 +Furthermore, as detailed in the following sections, this work comprises the +development of a fully-functional prototype to be used in clinical practice. +Leveraging this prototype, it was possible to further collect a new set with +100 slides. It differs from the CRS10K dataset, in the sense that they were +not carefully selected from the archives. Instead, these cases were actively +collected from the current year’s routine exams. We argue that this might +8 + +better reflect the real-world data distribution. Hence, we introduce this set as +a distinct dataset to evaluate the robustness of the presented methodology. +Differently from the datasets discussed below, the CRS Prototype dataset +has a more balanced distribution of the slide labels. Although it is useful +in practice, the usage of the fragment counting and selection algorithm for +the evaluation could potentiate the propagation of errors from one system +to another. Thus, in this evaluation, we did not use the fragment selection +algorithm, and as shown in Table 1, the number of tiles per slide doubles +when compared to CRS10K, which had its fragments carefully selected. +To evaluate the domain generalisation of the proposed approach, two +external datasets were used. We evaluate the proposed approaches on two +external datasets publicly available. The first dataset is composed of samples +of the TCGA-COAD [35] and TCGA-READ [36] collections from The Can- +cer Imaging Archive [34], which are composed in general by resection samples +(in contrast to our dataset, composed only of biopsies and polypectomies). +Samples containing pen markers, large air bubbles over tissue, tissue folds, +and other artefacts affecting large areas of the slide were excluded. The fi- +nal selection includes 232 whole-slide images reviewed and validated by the +same pathologists that reviewed the in-house datasets. 230 of those sam- +ples were diagnosed as high-grade lesions, whereas the remaining two have +been diagnosed as low-grade and non-neoplastic. For this dataset, the spe- +cific model of the scanner used to digitise the images is unknown, but the +file type (”.svs”) matches the file type of the training data. The second ex- +ternal dataset used to evaluate the model contains 100H&E slides from the +Pathology AI Platform [33] colorectal cohort, which contains all the cases +with a more superficial sampling of the lesion, for a better comparison with +our datasets. All the whole slide images in this dataset were digitised with +an Aperio AT2 at 20X magnification. Finally, the pathologists’ team fol- +lowed the same guidelines to review and validate all the WSI, which were all +classified as high-grade lesions. It is interesting to note that while the PAIP +contains significantly fewer tiles per slide, around 973, than the CRS10K +dataset, around 1293, the TCGA dataset shows the largest amount of tissue +per slide with an average of 6761 tiles as seen in Table 1. +2.3. Data pre-processing +H&E slides are composed of two distinct elements, white background +and colourful tissue. Since the former is not meaningful for the diagnostic, +9 + +the pre-processing of these slides incorporates an automatic tissue segmenta- +tion with Otsu’s thresholding [37] on the saturation (S) channel of the HSV +colour space, resulting in a separation between the tissue regions and the +background. The result of this step, which receives as input a 32× down- +sampled slide, is the mask used for the following steps. +Leveraging this +previous output, tiles with a dimension of 512 × 512 pixels (Figure 2) were +extracted from the original slide (without any downsampling) at its maxi- +mum magnification (40×), if they did not include any portion of background +(i.e. a 100% tissue threshold was used). Following previous experiments in +the literature, our empirical assessment, and the confirmation that smaller +tiles would significantly increase the number of tiles and the complexity of +the task, 512 × 512 was chosen as the tile size. Moreover, it is believed that +512 × 512 is the smallest tile size that still incorporates enough information +to make a good diagnostic with the possibility of visually explaining the de- +cision [17]. The selected threshold of 100% further reduces the number of +tiles by not including the tissue at the edges and decreases the complexity +of the task, since the model does not see the background at any moment. +Due to tissue variations in different images, there is also a different number +of tiles extracted per image. +2.4. Methodology +The massive size of images, which translates to thousands of tiles per +image, allied to a large number of samples in the CRS10K dataset, bottle- +necks the training of weakly-supervised models based on multiple instance +learning (MIL). Hence, in this document, we propose a mix-supervision ap- +proach with self-contained tile sampling to diagnose colorectal cancer samples +from whole-slide images. This subsection comprises the methodology, which +includes supervised training, sampling and weakly-supervised learning. +2.4.1. Supervised Training +As mentioned in previous sections, spatial annotations are rare in large +quantities. However, these include domain information, given by the expert +annotator, concerning the most meaningful areas and what are the most and +less severe tiles. Thus, they can facilitate the initial optimisation of a deep +neural network. As shown in the literature, there has been some research on +the impact of starting the training with a few iterations of fully-supervised +training [17, 38]. We further explore this in three different ways. First, we +have 967 annotated slides resulting in more than one million annotated tiles +10 + +Figure 2: Examples of regions with and a sample tile with 512 × 512 pixels (40× mag- +nification), representing each class: non-neoplastic (on top), low-grade dysplasia (on the +middle) and high-grade dysplasia (on the bottom). +for supervised training. Secondly, attending to the size of our dataset and +the need for a stronger initial supervised training, the models are trained +for 50 epochs, and their performance was monitored over specific checkpoint +epochs. Finally, we explore this pre-trained model as the main tool to sample +useful tiles for the weakly-supervised task. +11 + +Figure 3: Overall scheme of the proposed methodology containing the mix-supervision +framework that is responsible for diagnosing colorectal samples from WSI. +2.4.2. Tile Sampling +Our scenario presents a particularly difficult condition for scaling the +training data. First, let’s consider the structure of the data, which consists +of, on average, more than one thousand tiles per slide. Within this set of +tiles, some tiles provide meaningful value for the prediction, and others do +not add extra information. In other words, for the CRS10K dataset, the +extensive, lengthy, time and energy-consuming process of going through 13 +million tiles every epoch can be avoided, and as result, these models can be +trained for more epochs. Nowadays, there is an increasing concern regarding +energy and electricity consumption. Thus, these concerns, together with the +sustainability goals, further support the importance of more efficient training +processes. +Let T be the original set of tiles, and Ts be the original set of tiles from +the slide s, the former is composed by a union of the latter of all the slides +(Eq. 1). We propose to map T to a smaller set of tiles M without affecting +the overall performance and behaviour of the trained algorithm. +12 + +85 +annotated WsI +annotatedtiles (512x512px) +tiles classifier +non-annotated WSI +tiles set (512x512 px) +tiles classifier +tiles ranking +tiles sampling +(inference) +(top 200) +orw grad +T +sampled tiles +tiles classifier +tiles ranking +tiles classifier +slide diagnosis +(inference) +(training with top 5) +(top tile prediction)T = +S� +s=1 +Ts +(1) +The model trained in a fully supervised task, previously described, pro- +vides a good estimation of the utility of each tile. +Hence, we utilise the +function (Φ) learned by the model to compute the predicted severity of each +tile. As will be shown below, the weakly-supervised method utilises only the +five most severe tiles per slide to train in each epoch. As such, we select M +tiles per slide (M=200 in our experimental setup) utilising a Top-k function +(with k set to 200) to be retained for the weakly-supervised training. As in- +dicated by the results presented in the following sections, the value of M was +selected in accordance with a trade-off between information lost and training +time. This is formalised in Eq. 2. +Ms = Top-k(Φ(Ts)) +(2) +For instance, in the CRS10K dataset, the total number of tiles after +sampling would be at most 2,099,200, which represents a reduction of 6.46× +when compared to the total number of slides. Despite this upper bound on +the number of tiles, there are WSI samples that contain less than M tiles, and +as such, they remain unsampled and the actual total number of tiles after +sampling is potentially lower. During the evaluation and test time, there is +no sampling. +We conducted extensive studies on the performance of our methodology, +without sampling, with sampling on the training data, and with sampling +on training and validation. The results on the CRS4K dataset validate our +proposal. The number of selected tiles considers a trade-off between compu- +tational cost and information potentially lost, and for that reason, it is the +success of empirical optimization. +2.4.3. Weakly-Supervised Learning +The weakly-supervised learning approach designed for our methodology +follows the same principles of recent work [31]. It is divided into two distinct +stages, tile severity analysis and training. The former utilises the pre-trained +model to evaluate the severity of every tile in a set of tiles. In the first epoch, +T , the set of all the tiles in the complete dataset is used. This is possible +since the model used to assess the severity in this epoch is the same one used +for sampling. Hence, both tasks are integrated with the initial epoch. The +13 + +following epochs utilise the sampled tile set M instead of the original set. +This overall structure is represented in Figure 3. +The link between both stages is guided by a slide-wise tile ranking ap- +proach based on the expected severity. For tile Ts,n, the expected severity is +defined as +E( ˆCs,n) = +K +� +i=1 +i × p +� +ˆCs,n = C(i)� +(3) +where ˆCs,n is a random variable on the set of possible class labels {C(1), · · · , C(K)} +and p +� +ˆCs,n = C(i)� +are the K output values of the neural network. After +this analysis, the five most severe tiles are selected for training. The number +of selected tiles was chosen in accordance with previous studies [31]. These +five tiles per slide are used to train the proposed model for one more epoch. +Each epoch is composed of both stages, which means that the tiles used for +training vary across epochs. The slide label is used as the ground truth of +all five tiles of that same slide used for network optimisation. For validation +and evaluation, only the most severe tile is used for diagnostics. Although +it might lead to an increase in false positives, it shall significantly reduce +false negatives. Furthermore, we argue that increasing the variability and +quantity of data available leads to a better balance between the reduction of +these two types of errors. +2.5. Confidence Interval +In order to quantify the uncertainty of a result, it is common to compute +the 95 percent confidence interval. In this way, two different models can +be easily understood and compared based on the overlap of their confidence +intervals. The standard approach to calculating these intervals requires sev- +eral runs of a single experiment. As we increase the number of runs, our +interval becomes narrower. However, this procedure is impractical for the +computationally intensive experiments presented in this document. Hence, +we use an independent test set to approximate the confidence interval as a +Gaussian function [39]. To do so, we compute the standard error (SE) of an +evaluation metric m, which is dependent on the number of samples (n), as +seen in Equation 4. +SE = +� +1 +n × m × (1 − m) +(4) +14 + +For the SE computation to be mathematically correct, the metric m must +originate from a set of Bernoulli trials. In other words, if each prediction is +considered a Bernoulli trial, then the metric should classify them as correct +or incorrect. The number of correct samples is then given by a Binomial +distribution X ∼ (n, p), where p is the probability of correctly predicting a +label, and n is the number of samples. For instance, the accuracy is a metric +that fits all these constraints. +Following the definition and the properties of a Normal distribution, we +compute the number of standard deviations (z), known as a standard score, +that can be translated to the desired confidence (c) set to 95% of the area +under a normal distribution. This is a well-studied value, which is approx- +imately z ≈ 1.96. +This value z is then used to calculate the confidence +interval, calculated as the product of z and SE as seen in Equation 5. +M ± z ∗ +� +1 +n × m × (1 − m) +(5) +2.6. Experimental setup +For our experimental setup, we divide our data into training and valida- +tion sets. Besides, we further evaluate the performance of the former in our +test set composed of slides never seen by any of the methods presented or +in the literature. Following the split of these three sets, we have 8587, 1009 +and 900 stratified non-overlapping samples in the training, validation and +test set, respectively. +In an attempt to also contribute to reproducible research, the training of +all the versions of the proposed algorithm uses the deterministic constraints +available on Pytorch. The usage of deterministic constraints implies a trade- +off between performance, either in terms of algorithmic efficiency or on its +predictive power, and the complement with reproducible research guidelines. +As such, due to the current progress in the field, we have chosen to comply +with the reproducible research guidelines. +All the trained backbone networks were ResNet-34 [40]. Pytorch was used +to train these networks with the Adaptive Moment Estimation (Adam) [41] +optimiser, a learning rate of 6 × 10−6 and a weight decay of 3 × 10−4. The +training batch size was set to 32 for both fully and weakly supervised train- +ing, while the test and inference batch size was 256. The performance of the +model was evaluated on the validation set used for model selection in terms +of the best accuracies and quadratic weighted kappa (QWK). The training +15 + +was accelerated by an Nvidia Tesla V100 (32GB) GPU for 50 epochs of both +weakly and fully supervised learning. In addition to the proposed method- +ology, we extended our experiments to include the aggregation approach +proposed by Neto et al. [31] on top of our best-performing method. +2.7. Label correction +The complex process of labelling thousands of whole-slide images with +colorectal cancer diagnostic grades is a task of increased difficulty. It should +also be noted and taken into account that grading colorectal dysplasia is hur- +dled by considerable subjectivity, so it is to be expected that some borderline +cases will be classified by some pathologists as low-grade and others as high- +grade. Moreover, as the number of cases increases, it becomes increasingly +difficult to maintain perfect criteria and avoid mislabelling. For this reason, +we have extended the analysis of the model’s performance to understand its +errors and its capability to detect mislabelled slides. +After training the proposed model, it was evaluated on the test data. Fol- +lowing this evaluation, we identified the misclassified slides and conducted +a second round of labelling. These cases were all blindly reviewed by two +pathologists, and discordant cases from the initial ground truth were dis- +cussed and classified by both pathologists (and in case of doubt/complexity, +a third pathologist was also consulted). We tried to maintain similar criteria +between the graders and always followed the same guidelines. These new +labels were used to rectify the performance of all the algorithms evaluated in +the test set. We argue that the information regarding the strength/confidence +of predictions of a model used as a second opinion is of utter importance. A +correct integration of this feature can be shown as extremely insightful for +the pathologists using the developed tool. +2.8. Prototype and Interpretability Assessment +The proposed algorithm was integrated into a fully functional prototype +to enable its use and validation in a real clinical workflow. +This system +was developed as a server-side web application that can be accessed by any +pathologist in the lab. The system supports the evaluation of either a sin- +gle slide or a batch of slides simultaneously and in real time. It also caches +the most recent results, allowing re-evaluation without the need to re-upload +slides. In addition to displaying the slide diagnosis, and confidence level for +each class, a visual explanation map is also retrieved, to draw the patholo- +gist’s attention to key tissue areas within each slide (all seen in Figure 4). +16 + +The opaqueness of the map can be set to different thresholds, allowing the +pathologist to control its overlay over the tissue. An example of the zoomed +version of a slide with lower overlay of the map is shown in Figure 5. +Figure 4: Main view of the CAD system prototype for CRS: Slide identification, confidence +per class, diagnostic, mask overlay controller, results download as csv and slide search are +some of the features visible. Slide identification is anonymised. +Furthermore, the prototype also allows user feedback where the user can +accept/reject a result and provide a justification (Figure 6), an important +feature for software updates, research development and possible active learn- +ing frameworks that can be developed in the future. These results can be +downloaded with the corrected labels to allow for further retraining of the +model. +There are several advantages to developing such a system as a server- +side web application. First, it does not require any specific installation or +dedicated local storage in the user’s device. Secondly, it can be accessed at +the same time by several pathologists from different locations, allowing for +a quick review of a case by multiple pathologists without data transference. +Moreover, the lack of local storage of clinical data increases the privacy of +patient data, which can only be accessed through a highly encrypted virtual +private network (VPN). Finally, all the processing can be moved to an effi- +cient graphics processing unit (GPU), thus reducing the processing time by +17 + + CADPath ++ +1 +人 +CADPath + Search +Search +Download Results +00000000001-A- +OOOO +01-001_xxx_000 +000001.svs +processed +Processed +00000000002-A- +Case: 00000000003 +02-002_xxx_000 +Specimen: A +_000002.svs +Block: 03 + processed +Slide: 003 +Mask +00000000003-A +03-003_xxx_000 +_000003.svs +Model Results +processed +High grade +100% +Low grade +00000000004-A- +Normal +04-004_xxx_000 +_000004.svs +processed +High Grade +00000000005-A- +Results approved +05-005_xxx_000 +_000005.svs +processed +00000000006-A. +06-006_xxx_000 +_000006.svs +Upload new slidesFigure 5: Zoomed view of a slide from the CAD system prototype, with the predictions +map with a small overlay threshold. Slide identification is anonymised. +several orders of magnitude. Similar behaviour on a local machine would +require the installation of dedicated GPUs in the pathologists’ personal de- +vices. This platform is the first Pathology prototype for colorectal diagnosis +developed in Portugal, and, as far as we know, one of the pioneers in the +world. Its design was also carefully thought to be aligned with the needs of +the pathologists. +3. Results +In this section, we present the results of the proposed method. The results +are organised to first demonstrate the effectiveness of sampling, followed by +an evaluation of the model in the two internal datasets (CRS10K and the +prototype dataset), and finalise with the results on external datasets. We also +discuss the advantages and disadvantages of the proposed approach, perform +an analysis of the results from a clinical perspective, provide pathologists’ +feedback on the use of the prototype, and finally discuss future directions. +18 + + CADPath ++ +人☆ +CADPath +Search +Search +Download Results +processed +00000000006-A +06-006_xxx_000 +Upload new slides ++0:00000000Figure 6: CAD system prototype report tool: the user can report if the result is either +correct, wrong or inconclusive and leave a comment for each case individually. +Slide +identification is anonymised. +3.1. On the effectiveness of sampling +To find the most suitable threshold for sampling the tiles used in the +weakly supervised training, we evaluated the percentage of relevant tiles that +would be left out of the selection, if the original set was reduced to 75, 100, +150 or 200 tiles, over the first five inference epochs. +A tile is considered +relevant if it shares the same label as the slide, or if it would take part in +the learning process in the weakly-supervised stage. As it is possible to see +in Figure 7, if we set the maximum number of tiles to 200 after the second +loop of inference, we would discard only 3.5% of the potentially informative +tiles, in the worst-case scenario. On the other side of the spectrum, a more +radical sampling of only 50 tiles would lead to a cut of up to 8%. +Moreover, to assess the impact of this sampling on the model’s perfor- +mance, we also evaluated the accuracy and the QWK with and without +sampling the top 200 tiles after the first inference iteration (Table 2). This +evaluation considered sampling applied only to the training tile set, and to +both the training and validation tile sets. As can be noticed, the performance +is not degraded and the model is trained in a much faster way. In fact, using +the setup previously mentioned, the first epoch of inference, with the full set +19 + +CADPath +x +人☆ +CADPath + Search + Download Results +ownload +Feedback +Approval State: +00000000001-A- +01-001_xxx_000 +000001.svs +Expected: +(Low Grade + Normal +High Grade +processed +Processed +Comments +00000000002-A- +Case: 00000000002 +02-002_xxx_000 +Specimen: A +000002.svS +Block: 02 +processed +Slide: 002 +Mask +00000000003-A- +03-003_xxx_000 +000003.SvS +Model Results +processed +80% +Low grade +20% +00000000004-A- + Save changes +Normal +04-004_xxx_000 +000004.svs +High Grade +processed +00000000005-A. +Results rejected! +05-005_xxx_000 +000005.SVS +processed +O +00000000006-A- +06-006_xxx_000 +000006.svs +Upload new slidesFigure 7: Tile sampling impact on information loss: percentage of tiles not selected due +to sampling with different thresholds, over the first four inference epochs. +of tiles takes 28h to be completed, while from the second loop the training +time decreases to only 5h per epoch. Without sampling, training the model +for 50 epochs would take around 50 days, whereas with sampling it takes +around 10. +3.2. CRS10K and Prototype +CRS10K test set and the prototype dataset were collected through dif- +ferent procedures. The first followed the same data collection process as the +complete dataset, whereas the second originated from routine samples. Thus, +the evaluation of both these sets is done separately. +The first experiment was conducted on the CRS10K test set. +As ex- +pected, the steep increase in the number of training samples led to a signif- +icantly better algorithm in this test set. Initially, the model trained on the +CRS10K correctly predicted the class of 819 out of 900 samples. For the +20 + +8 +Info +Samplingof 20o tiles +sampling +Sampling of 1oo tiles +selected +Sampling of 75 tiles +Sampling of 50 tiles +01 +due +tiles +not selected +of +total number +4 +3 +tiles +2 +1 +0 +1 +2 +3 +4 +5 +EpochsTable 2: Model performance comparison with and without tile sampling of the top 200 +tiles from the first inference iteration. Compared the best epoch of the initial five epochs +and of the initial ten epochs. Validation is represented as Val. +Best Accuracy at +Best QWK at +Sampling +5th epoch +10th epoch +5th epoch +10th epoch +No +84.94% ± 2.20 +86.42% ± 2.11 +0.809 ± 0.024 +0.829 ± 0.023 +Train +85.43% ± 2.18 +86.82% ± 2.08 +0.817 ± 0.024 +0.828 ± 0.023 +Train and Val. +86.12% ± 2.13 +86.92% ± 2.08 +0.824 ± 0.023 +0.829 ± 0.023 +Table 3: Model performance evaluation on the CRS10K test set. The binary accuracy is +calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results +per column. +Method +ACC +Binary ACC +Sensitivity +iMIL4Path +91.33% ± 1.84 +97.00% ± 1.11 +0.997 ± 0.004 +Ours (CRS4K) +89.44% ± 2.01 +96.11% ± 1.26 +0.997 ± 0.004 +Ours (CRS10K) wo/ Agg +93.44% ± 1.62 +97.78% ± 0.96 +0.996 ± 0.005 +Ours (CRS10K) w/ Agg +90.67% ± 1.90 +97.55% ± 1.01 +0.985 ± 0.009 +wrong 81 cases, the pathologists performed a blind review of these cases and +found that the algorithm was indeed correct in 22 of them. This led to a +correction in the labels of the test set, and the appropriate adjustment of +the metrics. In Table 3, the performance of the different algorithms is pre- +sented. CRS10K outperforms the other approaches by a reasonable margin. +We further applied the aggregation proposed by Neto et al. [31] to the best +performing method, but without gains in performance. Despite being trained +on the same dataset iMIL4Path and the proposed methodology trained on +CRS4K, utilise different splits for training and validation, as well as different +optimisation techniques due to the deterministic approach. +In addition to examining quantitative metrics, such as the accuracy of +the model, we extended our study to include an analysis of the confidence in +the model when it correctly predicts a class and when it makes an incorrect +prediction. To this end, we recorded the confidence of the model for the +predicted class and divided it into the set of correct and incorrect predictions. +These were then used to fit a kernel density estimator (KDE). Figure 8 shows +the density estimation of the confidence values for the three different models. +It is worth noting that, when correct, the model trained on the CRS10K, +21 + +Figure 8: Kernel density estimation of the confidences of correct and incorrect predic- +tions performed on the three-class classification problem by three distinct models on the +CRS10K test set. The plots represent, from left to right, the proposed method trained on +CRS10K, the proposed method trained on CRS4K and iMIL4Path. +returns higher confidence levels as shown by the shift of its mean towards +values close to one. On the other hand, the confidence values of its incorrect +predictions decrease significantly, and although it does not present the lowest +values, it shows the largest gap between correct and incorrect means. +Table 4: Model performance evaluation on the prototype test set. Accuracy is represented +as (ACC). The binary accuracy is calculated as NNeo vs all. In bold are the best results +per column. +Method +ACC +Binary ACC +Sensitivity +iMIL4Path +89.00% ± 6.13 +96.00% ± 3.84 +1.000 ± 0.000 +Ours (CRS4K) +85.00% ± 6.99 +93.00% ± 5.00 +1.000 ± 0.000 +Ours (CRS10K) wo/ Agg +89.00% ± 6.13 +98.00% ± 2.74 +0.986 ± 0.026 +Ours (CRS10K) w/ Agg +85.00% ± 6.99 +98.00% ± 2.74 +0.986 ± 0.026 +When tested on the prototype data (n=100), the importance of a higher +volume of data remains visible (Table 4). Nonetheless, the performance of +iMIL4Path [31] approach is comparable to the proposed approach trained on +CRS10K. It is worth noting that the latter achieves better performance on +the binary accuracy at the cost of a decrease in sensibility. In other words, +the capability to detect negatives increases significantly. Due to the smaller +22 + +CRS10KTest Set-Ours(CRS10K) +CRS10KTestSet-Ours(CRS4K) +CRS10K Test Set - iMIL4Path +8 +8 +8 +Correct +Correct +Correct +Mean = 0.961 +Mean = 0.953 +. +Mean = 0.956 +7 +Incorrect +7 +Incorrect +7 +Incorrect +Mean = 0.769 +Mean = 0.762 +Mean = 0.773 +6 +6 +6 +5 +5 +5 +/p(c) +/p(c) +........ +3 +3 +3 +2 +2 +2 +1 +1 +1 +0 +0 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidencec +Confidencec +Confidence cset of slides, the confidence interval is much wider, as such, the performance +on the CRS10K test set is a good indication of how these values would shift +if more data was added. Similar performance drops were linked with the +introduction of aggregation. +Figure 9: Kernel density estimation of the confidences of correct and incorrect predictions +performed on the three-class classification problem by three distinct models on the proto- +type set. The plots represent, from left to right, the proposed method trained on CRS10K, +the proposed method trained on CRS4K and iMIL4Path. +Despite similar results, the confidence of the model in its predictions is +distinct in all three approaches, as seen in Figure 9. The proposed approach +when trained on the CRS10K dataset has a larger density on values close +to one when the predictions are correct, and the mean confidence of those +predictions is, once more, higher than the other approaches. However, espe- +cially when compared to the proposed approach trained on the CRS4K, the +confidence of wrong predictions is also higher. It can be a result of a larger +set of wrong predictions available on the latter approach. Nonetheless, the +steep increase in the density of values closer to one further indicates that +there is room to explore other effects of extending the number of training +samples, besides benefits in quantitative metrics. +3.3. Domain Generalisation Evaluation +To ensure the generalisation of the proposed approach across external +datasets, we have evaluated their performance on TCGA and PAIP. More- +23 + +Prototype-Ours(CRS10K) +Prototype-Ours(CRS4K) +Prototype - iMIL4Path +8 +8 +8 +Correct +Correct +Correct +Mean = 0.943 +Mean = 0.923 +Mean = 0.916 +7 +Incorrect +7 +Incorrect +7 +Incorrect +.... +Mean = 0.84 +Mean = 0.757 +Mean =0.818 +9 +6 +6 +5 +5 +5 +p(c) +/ p(c) +3 +3 +3 +2 +2 +2 +1 +1 +1 +0 +0 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidencec +Confidencec +Confidence cover, we conducted a similar analysis of both of these datasets, as the one +done for the internal datasets. +Table 5: Model performance evaluation on the PAIP test set. The binary accuracy is +calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results +per column. +Method +ACC +Binary ACC +Sensitivity +iMIL4Path +99.00% ± 1.95 +100.00% ± 0.00 +1.000 ± 0.000 +Ours (CRS4K) +69.00% ± 9.06 +100.00% ± 0.00 +1.000 ± 0.000 +Ours (CRS10K) wo/ Agg +100.00% ± 0.00 +100.00% ± 0.00 +1.000 ± 0.000 +Ours (CRS10K) w/ Agg +52.00 ± 9.79 +100.00% ± 0.00 +1.000 ± 0.000 +From the two datasets, PAIP is arguably the closest to CRS10K. It con- +tains similar tissue, despite its colour differences. The performances of the +proposed approaches were expected to match the performance of iMIL4Path +in this dataset. However, it did not happen for the version trained on the +CRS4K dataset, as seen in Table 5. A viable explanation concerns potential +overfitting to the training data potentiated by an increase in the number of +epochs of fully and weakly supervised training, a slight decrease in the tile +variability in the latter approach, and a smaller number of samples when +compared to the version trained on CRS10K. This version, trained on the +larger set, mitigates the problems of the other method due to a significant +increase in the training samples. Moreover, it is worth noting that in all +three approaches, the errors corresponded only to a divergence between low +and high-grade cases, with no non-neoplastic cases being classified as high- +grade or vice-versa. As in previous sets, the version trained on the CRS10K +dataset outperforms the remaining approaches. Using aggregation in this +dataset leads to a discriminative power to distinguish between high- and +low-grade lesions that is close to random. +In two of the three approaches, the number of incorrect samples is one +or zero, as such, there is no density estimation for wrong samples in their +confidence plot as seen in Figure 10. Yet, it is visible the shift towards higher +values of confidence in the proposed approach trained on the CRS10K when +compared to the method of iMIL4Path. +The version trained on CRS4K +shows very little separability between the confidence of correct and incorrect +predictions. +The TCGA dataset has established itself as the most challenging for the +proposed approaches. Besides the expected differences in colour and other +24 + +Figure 10: Kernel density estimation of the confidences of correct and incorrect predictions +performed on the three-class classification problem by three distinct models on the PAIP +dataset. The plots represent, from left to right, the proposed method trained on CRS10K, +the proposed method trained on CRS4K and iMIL4Path. +Table 6: Model performance evaluation on the TCGA test set. The binary accuracy is +calculated as NNeo vs all. Accuracy is represented as (ACC). In bold are the best results +per column. +Method +ACC +Binary ACC +Sensitivity +iMIL4Path +71.55% ± 5.80 +80.60% ± 5.05 +0.805 ± 0.051 +Ours (CRS4K) wo/ Agg +70.69% ± 5.86 +98.71% ± 1.45 +0.991 ± 0.012 +Ours (CRS10K) wo/ Agg +84.91% ± 4.61 +99.13% ± 1.19 +0.996 ± 0.008 +Ours (CRS10K) w/ Agg +69.83% ± 5.91 +97.41% ± 2.04 +0.983 ± 0.017 +elements, this dataset is mostly composed of resection samples, which are not +present in the training dataset. As such, this presents itself as an excellent +dataset to assess the capability of the model to handle these different types of +samples. Both iMIL4Path and the proposed method trained on CRS4K have +shown substantial problems in correctly classifying TCGA slides, as shown +in Table 6. Despite having a lower performance on the general accuracy, +the binary accuracy shows that our proposed method trained on CRS4K has +much lower misclassification errors regarding the classification of high-grade +samples as normal, demonstrating higher robustness of the new training ap- +proach against errors with a gap of two classes. As with other datasets, the +proposed approach trained on CRS10K shows better results, this time by a +25 + +PAIP -Ours(CRS10K) +PAIP - Ours(CRS4K) +PAIP - iMIL4Path +8 +8 +8 +Correct +Correct +Correct +Mean = 0.99 +Mean = 0.843 +Mean = 0.964 +7 +7 +Incorrect +7 +Mean =0.835 +9 +6 +6 +5 +5 +5 +/ p(c) +p(c) +3 +3 +3 +2 +2 +2 +1 +1 +1 +0 +: +0 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidencec +Confidencec +Confidence csignificant margin with no overlapping between the confidence intervals. +Figure 11: Kernel density estimation of the confidences of correct and incorrect predictions +performed on the three-class classification problem by three distinct models on the TCGA +dataset. The plots represent, from left to right, the proposed method trained on CRS10K, +the proposed method trained on CRS4K and iMIL4Path. +Inspecting the predictions’ confidence for the three models indicates a be- +haviour in line with the accuracy-based performance (Figure 11). Moreover, +a confidence shift of wrong predictions’ confidence towards smaller values is +clearly visible in the plot corresponding to the model trained on CRS10K. +The shown gap of 0.2 between the confidence of correct and wrong predic- +tions, indicates that it is possible to quantify the uncertainty of the model +and avoid the majority of the wrong predictions. In other words, when the +uncertainty is above a learnt threshold, then the model refuses to make any +prediction. It is extremely useful in models designed as a second opinion +system. +3.4. Prototype usability in clinical practice +As it is currently designed, the algorithm works preferentially as a “second +opinion”, allowing the assessment of difficult and troublesome cases, without +the immediate need for the intervention of a second pathologist. Due to its +“user-friendly” nature and very practical interface, the cases can be easily +introduced into the system and results are rapidly shown and easily accessed. +Also, by not only providing results but presenting visualisation maps (cor- +responding to each diagnostic class), the pathologist is able to compare his +26 + +TCGA-Ours(CRS10K) +TCGA-Ours(CRS4K) +TCGA- iMIL4Path +8 +8 +8 +Correct +Correct +Correct +Mean=0.965 +Mean=0.956 +. +Mean=0.932 +7 +Incorrect +7 +Incorrect +7 +Incorrect +Mean = 0.764 +Mean = 0.876 +Mean = 0.815 +6 +6 +6 +5 +5 +5 +p(c) +p(c) +3 +3 +3 +2 +2 +2 +1 +1 +1 +0 +0 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidencec +Confidencec +Confidencecown remarks to those of the algorithm itself, towards a future “AI-assisted +diagnosis”. Another relevant aspect is the fact that the prototype allows for +user feedback (agreeing or not with the model’s proposed result), which can +be further integrated into further updates of the software. Also interesting, +is the possibility of using the prototype as a triage system on a pathologist’s +daily workflow (by running front, before the pathologist checks the cases). +Signalling the cases that would need to be more urgently observed (namely +high-risk lesions) would allow the pathologists to prioritise their workflow. +Further, by providing a previous assessment of the cases, it would also con- +tribute to enhancing the pathologists’ efficiency. Although it is possible to +use the model as it is upfront, it would classify some samples incorrectly +(since it was not trained on the full spectrum of colorectal pathology). As +such, the uncertainty quantification based on the provided confidence given +in the user interface could also be extremely useful. Presently, there is no rec- +ommendation for dual independent diagnosis of colorectal biopsies (contrary +to gastric biopsies, where, in cases in which surgical treatment is considered, +it is recommended to obtain a pre-treatment diagnostic second opinion [42]), +but, in case that in the future this also becomes a requirement, a tool such as +CADPath.AI prototype could assist in this task. This has increased impor- +tance due to the worldwide shortage of pathologists and so, such CAD tools +can really make a difference in patient care (in similarity, for example, with +Google Health’s research, using deep learning to screen diabetic retinopa- +thy in low/middle-income countries, in which the system showed real-time +retinopathy detection capability similar to retina specialists, alleviating the +significant manpower constrictions in this setting [43]). Lastly, we also an- +ticipate that this prototype, and similar tools, can be used in a teaching +environment since its easy use and explainable capability (through the visu- +alisation maps) allows for easy understanding of the given classifications and +having a web-based interface allows for easy sharing. +3.5. Future work +The proposed algorithm still has potential for improvement. +We aim +to include the recognition of serrated lesions, to distinguish normal mucosa +from significant inflammatory alterations/diseases, to stratify high-risk le- +sions into high-grade dysplasia and invasive carcinomas and to identify other +neoplasia subtypes. Further, we would like to leverage the model to be able +to evaluate also surgical specimens. Another relevant step will be the merge +of our dataset and external ones for training, besides only testing it on ex- +27 + +ternal samples. This will enhance its generalisation capabilities and provide +a more robust system. Lastly, we intend to measure the “user experience” +and feedback from the pathologists, by its gradual implementation in general +laboratory routine work. +The following goals comprise a more extensive evaluation of the model +across more scanner brands and labs. +We also want to promote certain +behaviours that would allow for more direct and integrated uncertainty esti- +mation. We have also been looking towards aggregation methods, but, since +in the majority of them there is an increased risk of false negatives, we have +work to do in that research direction. +4. Discussion +In this document, we have redesigned the previous methodology on MIL +for colorectal cancer diagnosis. First, we extended and leveraged the mixed +supervision approach to design a sampling strategy, which utilises the knowl- +edge from the full supervision training as a proxy to tile utility. Secondly, we +studied the confidence that the model shows in its predictions when they are +correct and when they are incorrect. Additionally, this confidence is shown to +be a potential resource to quantify uncertainty and avoid wrong predictions +on low-certainty scenarios. This is entirely integrated within a web-based +prototype to aid pathologists in their routine work. +The proposed methodology was evaluated on several datasets, including +two external sets. Through this evaluation, it was possible to infer that the +performance of the proposed methodology benefits from a larger dataset and +surpasses the performance of previous state-of-the-art models. As such, and +given the excelling results that originated from the increase in the dataset, +we are also publicly releasing the majority of the CRS10K dataset, one of +the largest publicly available colorectal datasets composed of H&E images in +the literature, including the test set for the benchmark of distinct approaches +across the literature. +Finally, we have clearly defined a set of potential future directions to be +explored, either for better model design, the development of useful prototypes +or even the integration of uncertainty in the predictions. +References +[1] International Agency for Research on Cancer (IARC), Global cancer +observatory, https://gco.iarc.fr/ (2022). +28 + +[2] H. Brody, Colorectal cancer, Nature 521 (2015) S1. +doi:10.1038/ +521S1a. +[3] D. Holmes, A disease of growth, Nature 521 (2015) S2–S3. doi:10. +1038/521S2a. +[4] Digestive Cancers Europe (DiCE), Colorectal screening in europe, +https://bit.ly/3rFxSEL. +[5] C. Hassan, G. Antonelli, J.-M. Dumonceau, J. Regula, M. Bretthauer, +S. Chaussade, E. Dekker, M. Ferlitsch, A. Gimeno-Garcia, R. Jover, +M. Kalager, M. Pellis´e, C. Pox, L. Ricciardiello, M. Rutter, L. M. +Helsingen, A. Bleijenberg, C. Senore, J. E. van Hooft, M. Dinis-Ribeiro, +E. Quintero, Post-polypectomy colonoscopy surveillance: European so- +ciety of gastrointestinal endoscopy guideline - update 2020, Endoscopy +52 (8) (2020) 687–700. doi:10.1055/a-1185-3109. +[6] D. Mahajan, E. Downs-Kelly, X. Liu, R. Pai, D. Patil, L. Rybicki, +A. Bennett, T. Plesec, O. Cummings, D. Rex, J. Goldblum, Repro- +ducibility of the villous component and high-grade dysplasia in col- +orectal adenomas <1 cm: +Implications for endoscopic surveillance, +American Journal of Surgical Pathology 37 (3) (2013) 427–433. doi: +10.1097/PAS.0b013e31826cf50f. +[7] S. Gupta, D. Lieberman, J. C. Anderson, C. A. Burke, J. A. Dominitz, +T. Kaltenbach, D. J. Robertson, A. Shaukat, S. Syngal, D. K. Rex, +Recommendations for follow-up after colonoscopy and polypectomy: A +consensus update by the us multi-society task force on colorectal cancer, +Gastrointestinal Endoscopy (2020). doi:10.1016/j.gie.2020.01.014. +[8] C. Eloy, J. Vale, M. Curado, A. Pol´onia, S. Campelos, A. Caramelo, +R. Sousa, M. Sobrinho-Sim˜oes, Digital pathology workflow imple- +mentation at ipatimup, Diagnostics 11 (11) (2021). +doi:10.3390/ +diagnostics11112111. +URL https://www.mdpi.com/2075-4418/11/11/2111 +[9] F. Fraggetta, A. Caputo, R. Guglielmino, M. G. Pellegrino, G. Runza, +V. L’Imperio, A survival guide for the rapid transition to a fully digital +workflow: The “caltagirone example”, Diagnostics 11 (10) (2021). doi: +29 + +10.3390/diagnostics11101916. +URL https://www.mdpi.com/2075-4418/11/10/1916 +[10] D. Montezuma, A. Monteiro, J. Fraga, L. Ribeiro, S. Gon¸calves, +A. Tavares, J. Monteiro, I. Macedo-Pinto, Digital pathology implemen- +tation in private practice: Specific challenges and opportunities, Diag- +nostics 12 (2) (2022) 529. +[11] A. Madabhushi, G. Lee, Image analysis and machine learning in digital +pathology: challenges and opportunities, Medical Image Analysis 33 +(2016) 170–175. doi:10.1016/j.media.2016.06.037. +[12] E. A. Rakha, M. Toss, S. Shiino, P. Gamble, R. Jaroensri, C. H. Mermel, +P.-H. C. Chen, Current and future applications of artificial intelligence in +pathology: a clinical perspective, Journal of Clinical Pathology (2020). +doi:10.1136/jclinpath-2020-206908. +[13] M. Veta, P. J. van Diest, S. M. Willems, H. Wang, A. Madabhushi, +A. Cruz-Roa, F. Gonzalez, A. B. Larsen, J. S. Vestergaard, A. B. Dahl, +D. C. Cire¸san, J. Schmidhuber, A. Giusti, L. M. Gambardella, F. B. +Tek, T. Walter, C.-W. Wang, S. Kondo, B. J. Matuszewski, F. Precioso, +V. Snell, J. Kittler, T. E. de Campos, A. M. Khan, N. M. Rajpoot, +E. Arkoumani, M. M. Lacle, M. A. Viergever, J. P. Pluim, Assessment of +algorithms for mitosis detection in breast cancer histopathology images, +Medical Image Analysis 20 (1) (2015) 237–248. doi:10.1016/j.media. +2014.11.010. +[14] G. Campanella, M. G. Hanna, L. Geneslaw, A. Miraflor, V. Werneck +Krauss Silva, K. J. Busam, E. Brogi, V. E. Reuter, D. S. Klimstra, T. J. +Fuchs, Clinical-grade computational pathology using weakly supervised +deep learning on whole slide images, Nat. Med. 25 (8) (2019) 1301–1309. +doi:10.1038/s41591-019-0508-1. +[15] S. P. Oliveira, J. Ribeiro Pinto, T. Gon¸calves, R. Canas-Marques, M.- +J. Cardoso, H. P. Oliveira, J. S. Cardoso, Weakly-supervised classifi- +cation of HER2 expression in breast cancer haematoxylin and eosin +stained slides, Applied Sciences 10 (14) (2020) 4728. +doi:10.3390/ +app10144728. +30 + +[16] T. Albuquerque, A. Moreira, J. S. Cardoso, Deep ordinal focus assess- +ment for whole slide images, in: Proceedings of the IEEE/CVF Inter- +national Conference on Computer Vision, 2021, pp. 657–663. +[17] S. P. Oliveira, P. C. Neto, J. Fraga, D. Montezuma, A. Monteiro, +J. Monteiro, L. Ribeiro, S. Gon¸calves, I. M. Pinto, J. S. Cardoso, CAD +systems for colorectal cancer from WSI are still not ready for clini- +cal acceptance, Scientific Reports 11 (1) (2021) 14358. doi:10.1038/ +s41598-021-93746-z. +[18] N. Thakur, H. Yoon, Y. Chong, Current trends of artificial intelligence +for colorectal cancer pathology image analysis: a systematic review, +Cancers 12 (7) (2020). doi:10.3390/cancers12071884. +[19] Y. Wang, X. He, H. Nie, J. Zhou, P. Cao, C. Ou, Application of artificial +intelligence to the diagnosis and therapy of colorectal cancer, Am. J. +Cancer Res. 10 (11) (2020) 3575–3598. +[20] A. Davri, E. Birbas, T. Kanavos, G. Ntritsos, N. Giannakeas, A. T. +Tzallas, A. Batistatou, Deep learning on histopathological images for +colorectal cancer diagnosis: A systematic review, Diagnostics 12 (4) +(2022) 837. +[21] O. Iizuka, F. Kanavati, K. Kato, M. Rambeau, K. Arihiro, M. Tsuneki, +Deep learning models for histopathological classification of gastric and +colonic epithelial tumours, Scientific Rep. 10 (2020). +doi:10.1038/ +s41598-020-58467-9. +[22] H. Tizhoosh, L. Pantanowitz, Artificial intelligence and digital pathol- +ogy: +challenges and opportunities, J. Pathol. Inform 9 (1) (2018). +doi:10.4103/jpi.jpi\_53\_18. +[23] J. W. Wei, A. A. Suriawinata, L. J. Vaickus, B. Ren, X. Liu, M. Lisovsky, +N. Tomita, B. Abdollahi, A. S. Kim, D. C. Snover, J. A. Baron, E. L. +Barry, S. Hassanpour, Evaluation of a deep neural network for auto- +mated classification of colorectal polyps on histopathologic slides, JAMA +Network Open 3 (4) (2020). +doi:10.1001/jamanetworkopen.2020. +3398. +[24] Z. Song, C. Yu, S. Zou, W. Wang, Y. Huang, X. Ding, J. Liu, L. Shao, +J. Yuan, X. Gou, W. Jin, Z. Wang, X. Chen, H. Chen, C. Liu, G. Xu, +31 + +Z. Sun, C. Ku, Y. Zhang, X. Dong, S. Wang, W. Xu, N. Lv, H. Shi, +Automatic deep learning-based colorectal adenoma detection system and +its similarities with pathologists, BMJ Open 10 (9) (2020). doi:10. +1136/bmjopen-2019-036423. +[25] L. Xu, B. Walker, P.-I. Liang, Y. Tong, C. Xu, Y. Su, A. Karsan, Col- +orectal cancer detection based on deep learning, J. Pathol. Inf. 11 (1) +(2020). doi:10.4103/jpi.jpi\_68\_19. +[26] K.-S. Wang, G. Yu, C. Xu, X.-H. Meng, J. Zhou, C. Zheng, Z. Deng, +L. Shang, R. Liu, S. Su, et al., Accurate diagnosis of colorectal can- +cer based on histopathology images using artificial intelligence, BMC +medicine 19 (1) (2021) 1–12. doi:10.1186/s12916-021-01942-5. +[27] G. Yu, K. Sun, C. Xu, X.-H. Shi, C. Wu, T. Xie, R.-Q. Meng, X.-H. +Meng, K.-S. Wang, H.-M. Xiao, et al., Accurate recognition of colorec- +tal cancer with semi-supervised deep learning on pathological images, +Nature communications 12 (1) (2021) 1–13. +[28] N. Marini, S. Ot´alora, F. Ciompi, G. Silvello, S. Marchesin, S. Vatrano, +G. Buttafuoco, M. Atzori, H. M¨uller, Multi-scale task multiple instance +learning for the classification of digital pathology images with global +annotations, in: M. Atzori, N. Burlutskiy, F. Ciompi, Z. Li, F. Minhas, +H. M¨uller, T. Peng, N. Rajpoot, B. Torben-Nielsen, J. van der Laak, +M. Veta, Y. Yuan, I. Zlobec (Eds.), Proceedings of the MICCAI Work- +shop on Computational Pathology, Vol. 156 of Proceedings of Machine +Learning Research, PMLR, 2021, pp. 170–181. +[29] C. Ho, Z. Zhao, X. F. Chen, J. Sauer, S. A. Saraf, R. Jialdasani, +K. Taghipour, A. Sathe, L.-Y. Khor, K.-H. Lim, et al., A promising +deep learning-assistive algorithm for histopathological screening of col- +orectal cancer, Scientific Reports 12 (1) (2022) 1–9. +[30] T. Albuquerque, A. Moreira, B. Barros, D. Montezuma, S. P. Oliveira, +P. C. Neto, J. Monteiro, L. Ribeiro, S. Gon¸calves, A. Monteiro, I. M. +Pinto, J. S. Cardoso, Quality control in digital pathology: Automatic +fragment detection and counting, in: 2022 44th Annual International +Conference of the IEEE Engineering in Medicine & Biology Society +(EMBC), 2022, pp. 588–593. doi:10.1109/EMBC48229.2022.9871208. +32 + +[31] P. C. Neto, S. P. Oliveira, D. Montezuma, J. Fraga, A. Monteiro, +L. Ribeiro, S. Gon¸calves, I. M. Pinto, J. S. Cardoso, imil4path: A +semi-supervised interpretable approach for colorectal whole-slide images, +Cancers 14 (10) (2022) 2489. +[32] Pathcore, Sedeen viewer, https://pathcore.com/sedeen (2020). +[33] P. A. Platform, Paip, http://www.wisepaip.org, last accessed on +20/01/22 (2020). +[34] K. Clark, B. Vendt, K. Smith, J. Freymann, J. Kirby, P. Koppel, +S. Moore, S. Phillips, D. Maffitt, M. Pringle, L. Tarbox, F. Prior, The +cancer imaging archive (TCIA): Maintaining and operating a public in- +formation repository, Journal of Digital Imaging 26 (2013) 1045–1057. +doi:10.1007/s10278-013-9622-7. +[35] S. Kirk, Y. Lee, C. A. Sadow, S. Levine, C. Roche, E. Bonaccio, J. Fil- +iippini, Radiology data from the cancer genome atlas colon adenocarci- +noma [TCGA-COAD] collection. (2016). doi:10.7937/K9/TCIA.2016. +HJJHBOXZ. +[36] S. Kirk, Y. Lee, C. A. Sadow, S. Levine, Radiology data from the cancer +genome atlas rectum adenocarcinoma [TCGA-READ] collection. (2016). +doi:10.7937/K9/TCIA.2016.F7PPNPNU. +[37] N. Otsu, A threshold selection method from gray-level histograms, IEEE +Transactions on Systems, Man, and Cybernetics 9 (1) (1979) 62–66. +doi:10.1109/TSMC.1979.4310076. +[38] J. Boˇziˇc, D. Tabernik, D. Skoˇcaj, Mixed supervision for surface-defect +detection: From weakly to fully supervised learning, Computers in In- +dustry 129 (2021) 103459. +[39] S. Raschka, Model evaluation, model selection, and algorithm selection +in machine learning, arXiv preprint arXiv:1811.12808 (2018). +[40] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image +recognition, in: Proceedings of the IEEE conference on computer vision +and pattern recognition, 2016, pp. 770–778. doi:10.1109/CVPR.2016. +90. +33 + +[41] D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, in: +ICLR (Poster), 2015. +[42] W. C. of Tumours Editorial Board, WHO classification of tumours of +the digestive system, no. Ed. 5, World Health Organization, 2019. +[43] P. Ruamviboonsuk, R. Tiwari, R. Sayres, V. Nganthavee, K. Hemarat, +A. Kongprayoon, R. Raman, B. Levinstein, Y. Liu, M. Schaekermann, +et al., Real-time diabetic retinopathy screening by deep learning in a +multisite national screening programme: a prospective interventional +cohort study, The Lancet Digital Health 4 (4) (2022) e235–e244. +34 + diff --git a/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/load_file.txt b/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69ee18f8e41e60e0cb61f9519816ccbbbe405fba --- /dev/null +++ b/2tE0T4oBgHgl3EQfuwEX/content/tmp_files/load_file.txt @@ -0,0 +1,1276 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf,len=1275 +page_content='A CAD System for Colorectal Cancer from WSI: A Clinically Validated Interpretable ML-based Prototype Pedro C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Netoa,b,1, Diana Montezumac,f,d,1, Sara P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveiraa,b,1, Domingos Oliveirac, Jo˜ao Fragae, Ana Monteiroc, Jo˜ao Monteiroc, Liliana Ribeiroc, Sofia Gon¸calvesc, Stefan Reinhardg, Inti Zlobecg, Isabel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pintoc, Jaime S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardosoa,b aInstitute for Systems and Computer Engineering, Technology and Science (INESC TEC), R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Roberto Frias, Porto, 4200-465, Porto, Portugal bFaculty of Engineering, University of Porto (FEUP), R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Roberto Frias, Porto, 4200-465, Porto, Portugal cIMP Diagnostics, Praca do Bom Sucesso, 61, sala 809, Porto, 4150-146, Porto, Portugal dCancer Biology and Epigenetics Group, IPO-Porto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ant´onio Bernardino de Almeida 865, Porto, 4200-072, Porto, Portugal eDepartment of Pathology, IPO-Porto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ant´onio Bernardino de Almeida 865, Porto, 4200-072, Porto, Portugal fSchool of Medicine and Biomedical Sciences, University of Porto (ICBAS), R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Jorge de Viterbo Ferreira 228, Porto, 4050-313, Porto, Portugal gInstitute of Pathology, University of Bern, Uni Bern, Murtenstrasse 31, Bern, 3008, Bern, Switzerland Abstract The integration of Artificial Intelligence (AI) and Digital Pathology has been increasing over the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nowadays, applications of deep learn- ing (DL) methods to diagnose cancer from whole-slide images (WSI) are, more than ever, a reality within different research groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nonetheless, the development of these systems was limited by a myriad of constraints re- garding the lack of training samples, the scaling difficulties, the opaqueness of DL methods, and, more importantly, the lack of clinical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, we propose a system designed specifically for the diagnosis of colorectal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The construction of such a system consisted of four stages: (1) a careful data collection and annotation process, which resulted in one of the largest WSI colorectal samples datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' (2) the design of an interpretable 1These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Preprint submitted to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' January 9, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='02608v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='IV] 6 Jan 2023 mixed-supervision scheme to leverage the domain knowledge introduced by pathologists through spatial annotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' (3) the development of an effective sampling approach based on the expected severeness of each tile, which de- creased the computation cost by a factor of almost 6x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' (4) the creation of a prototype that integrates the full set of features of the model to be eval- uated in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' During these stages, the proposed method was evaluated in four separate test sets, two of them are external and completely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the largest of those sets, the proposed approach achieved an accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' DL for colorectal samples is a few steps closer to stop being research exclusive and to become fully integrated in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Keywords: Clinical Prototype, Colorectal Cancer, Interpretable Artificial Intelligence, Deep Learning, Whole-Slide Images 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Introduction Colorectal cancer (CRC) incidence and mortality are increasing, and it is estimated that they will keep growing at least until 2040 [1], according to estimations of the International Agency for Research on Cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nowadays, it is the third most incident (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='7% of all cancer diagnoses) and the second most deadly type of cancer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to the effect of lifestyle, genetics, envi- ronmental factors and an increase in life expectancy, the current increase in world wealth and the adoption of western lifestyles further advocates for an increase in the capabilities to perform more CRC evaluations for potential diagnosis [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite the pessimist predictions for an increase in the in- cidence, CRC is preventable and curable when detected in its earlier stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, effective screening through medical examination, imaging techniques and colonoscopy are of utmost importance [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite the CRC detection capabilities shown by imaging techniques, the diagnosis of cancer is always based on the pathologist’s evaluation of biopsies/surgical specimen samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The stratification of neoplasia develop- ment stages consists of non-neoplastic (NNeo), low-grade dysplasia (LGD), high-grade dysplasia (HGD, including intramucosal carcinomas), and inva- sive carcinomas, from the initial to the latest stage of cancer progression, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In spite of the inherent subjectivity of the dysplasia grading system [6], recent guidelines from the European Society of Gastrointestinal Endoscopy (ESGE), as well as those from the US multi-society task force on CRC, consistently recommend shorter surveillance intervals for patients with 2 polyps with high-grade dysplasia, regardless of their dimension [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, grading dysplasia is still routinely performed by pathologists worldwide when assessing colorectal tissue samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Private datasets of digitised slides are becoming widely available, in the form of whole-slide images (WSI), with an increase in the adoption of digital workflows [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite the burden of the additional scanning step, WSI eases the revision of old cases, data sharing and quick peer-review [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It has also created several research opportunities within the computer vision domain, especially due to the complexity of the problem and the high dimen- sions of WSI [13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, robust and high-performance systems can be valuable assets to the digital workflow of a laboratory, especially if they are transparent and interpretable [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' However, some limitations still affect the applicability of such solutions in practice [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The majority of the works on CRC diagnosis direct their focus towards the classification of cropped regions of interest, or small tiles, instead of tackling the challenging task of diagnosing the entire WSI [18, 19, 17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Notwith- standing, some authors already presented methods to assess the grading of the complete slide of colorectal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In 2020, Iizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [21] used a re- current neural network (RNN) to aggregate the predictions of individual tiles processed by an Inception-v3 network into non-neoplastic, adenoma (AD) and adenocarcinoma (ADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to the large dimensions of WSI related to their pyramidal format (with several magnification levels) [22], usually over 50,000 × 50,000 pixels, it is usual to use a scheme consisting of a grid of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This scheme permits the acceleration of the processing steps since the tiles are small enough to fit in the memory of the graphics processing units (GPU), popular units for the training of deep learning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [23] studied the usage of an ensemble of five distinct ResNet networks, in order to distinguish the types of CRC adenomas H&E stained slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [24] experimented with a modified DeepLab-v2 network for tile classification, and proposed pixel probability thresholding to detect CRC adenomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Both Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [25] and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [26, 27] looked into the performance of the Inception- v3 architecture to detect CRC, with the latter also retrieving a cluster-based slide classification and a map of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The MuSTMIL [28] method, classifieds five colon-tissue findings: normal glands, hyperplastic polyps, low- grade dysplasias, high-grade dysplasias and carcinomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This classification originates from a multitask architecture that leverages several levels of mag- nification of a slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [29] extended the experiments with multitask learning, but instead of leveraging the magnification, its model aims to jointly 3 segment glands, detect tumour areas and sort the slides into low-risk (benign, inflammation or reactive changes) and high-risk (adenocarcinoma or dyspla- sia) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The architecture of this model is considerably more complex, with regard to the number of parameters, and is known as Faster-RCNN with a ResNet-101 backbone network for the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Further to this task, a gradient-boosted decision tree completes the pipeline that results in the final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As stated in the previous paragraph, recent state-of-the-art computer- aided diagnosis systems are based on deep learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These sys- tems rely on large volumes of data to learn how to perform a particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Increasing the complexity of the task often demands an increase in the data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Collecting this data is expensive and tedious due to the annotation complexity and the need for expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite recent publications that present approaches using large volumes of data to train CAD systems, the majority do not publicly release the data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To reverse this trend, the novel and completely anonymised dataset introduced in this document will have the majority of the available slides publicly released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This dataset contains, approximately, 10500 high-quality slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The available slides origi- nate one of the largest colorectal samples (CRS) datasets to be made publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This high volume of data, in addition to the massive resolution of the images, creates a significant bottleneck of deep learning approaches that extract patches from the whole slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, we introduce an efficient sampling approach that is performed once without sacrificing prediction per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The proposed sampling leverages knowledge learnt from the data to create a proxy that reduces the rate of important information discarded, when compared to random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to the cost of annotating the en- tire dataset, it is only annotated for a portion at the pixel level, while the remaining portion is labelled for a portion at the slide level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To leverage these two levels of supervision, we propose a mixed supervision training scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' One other increasingly relevant issue with current research is the lack of external validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is not uncommon to observe models that perform extremely well on a test set collected from the same data distribution used to train, but fail on samples from other laboratories or scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, we validate our proposed model in two different external datasets that vary in quality, country of origin and laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' While the results on a similar dataset show the performance of the model if implemented in the institution that collects that data, the test on external samples indicates its capabilities to be deployed in other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 4 In order to bring this CAD system into production, and to infer its ca- pabilities within clinical practice, we developed a prototype that has been used by pathologists in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We further collected information on the misdiagnoses and the pathologists’ feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Since it is expected that the model indicates some rationale behind its predictions, we developed a visual approach to explain the decision and guide pathologists’ focus toward more aggressive areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This mechanism increased the acceptance of the al- gorithm in clinical practice, and its usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' When collected from routine the slides can be digitised with duplicated tissue areas, known as fragments, which might be of lower quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, the workflow for the automatic diagnostic also included an automatic fragment detection and counting sys- tem [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, it was possible to utilize this prototype to do the second round of labelling based on the test set prediction given by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This second round led to the correction of certain labels (that the model pre- dicted correctly and were mislabeled by the expert pathologist) and insights regarding the areas where the model has to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To summarise, in this paper we propose a novel dataset with more than thirteen million tiles, a sampling approach to reduce the difficulty of using large datasets, a deep learning model that is trained with mixed supervision, evaluated on two external datasets, and incorporated in a prototype that provides a simple integration in clinical practice and visual explanations of the model’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This way, we come a step closer to making CAD tools a reality for colorectal diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Methods In this section, after defining the problem at hand, we introduce the pro- posed dataset used to train, validate and test the model, the external datasets to evaluate the generalisation capabilities of the model and the pre-processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Afterwards, we describe in detail the methodology followed to cre- ate the deep learning model and to design the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, we also detail the clinical assessment and evaluation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Problem definition Digitised colorectal cancer histological samples have large dimensions, which are far larger than the dimensions of traditional images used in medi- cal or general computer vision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Labelling such images is expensive and highly dependent on the availability of expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It limits the 5 availability of whole slide images, and, in scenarios where these are available, meaningful annotations are usually lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the other hand, it is easier to label the dataset at the slide level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The inclusion of detailed spatial an- notations on approximately 10% of the dataset has been shown to positively impact the performance of deep learning algorithms [17, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Figure 1: Problem definition as a fully supervised task (on top), and as a weakly-supervised task (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To fully leverage the potential of spatial and slide labels, we propose a deep learning pipeline, based on previous approaches [17, 31], using mixed supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Each slide, S is composed of a set of tiles Ts,n, where s represents the index of the slide and n ∈ {1, · · · , ns} the tile number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Furthermore, there is an inherent order in the grading used to classify the input into one of the C(k) classes, which represents a variation in severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For fully supervised learning, only strongly annotated slides are useful, and for those, the label of each tile Cs,n is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The remaining slides are deprived of these detailed labels, hence, they can only be leveraged by training algorithms with weakly supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To be used by these algorithms, the weakly annotated slides have only a single label for the entire bag (set) of tiles, as seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 6 Following the order of the labels and the clinical knowledge, we assume that the predicted slide label Cs is the most severe case of the tile labels: Cs = maxn{Cs,n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In other words, if there is at least one tile classified as containing high- grade dysplasia, then the entire slide that contains the tile is classified ac- cordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the other end of the spectrum, if the worst tile is classified as non-neoplastic, then it is assumed that there is no dysplasia in the entire set of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This is a generalisation of multiple-instance learning (MIL) to an ordinal classification problem, as proposed by Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Datasets The spectrum of large-scale CRC/CRS datasets is slowly increasing due to the contributions of several researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Two datasets that have been recently introduced in the literature are the CRS1K [17] and CRS4K [31] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Since the latter is an extension of the former with roughly four times more slides, it will be the baseline dataset for the remaining of this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, we further extend these with the CRS10K dataset, which contains 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='26x and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='36x more slides than CRS1K and CRS4K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Similarly, the number of tiles is multiplied by a factor of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='58 (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This volume of slides is translated into an increase in the flexibility to design experiments and infer the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, the inclusion of a test set separated from the validation set is now facilitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The set is composed of colorectal biopsies and polypectomies (excluding surgical specimens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Following the same annotation process as the previ- ous datasets, CRS10K slides are labelled according to three main categories: non-neoplastic (NNeo), low-grade lesions (LG), and high-grade lesions (HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The first, contains normal colorectal mucosa, hyperplasia and non-specific inflammation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' LG lesions categorise conventional adenomas with low-grade dysplasia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, HG lesions are composed of adenomas with high-grade dysplasia (including intra-mucosal carcinomas) and invasive adenocarcino- mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In order to avoid diversions from the main goal, slides with suspicion of known history of inflammatory bowel disease/infection, serrated lesions or other polyp types were not included in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The slides, retrieved from an archive of previous cases, were digitised with Leica GT450 WSI scanners, at 40× magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The cases were initially seen and classified (labelled) by one of three pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The pathologist revised and classified the slides, and then compared them with the initial report diagnosis (which served as a second-grader).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' If there was a match 7 between both, no further steps were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In discordant cases, a third pathologist served as a tie-breaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Roughly 9% of the dataset (967 slides and over a million tiles) were manually annotated by a pathologist and rechecked by the other, in turn, using the Sedeen Viewer software [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For complex cases, or when the agreement for a joint decision could not be reached, a third pathologist reevaluated the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The CRS10K dataset was divided into train, validation and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The first includes all the strongly annotated slides and other slides randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Whereas the second is composed of only non-annotated slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, the test set was selected from the new data added to extend the previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, it is completely separated from the training and validation sets of previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The test set, will be publicly available, so that future research can directly compare their results and use that set as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Table 1: Dataset summary, with the number of slides (annotated samples are detailed in parentheses) and tiles distributed by class for all the datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' NNeo LG HG Total # slides 300 (6) 552 (35) 281 (59) 1133 (100) CRS1K dataset [17] # annotated tiles 49,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='640 77,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='946 83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='649 211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='235 # non-annotated tiles 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='361 # slides 663 (12) 2394 (207) 1376 (181) 4433 (400) CRS4K dataset [31] # annotated tiles 145,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='898 196,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='116 163,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='603 505,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='617 # non-annotated tiles 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='265,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='362 # slides 1740 (12) 5387 (534) 3369 (421) 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='496 (967) CRS10K dataset # annotated tiles 338,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='979 371,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='587 341,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='268 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='051,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='834 # non-annotated tiles 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='571,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='871 CRS Prototype # slides 28 44 28 100 # non-annotated tiles 244,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='160 PAIP [33] # slides 100 100 # non-annotated tiles 97,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='392 TCGA [34] # slides 1 1 230 232 # non-annotated tiles 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='568,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='584 Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' as detailed in the following sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' this work comprises the development of a fully-functional prototype to be used in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Leveraging this prototype, it was possible to further collect a new set with 100 slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It differs from the CRS10K dataset, in the sense that they were not carefully selected from the archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Instead, these cases were actively collected from the current year’s routine exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We argue that this might 8 better reflect the real-world data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, we introduce this set as a distinct dataset to evaluate the robustness of the presented methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Differently from the datasets discussed below, the CRS Prototype dataset has a more balanced distribution of the slide labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Although it is useful in practice, the usage of the fragment counting and selection algorithm for the evaluation could potentiate the propagation of errors from one system to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, in this evaluation, we did not use the fragment selection algorithm, and as shown in Table 1, the number of tiles per slide doubles when compared to CRS10K, which had its fragments carefully selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To evaluate the domain generalisation of the proposed approach, two external datasets were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We evaluate the proposed approaches on two external datasets publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The first dataset is composed of samples of the TCGA-COAD [35] and TCGA-READ [36] collections from The Can- cer Imaging Archive [34], which are composed in general by resection samples (in contrast to our dataset, composed only of biopsies and polypectomies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Samples containing pen markers, large air bubbles over tissue, tissue folds, and other artefacts affecting large areas of the slide were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The fi- nal selection includes 232 whole-slide images reviewed and validated by the same pathologists that reviewed the in-house datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 230 of those sam- ples were diagnosed as high-grade lesions, whereas the remaining two have been diagnosed as low-grade and non-neoplastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For this dataset, the spe- cific model of the scanner used to digitise the images is unknown, but the file type (”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs”) matches the file type of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The second ex- ternal dataset used to evaluate the model contains 100H&E slides from the Pathology AI Platform [33] colorectal cohort, which contains all the cases with a more superficial sampling of the lesion, for a better comparison with our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' All the whole slide images in this dataset were digitised with an Aperio AT2 at 20X magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, the pathologists’ team fol- lowed the same guidelines to review and validate all the WSI, which were all classified as high-grade lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is interesting to note that while the PAIP contains significantly fewer tiles per slide, around 973, than the CRS10K dataset, around 1293, the TCGA dataset shows the largest amount of tissue per slide with an average of 6761 tiles as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Data pre-processing H&E slides are composed of two distinct elements, white background and colourful tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Since the former is not meaningful for the diagnostic, 9 the pre-processing of these slides incorporates an automatic tissue segmenta- tion with Otsu’s thresholding [37] on the saturation (S) channel of the HSV colour space, resulting in a separation between the tissue regions and the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The result of this step, which receives as input a 32× down- sampled slide, is the mask used for the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Leveraging this previous output, tiles with a dimension of 512 × 512 pixels (Figure 2) were extracted from the original slide (without any downsampling) at its maxi- mum magnification (40×), if they did not include any portion of background (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' a 100% tissue threshold was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Following previous experiments in the literature, our empirical assessment, and the confirmation that smaller tiles would significantly increase the number of tiles and the complexity of the task, 512 × 512 was chosen as the tile size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, it is believed that 512 × 512 is the smallest tile size that still incorporates enough information to make a good diagnostic with the possibility of visually explaining the de- cision [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The selected threshold of 100% further reduces the number of tiles by not including the tissue at the edges and decreases the complexity of the task, since the model does not see the background at any moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to tissue variations in different images, there is also a different number of tiles extracted per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Methodology The massive size of images, which translates to thousands of tiles per image, allied to a large number of samples in the CRS10K dataset, bottle- necks the training of weakly-supervised models based on multiple instance learning (MIL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, in this document, we propose a mix-supervision ap- proach with self-contained tile sampling to diagnose colorectal cancer samples from whole-slide images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This subsection comprises the methodology, which includes supervised training, sampling and weakly-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Supervised Training As mentioned in previous sections, spatial annotations are rare in large quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' However, these include domain information, given by the expert annotator, concerning the most meaningful areas and what are the most and less severe tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, they can facilitate the initial optimisation of a deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As shown in the literature, there has been some research on the impact of starting the training with a few iterations of fully-supervised training [17, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We further explore this in three different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' First, we have 967 annotated slides resulting in more than one million annotated tiles 10 Figure 2: Examples of regions with and a sample tile with 512 × 512 pixels (40× mag- nification), representing each class: non-neoplastic (on top), low-grade dysplasia (on the middle) and high-grade dysplasia (on the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' for supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Secondly, attending to the size of our dataset and the need for a stronger initial supervised training, the models are trained for 50 epochs, and their performance was monitored over specific checkpoint epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, we explore this pre-trained model as the main tool to sample useful tiles for the weakly-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 11 Figure 3: Overall scheme of the proposed methodology containing the mix-supervision framework that is responsible for diagnosing colorectal samples from WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tile Sampling Our scenario presents a particularly difficult condition for scaling the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' First, let’s consider the structure of the data, which consists of, on average, more than one thousand tiles per slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Within this set of tiles, some tiles provide meaningful value for the prediction, and others do not add extra information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In other words, for the CRS10K dataset, the extensive, lengthy, time and energy-consuming process of going through 13 million tiles every epoch can be avoided, and as result, these models can be trained for more epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nowadays, there is an increasing concern regarding energy and electricity consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, these concerns, together with the sustainability goals, further support the importance of more efficient training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Let T be the original set of tiles, and Ts be the original set of tiles from the slide s, the former is composed by a union of the latter of all the slides (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We propose to map T to a smaller set of tiles M without affecting the overall performance and behaviour of the trained algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 12 85 annotated WsI annotatedtiles (512x512px) tiles classifier non-annotated WSI tiles set (512x512 px) tiles classifier tiles ranking tiles sampling (inference) (top 200) orw grad T sampled tiles tiles classifier tiles ranking tiles classifier slide diagnosis (inference) (training with top 5) (top tile prediction)T = S� s=1 Ts (1) The model trained in a fully supervised task, previously described, pro- vides a good estimation of the utility of each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, we utilise the function (Φ) learned by the model to compute the predicted severity of each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As will be shown below, the weakly-supervised method utilises only the five most severe tiles per slide to train in each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, we select M tiles per slide (M=200 in our experimental setup) utilising a Top-k function (with k set to 200) to be retained for the weakly-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As in- dicated by the results presented in the following sections, the value of M was selected in accordance with a trade-off between information lost and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This is formalised in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ms = Top-k(Φ(Ts)) (2) For instance, in the CRS10K dataset, the total number of tiles after sampling would be at most 2,099,200, which represents a reduction of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='46× when compared to the total number of slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite this upper bound on the number of tiles, there are WSI samples that contain less than M tiles, and as such, they remain unsampled and the actual total number of tiles after sampling is potentially lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' During the evaluation and test time, there is no sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We conducted extensive studies on the performance of our methodology, without sampling, with sampling on the training data, and with sampling on training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The results on the CRS4K dataset validate our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The number of selected tiles considers a trade-off between compu- tational cost and information potentially lost, and for that reason, it is the success of empirical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Weakly-Supervised Learning The weakly-supervised learning approach designed for our methodology follows the same principles of recent work [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is divided into two distinct stages, tile severity analysis and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The former utilises the pre-trained model to evaluate the severity of every tile in a set of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In the first epoch, T , the set of all the tiles in the complete dataset is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This is possible since the model used to assess the severity in this epoch is the same one used for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, both tasks are integrated with the initial epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The 13 following epochs utilise the sampled tile set M instead of the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This overall structure is represented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The link between both stages is guided by a slide-wise tile ranking ap- proach based on the expected severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For tile Ts,n, the expected severity is defined as E( ˆCs,n) = K � i=1 i × p � ˆCs,n = C(i)� (3) where ˆCs,n is a random variable on the set of possible class labels {C(1), · · · , C(K)} and p � ˆCs,n = C(i)� are the K output values of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' After this analysis, the five most severe tiles are selected for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The number of selected tiles was chosen in accordance with previous studies [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These five tiles per slide are used to train the proposed model for one more epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Each epoch is composed of both stages, which means that the tiles used for training vary across epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The slide label is used as the ground truth of all five tiles of that same slide used for network optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For validation and evaluation, only the most severe tile is used for diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Although it might lead to an increase in false positives, it shall significantly reduce false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Furthermore, we argue that increasing the variability and quantity of data available leads to a better balance between the reduction of these two types of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Confidence Interval In order to quantify the uncertainty of a result, it is common to compute the 95 percent confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In this way, two different models can be easily understood and compared based on the overlap of their confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The standard approach to calculating these intervals requires sev- eral runs of a single experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As we increase the number of runs, our interval becomes narrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' However, this procedure is impractical for the computationally intensive experiments presented in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hence, we use an independent test set to approximate the confidence interval as a Gaussian function [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To do so, we compute the standard error (SE) of an evaluation metric m, which is dependent on the number of samples (n), as seen in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' SE = � 1 n × m × (1 − m) (4) 14 For the SE computation to be mathematically correct, the metric m must originate from a set of Bernoulli trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In other words, if each prediction is considered a Bernoulli trial, then the metric should classify them as correct or incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The number of correct samples is then given by a Binomial distribution X ∼ (n, p), where p is the probability of correctly predicting a label, and n is the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For instance, the accuracy is a metric that fits all these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Following the definition and the properties of a Normal distribution, we compute the number of standard deviations (z), known as a standard score, that can be translated to the desired confidence (c) set to 95% of the area under a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This is a well-studied value, which is approx- imately z ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This value z is then used to calculate the confidence interval, calculated as the product of z and SE as seen in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M ± z ∗ � 1 n × m × (1 − m) (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Experimental setup For our experimental setup, we divide our data into training and valida- tion sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Besides, we further evaluate the performance of the former in our test set composed of slides never seen by any of the methods presented or in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Following the split of these three sets, we have 8587, 1009 and 900 stratified non-overlapping samples in the training, validation and test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In an attempt to also contribute to reproducible research, the training of all the versions of the proposed algorithm uses the deterministic constraints available on Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The usage of deterministic constraints implies a trade- off between performance, either in terms of algorithmic efficiency or on its predictive power, and the complement with reproducible research guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, due to the current progress in the field, we have chosen to comply with the reproducible research guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' All the trained backbone networks were ResNet-34 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pytorch was used to train these networks with the Adaptive Moment Estimation (Adam) [41] optimiser, a learning rate of 6 × 10−6 and a weight decay of 3 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The training batch size was set to 32 for both fully and weakly supervised train- ing, while the test and inference batch size was 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The performance of the model was evaluated on the validation set used for model selection in terms of the best accuracies and quadratic weighted kappa (QWK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The training 15 was accelerated by an Nvidia Tesla V100 (32GB) GPU for 50 epochs of both weakly and fully supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In addition to the proposed method- ology, we extended our experiments to include the aggregation approach proposed by Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [31] on top of our best-performing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Label correction The complex process of labelling thousands of whole-slide images with colorectal cancer diagnostic grades is a task of increased difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It should also be noted and taken into account that grading colorectal dysplasia is hur- dled by considerable subjectivity, so it is to be expected that some borderline cases will be classified by some pathologists as low-grade and others as high- grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, as the number of cases increases, it becomes increasingly difficult to maintain perfect criteria and avoid mislabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For this reason, we have extended the analysis of the model’s performance to understand its errors and its capability to detect mislabelled slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' After training the proposed model, it was evaluated on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fol- lowing this evaluation, we identified the misclassified slides and conducted a second round of labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These cases were all blindly reviewed by two pathologists, and discordant cases from the initial ground truth were dis- cussed and classified by both pathologists (and in case of doubt/complexity, a third pathologist was also consulted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We tried to maintain similar criteria between the graders and always followed the same guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These new labels were used to rectify the performance of all the algorithms evaluated in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We argue that the information regarding the strength/confidence of predictions of a model used as a second opinion is of utter importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A correct integration of this feature can be shown as extremely insightful for the pathologists using the developed tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Prototype and Interpretability Assessment The proposed algorithm was integrated into a fully functional prototype to enable its use and validation in a real clinical workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This system was developed as a server-side web application that can be accessed by any pathologist in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The system supports the evaluation of either a sin- gle slide or a batch of slides simultaneously and in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It also caches the most recent results, allowing re-evaluation without the need to re-upload slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In addition to displaying the slide diagnosis, and confidence level for each class, a visual explanation map is also retrieved, to draw the patholo- gist’s attention to key tissue areas within each slide (all seen in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 16 The opaqueness of the map can be set to different thresholds, allowing the pathologist to control its overlay over the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' An example of the zoomed version of a slide with lower overlay of the map is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Figure 4: Main view of the CAD system prototype for CRS: Slide identification, confidence per class, diagnostic, mask overlay controller, results download as csv and slide search are some of the features visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Slide identification is anonymised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Furthermore, the prototype also allows user feedback where the user can accept/reject a result and provide a justification (Figure 6), an important feature for software updates, research development and possible active learn- ing frameworks that can be developed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These results can be downloaded with the corrected labels to allow for further retraining of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' There are several advantages to developing such a system as a server- side web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' First, it does not require any specific installation or dedicated local storage in the user’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Secondly, it can be accessed at the same time by several pathologists from different locations, allowing for a quick review of a case by multiple pathologists without data transference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, the lack of local storage of clinical data increases the privacy of patient data, which can only be accessed through a highly encrypted virtual private network (VPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, all the processing can be moved to an effi- cient graphics processing unit (GPU), thus reducing the processing time by 17 CADPath + 1 人 CADPath Search Search Download Results 00000000001-A- OOOO 01-001_xxx_000 000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs processed Processed 00000000002-A- Case: 00000000003 02-002_xxx_000 Specimen: A _000002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs Block: 03 processed Slide: 003 Mask 00000000003-A 03-003_xxx_000 _000003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs Model Results processed High grade 100% Low grade 00000000004-A- Normal 04-004_xxx_000 _000004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs processed High Grade 00000000005-A- Results approved 05-005_xxx_000 _000005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs processed 00000000006-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 06-006_xxx_000 _000006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs Upload new slidesFigure 5: Zoomed view of a slide from the CAD system prototype, with the predictions map with a small overlay threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Slide identification is anonymised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Similar behaviour on a local machine would require the installation of dedicated GPUs in the pathologists’ personal de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This platform is the first Pathology prototype for colorectal diagnosis developed in Portugal, and, as far as we know, one of the pioneers in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Its design was also carefully thought to be aligned with the needs of the pathologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Results In this section, we present the results of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The results are organised to first demonstrate the effectiveness of sampling, followed by an evaluation of the model in the two internal datasets (CRS10K and the prototype dataset), and finalise with the results on external datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We also discuss the advantages and disadvantages of the proposed approach, perform an analysis of the results from a clinical perspective, provide pathologists’ feedback on the use of the prototype, and finally discuss future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 18 CADPath + 人☆ CADPath Search Search Download Results processed 00000000006-A 06-006_xxx_000 Upload new slides +0:00000000Figure 6: CAD system prototype report tool: the user can report if the result is either correct, wrong or inconclusive and leave a comment for each case individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Slide identification is anonymised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the effectiveness of sampling To find the most suitable threshold for sampling the tiles used in the weakly supervised training, we evaluated the percentage of relevant tiles that would be left out of the selection, if the original set was reduced to 75, 100, 150 or 200 tiles, over the first five inference epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A tile is considered relevant if it shares the same label as the slide, or if it would take part in the learning process in the weakly-supervised stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As it is possible to see in Figure 7, if we set the maximum number of tiles to 200 after the second loop of inference, we would discard only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='5% of the potentially informative tiles, in the worst-case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the other side of the spectrum, a more radical sampling of only 50 tiles would lead to a cut of up to 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, to assess the impact of this sampling on the model’s perfor- mance, we also evaluated the accuracy and the QWK with and without sampling the top 200 tiles after the first inference iteration (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This evaluation considered sampling applied only to the training tile set, and to both the training and validation tile sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As can be noticed, the performance is not degraded and the model is trained in a much faster way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In fact, using the setup previously mentioned, the first epoch of inference, with the full set 19 CADPath x 人☆ CADPath Search Download Results ownload Feedback Approval State: 00000000001-A- 01-001_xxx_000 000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs Expected: (Low Grade Normal High Grade processed Processed Comments 00000000002-A- Case: 00000000002 02-002_xxx_000 Specimen: A 000002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svS Block: 02 processed Slide: 002 Mask 00000000003-A- 03-003_xxx_000 000003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='SvS Model Results processed 80% Low grade 20% 00000000004-A- Save changes Normal 04-004_xxx_000 000004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs High Grade processed 00000000005-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Results rejected!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 05-005_xxx_000 000005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='SVS processed O 00000000006-A- 06-006_xxx_000 000006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='svs Upload new slidesFigure 7: Tile sampling impact on information loss: percentage of tiles not selected due to sampling with different thresholds, over the first four inference epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' of tiles takes 28h to be completed, while from the second loop the training time decreases to only 5h per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Without sampling, training the model for 50 epochs would take around 50 days, whereas with sampling it takes around 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' CRS10K and Prototype CRS10K test set and the prototype dataset were collected through dif- ferent procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The first followed the same data collection process as the complete dataset, whereas the second originated from routine samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thus, the evaluation of both these sets is done separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The first experiment was conducted on the CRS10K test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As ex- pected, the steep increase in the number of training samples led to a signif- icantly better algorithm in this test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Initially, the model trained on the CRS10K correctly predicted the class of 819 out of 900 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' For the 20 8 Info Samplingof 20o tiles sampling Sampling of 1oo tiles selected Sampling of 75 tiles Sampling of 50 tiles 01 due tiles not selected of total number 4 3 tiles 2 1 0 1 2 3 4 5 EpochsTable 2: Model performance comparison with and without tile sampling of the top 200 tiles from the first inference iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Compared the best epoch of the initial five epochs and of the initial ten epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Validation is represented as Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Best Accuracy at Best QWK at Sampling 5th epoch 10th epoch 5th epoch 10th epoch No 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='94% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='20 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='42% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='809 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='023 Train 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='43% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='18 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='82% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='817 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='828 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='023 Train and Val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='12% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='13 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='92% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='824 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='023 Table 3: Model performance evaluation on the CRS10K test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The binary accuracy is calculated as NNeo vs all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Accuracy is represented as (ACC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In bold are the best results per column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Method ACC Binary ACC Sensitivity iMIL4Path 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='33% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='84 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='997 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='004 Ours (CRS4K) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='44% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='01 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='11% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='997 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='004 Ours (CRS10K) wo/ Agg 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='44% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='62 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='78% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='996 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='005 Ours (CRS10K) w/ Agg 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='67% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='90 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='55% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='985 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='009 wrong 81 cases, the pathologists performed a blind review of these cases and found that the algorithm was indeed correct in 22 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This led to a correction in the labels of the test set, and the appropriate adjustment of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In Table 3, the performance of the different algorithms is pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' CRS10K outperforms the other approaches by a reasonable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We further applied the aggregation proposed by Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [31] to the best performing method, but without gains in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite being trained on the same dataset iMIL4Path and the proposed methodology trained on CRS4K, utilise different splits for training and validation, as well as different optimisation techniques due to the deterministic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In addition to examining quantitative metrics, such as the accuracy of the model, we extended our study to include an analysis of the confidence in the model when it correctly predicts a class and when it makes an incorrect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' To this end, we recorded the confidence of the model for the predicted class and divided it into the set of correct and incorrect predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' These were then used to fit a kernel density estimator (KDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Figure 8 shows the density estimation of the confidence values for the three different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is worth noting that, when correct, the model trained on the CRS10K, 21 Figure 8: Kernel density estimation of the confidences of correct and incorrect predic- tions performed on the three-class classification problem by three distinct models on the CRS10K test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The plots represent, from left to right, the proposed method trained on CRS10K, the proposed method trained on CRS4K and iMIL4Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' returns higher confidence levels as shown by the shift of its mean towards values close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' On the other hand, the confidence values of its incorrect predictions decrease significantly, and although it does not present the lowest values, it shows the largest gap between correct and incorrect means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Table 4: Model performance evaluation on the prototype test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Accuracy is represented as (ACC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The binary accuracy is calculated as NNeo vs all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In bold are the best results per column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Method ACC Binary ACC Sensitivity iMIL4Path 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='13 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 Ours (CRS4K) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 Ours (CRS10K) wo/ Agg 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='13 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='986 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='026 Ours (CRS10K) w/ Agg 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='99 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='986 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='026 When tested on the prototype data (n=100), the importance of a higher volume of data remains visible (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nonetheless, the performance of iMIL4Path [31] approach is comparable to the proposed approach trained on CRS10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is worth noting that the latter achieves better performance on the binary accuracy at the cost of a decrease in sensibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In other words, the capability to detect negatives increases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to the smaller 22 CRS10KTest Set-Ours(CRS10K) CRS10KTestSet-Ours(CRS4K) CRS10K Test Set - iMIL4Path 8 8 8 Correct Correct Correct Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='961 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='953 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='956 7 Incorrect 7 Incorrect 7 Incorrect Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='769 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='762 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='773 6 6 6 5 5 5 /p(c) /p(c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='. 3 3 3 2 2 2 1 1 1 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 Confidencec Confidencec Confidence cset of slides, the confidence interval is much wider, as such, the performance on the CRS10K test set is a good indication of how these values would shift if more data was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Similar performance drops were linked with the introduction of aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Figure 9: Kernel density estimation of the confidences of correct and incorrect predictions performed on the three-class classification problem by three distinct models on the proto- type set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The plots represent, from left to right, the proposed method trained on CRS10K, the proposed method trained on CRS4K and iMIL4Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite similar results, the confidence of the model in its predictions is distinct in all three approaches, as seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The proposed approach when trained on the CRS10K dataset has a larger density on values close to one when the predictions are correct, and the mean confidence of those predictions is, once more, higher than the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' However, espe- cially when compared to the proposed approach trained on the CRS4K, the confidence of wrong predictions is also higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It can be a result of a larger set of wrong predictions available on the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nonetheless, the steep increase in the density of values closer to one further indicates that there is room to explore other effects of extending the number of training samples, besides benefits in quantitative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Domain Generalisation Evaluation To ensure the generalisation of the proposed approach across external datasets, we have evaluated their performance on TCGA and PAIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' More- 23 Prototype-Ours(CRS10K) Prototype-Ours(CRS4K) Prototype - iMIL4Path 8 8 8 Correct Correct Correct Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='943 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='923 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='916 7 Incorrect 7 Incorrect 7 Incorrect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='. Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='84 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='757 Mean =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='818 9 6 6 5 5 5 p(c) / p(c) 3 3 3 2 2 2 1 1 1 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 Confidencec Confidencec Confidence cover, we conducted a similar analysis of both of these datasets, as the one done for the internal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Table 5: Model performance evaluation on the PAIP test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The binary accuracy is calculated as NNeo vs all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Accuracy is represented as (ACC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In bold are the best results per column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Method ACC Binary ACC Sensitivity iMIL4Path 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='95 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 Ours (CRS4K) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='06 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 Ours (CRS10K) wo/ Agg 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 Ours (CRS10K) w/ Agg 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='79 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='000 From the two datasets, PAIP is arguably the closest to CRS10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It con- tains similar tissue, despite its colour differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The performances of the proposed approaches were expected to match the performance of iMIL4Path in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' However, it did not happen for the version trained on the CRS4K dataset, as seen in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A viable explanation concerns potential overfitting to the training data potentiated by an increase in the number of epochs of fully and weakly supervised training, a slight decrease in the tile variability in the latter approach, and a smaller number of samples when compared to the version trained on CRS10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This version, trained on the larger set, mitigates the problems of the other method due to a significant increase in the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, it is worth noting that in all three approaches, the errors corresponded only to a divergence between low and high-grade cases, with no non-neoplastic cases being classified as high- grade or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As in previous sets, the version trained on the CRS10K dataset outperforms the remaining approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Using aggregation in this dataset leads to a discriminative power to distinguish between high- and low-grade lesions that is close to random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In two of the three approaches, the number of incorrect samples is one or zero, as such, there is no density estimation for wrong samples in their confidence plot as seen in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yet, it is visible the shift towards higher values of confidence in the proposed approach trained on the CRS10K when compared to the method of iMIL4Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The version trained on CRS4K shows very little separability between the confidence of correct and incorrect predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The TCGA dataset has established itself as the most challenging for the proposed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Besides the expected differences in colour and other 24 Figure 10: Kernel density estimation of the confidences of correct and incorrect predictions performed on the three-class classification problem by three distinct models on the PAIP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The plots represent, from left to right, the proposed method trained on CRS10K, the proposed method trained on CRS4K and iMIL4Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Table 6: Model performance evaluation on the TCGA test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The binary accuracy is calculated as NNeo vs all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Accuracy is represented as (ACC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In bold are the best results per column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Method ACC Binary ACC Sensitivity iMIL4Path 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='55% ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='60% ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='805 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='051 Ours (CRS4K) wo/ Agg 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='69% ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='86 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='71% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='991 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='012 Ours (CRS10K) wo/ Agg 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='91% ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='61 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='13% ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='996 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='008 Ours (CRS10K) w/ Agg 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='83% ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='91 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='41% ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='983 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='017 elements, this dataset is mostly composed of resection samples, which are not present in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, this presents itself as an excellent dataset to assess the capability of the model to handle these different types of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Both iMIL4Path and the proposed method trained on CRS4K have shown substantial problems in correctly classifying TCGA slides, as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Despite having a lower performance on the general accuracy, the binary accuracy shows that our proposed method trained on CRS4K has much lower misclassification errors regarding the classification of high-grade samples as normal, demonstrating higher robustness of the new training ap- proach against errors with a gap of two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As with other datasets, the proposed approach trained on CRS10K shows better results, this time by a 25 PAIP -Ours(CRS10K) PAIP - Ours(CRS4K) PAIP - iMIL4Path 8 8 8 Correct Correct Correct Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='99 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='843 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='964 7 7 Incorrect 7 Mean =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='835 9 6 6 5 5 5 / p(c) p(c) 3 3 3 2 2 2 1 1 1 0 : 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 Confidencec Confidencec Confidence csignificant margin with no overlapping between the confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Figure 11: Kernel density estimation of the confidences of correct and incorrect predictions performed on the three-class classification problem by three distinct models on the TCGA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The plots represent, from left to right, the proposed method trained on CRS10K, the proposed method trained on CRS4K and iMIL4Path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Inspecting the predictions’ confidence for the three models indicates a be- haviour in line with the accuracy-based performance (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreover, a confidence shift of wrong predictions’ confidence towards smaller values is clearly visible in the plot corresponding to the model trained on CRS10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The shown gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 between the confidence of correct and wrong predic- tions, indicates that it is possible to quantify the uncertainty of the model and avoid the majority of the wrong predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' In other words, when the uncertainty is above a learnt threshold, then the model refuses to make any prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' It is extremely useful in models designed as a second opinion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Prototype usability in clinical practice As it is currently designed, the algorithm works preferentially as a “second opinion”, allowing the assessment of difficult and troublesome cases, without the immediate need for the intervention of a second pathologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Due to its “user-friendly” nature and very practical interface, the cases can be easily introduced into the system and results are rapidly shown and easily accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Also, by not only providing results but presenting visualisation maps (cor- responding to each diagnostic class), the pathologist is able to compare his 26 TCGA-Ours(CRS10K) TCGA-Ours(CRS4K) TCGA- iMIL4Path 8 8 8 Correct Correct Correct Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='965 Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='956 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Mean=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='932 7 Incorrect 7 Incorrect 7 Incorrect Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='764 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='876 Mean = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='815 6 6 6 5 5 5 p(c) p(c) 3 3 3 2 2 2 1 1 1 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0 Confidencec Confidencec Confidencecown remarks to those of the algorithm itself, towards a future “AI-assisted diagnosis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Another relevant aspect is the fact that the prototype allows for user feedback (agreeing or not with the model’s proposed result), which can be further integrated into further updates of the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Also interesting, is the possibility of using the prototype as a triage system on a pathologist’s daily workflow (by running front, before the pathologist checks the cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Signalling the cases that would need to be more urgently observed (namely high-risk lesions) would allow the pathologists to prioritise their workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Further, by providing a previous assessment of the cases, it would also con- tribute to enhancing the pathologists’ efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Although it is possible to use the model as it is upfront, it would classify some samples incorrectly (since it was not trained on the full spectrum of colorectal pathology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, the uncertainty quantification based on the provided confidence given in the user interface could also be extremely useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Presently, there is no rec- ommendation for dual independent diagnosis of colorectal biopsies (contrary to gastric biopsies, where, in cases in which surgical treatment is considered, it is recommended to obtain a pre-treatment diagnostic second opinion [42]), but, in case that in the future this also becomes a requirement, a tool such as CADPath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='AI prototype could assist in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This has increased impor- tance due to the worldwide shortage of pathologists and so, such CAD tools can really make a difference in patient care (in similarity, for example, with Google Health’s research, using deep learning to screen diabetic retinopa- thy in low/middle-income countries, in which the system showed real-time retinopathy detection capability similar to retina specialists, alleviating the significant manpower constrictions in this setting [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lastly, we also an- ticipate that this prototype, and similar tools, can be used in a teaching environment since its easy use and explainable capability (through the visu- alisation maps) allows for easy understanding of the given classifications and having a web-based interface allows for easy sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Future work The proposed algorithm still has potential for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We aim to include the recognition of serrated lesions, to distinguish normal mucosa from significant inflammatory alterations/diseases, to stratify high-risk le- sions into high-grade dysplasia and invasive carcinomas and to identify other neoplasia subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Further, we would like to leverage the model to be able to evaluate also surgical specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Another relevant step will be the merge of our dataset and external ones for training, besides only testing it on ex- 27 ternal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This will enhance its generalisation capabilities and provide a more robust system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lastly, we intend to measure the “user experience” and feedback from the pathologists, by its gradual implementation in general laboratory routine work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The following goals comprise a more extensive evaluation of the model across more scanner brands and labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We also want to promote certain behaviours that would allow for more direct and integrated uncertainty esti- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' We have also been looking towards aggregation methods, but, since in the majority of them there is an increased risk of false negatives, we have work to do in that research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Discussion In this document, we have redesigned the previous methodology on MIL for colorectal cancer diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' First, we extended and leveraged the mixed supervision approach to design a sampling strategy, which utilises the knowl- edge from the full supervision training as a proxy to tile utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Secondly, we studied the confidence that the model shows in its predictions when they are correct and when they are incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Additionally, this confidence is shown to be a potential resource to quantify uncertainty and avoid wrong predictions on low-certainty scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' This is entirely integrated within a web-based prototype to aid pathologists in their routine work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' The proposed methodology was evaluated on several datasets, including two external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Through this evaluation, it was possible to infer that the performance of the proposed methodology benefits from a larger dataset and surpasses the performance of previous state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' As such, and given the excelling results that originated from the increase in the dataset, we are also publicly releasing the majority of the CRS10K dataset, one of the largest publicly available colorectal datasets composed of H&E images in the literature, including the test set for the benchmark of distinct approaches across the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Finally, we have clearly defined a set of potential future directions to be explored, either for better model design, the development of useful prototypes or even the integration of uncertainty in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' References [1] International Agency for Research on Cancer (IARC), Global cancer observatory, https://gco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='iarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='fr/ (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 28 [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Brody, Colorectal cancer, Nature 521 (2015) S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1038/ 521S1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Holmes, A disease of growth, Nature 521 (2015) S2–S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 1038/521S2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [4] Digestive Cancers Europe (DiCE), Colorectal screening in europe, https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='ly/3rFxSEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hassan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Antonelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dumonceau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Regula, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Bretthauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chaussade, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dekker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ferlitsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gimeno-Garcia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Jover, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kalager, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pellis´e, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pox, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ricciardiello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rutter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Helsingen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Bleijenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Senore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' van Hooft, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dinis-Ribeiro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Quintero, Post-polypectomy colonoscopy surveillance: European so- ciety of gastrointestinal endoscopy guideline - update 2020, Endoscopy 52 (8) (2020) 687–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1055/a-1185-3109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Mahajan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Downs-Kelly, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Patil, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rybicki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Bennett, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Plesec, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cummings, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rex, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Goldblum, Repro- ducibility of the villous component and high-grade dysplasia in col- orectal adenomas <1 cm: Implications for endoscopic surveillance, American Journal of Surgical Pathology 37 (3) (2013) 427–433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1097/PAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='0b013e31826cf50f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gupta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lieberman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Anderson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Burke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dominitz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kaltenbach, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Robertson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shaukat, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Syngal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rex, Recommendations for follow-up after colonoscopy and polypectomy: A consensus update by the us multi-society task force on colorectal cancer, Gastrointestinal Endoscopy (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='gie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Eloy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Vale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Curado, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pol´onia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Campelos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Caramelo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sousa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sobrinho-Sim˜oes, Digital pathology workflow imple- mentation at ipatimup, Diagnostics 11 (11) (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3390/ diagnostics11112111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='com/2075-4418/11/11/2111 [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fraggetta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Caputo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Guglielmino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pellegrino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Runza, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' L’Imperio, A survival guide for the rapid transition to a fully digital workflow: The “caltagirone example”, Diagnostics 11 (10) (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi: 29 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3390/diagnostics11101916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='com/2075-4418/11/10/1916 [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Montezuma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fraga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ribeiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gon¸calves, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tavares, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Macedo-Pinto, Digital pathology implemen- tation in private practice: Specific challenges and opportunities, Diag- nostics 12 (2) (2022) 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Madabhushi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lee, Image analysis and machine learning in digital pathology: challenges and opportunities, Medical Image Analysis 33 (2016) 170–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rakha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Toss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shiino, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gamble, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Jaroensri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Mermel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chen, Current and future applications of artificial intelligence in pathology: a clinical perspective, Journal of Clinical Pathology (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1136/jclinpath-2020-206908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Veta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' van Diest, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Willems, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Madabhushi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cruz-Roa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gonzalez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Larsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Vestergaard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dahl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cire¸san, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Schmidhuber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Giusti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gambardella, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Walter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kondo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Matuszewski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Precioso, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Snell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kittler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' de Campos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Khan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rajpoot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Arkoumani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lacle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Viergever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pluim, Assessment of algorithms for mitosis detection in breast cancer histopathology images, Medical Image Analysis 20 (1) (2015) 237–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Campanella, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hanna, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Geneslaw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Miraflor, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Werneck Krauss Silva, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Busam, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Brogi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Reuter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Klimstra, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fuchs, Clinical-grade computational pathology using weakly supervised deep learning on whole slide images, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 25 (8) (2019) 1301–1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1038/s41591-019-0508-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ribeiro Pinto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gon¸calves, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Canas-Marques, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='- J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, Weakly-supervised classifi- cation of HER2 expression in breast cancer haematoxylin and eosin stained slides, Applied Sciences 10 (14) (2020) 4728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3390/ app10144728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 30 [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Albuquerque, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, Deep ordinal focus assess- ment for whole slide images, in: Proceedings of the IEEE/CVF Inter- national Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 657–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Neto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fraga, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Montezuma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ribeiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gon¸calves, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, CAD systems for colorectal cancer from WSI are still not ready for clini- cal acceptance, Scientific Reports 11 (1) (2021) 14358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1038/ s41598-021-93746-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Thakur, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yoon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chong, Current trends of artificial intelligence for colorectal cancer pathology image analysis: a systematic review, Cancers 12 (7) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='3390/cancers12071884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zhou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ou, Application of artificial intelligence to the diagnosis and therapy of colorectal cancer, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cancer Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 10 (11) (2020) 3575–3598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Davri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Birbas, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kanavos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ntritsos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Giannakeas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tzallas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Batistatou, Deep learning on histopathological images for colorectal cancer diagnosis: A systematic review, Diagnostics 12 (4) (2022) 837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [21] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Iizuka, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kanavati, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rambeau, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Arihiro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tsuneki, Deep learning models for histopathological classification of gastric and colonic epithelial tumours, Scientific Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 10 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1038/ s41598-020-58467-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tizhoosh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pantanowitz, Artificial intelligence and digital pathol- ogy: challenges and opportunities, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pathol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Inform 9 (1) (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4103/jpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='jpi\\_53\\_18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Suriawinata, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Vaickus, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lisovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tomita, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Abdollahi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Snover, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Baron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Barry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hassanpour, Evaluation of a deep neural network for auto- mated classification of colorectal polyps on histopathologic slides, JAMA Network Open 3 (4) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1001/jamanetworkopen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 3398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [24] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Song, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yuan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Jin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, 31 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ku, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lv, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shi, Automatic deep learning-based colorectal adenoma detection system and its similarities with pathologists, BMJ Open 10 (9) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 1136/bmjopen-2019-036423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Walker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Su, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Karsan, Col- orectal cancer detection based on deep learning, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pathol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 11 (1) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4103/jpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='jpi\\_68\\_19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Deng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Su, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=', Accurate diagnosis of colorectal can- cer based on histopathology images using artificial intelligence, BMC medicine 19 (1) (2021) 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1186/s12916-021-01942-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Shi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Meng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Meng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Xiao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=', Accurate recognition of colorec- tal cancer with semi-supervised deep learning on pathological images, Nature communications 12 (1) (2021) 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Marini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ot´alora, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ciompi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Silvello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Marchesin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Vatrano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Buttafuoco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Atzori, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M¨uller, Multi-scale task multiple instance learning for the classification of digital pathology images with global annotations, in: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Atzori, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Burlutskiy, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ciompi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Minhas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M¨uller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Peng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Rajpoot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Torben-Nielsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' van der Laak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Veta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Yuan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zlobec (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' ), Proceedings of the MICCAI Work- shop on Computational Pathology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 156 of Proceedings of Machine Learning Research, PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 170–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ho, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Saraf, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Jialdasani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Taghipour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sathe, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Khor, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=', A promising deep learning-assistive algorithm for histopathological screening of col- orectal cancer, Scientific Reports 12 (1) (2022) 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Albuquerque, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moreira, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Barros, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Montezuma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Neto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ribeiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gon¸calves, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, Quality control in digital pathology: Automatic fragment detection and counting, in: 2022 44th Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 588–593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1109/EMBC48229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='9871208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 32 [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Neto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Oliveira, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Montezuma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fraga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Monteiro, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ribeiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Gon¸calves, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pinto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Cardoso, imil4path: A semi-supervised interpretable approach for colorectal whole-slide images, Cancers 14 (10) (2022) 2489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [32] Pathcore, Sedeen viewer, https://pathcore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='com/sedeen (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Platform, Paip, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='wisepaip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='org, last accessed on 20/01/22 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Clark, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Vendt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Freymann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kirby, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Koppel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Moore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Phillips, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Maffitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Pringle, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tarbox, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Prior, The cancer imaging archive (TCIA): Maintaining and operating a public in- formation repository, Journal of Digital Imaging 26 (2013) 1045–1057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1007/s10278-013-9622-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kirk, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sadow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Levine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Roche, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Bonaccio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Fil- iippini, Radiology data from the cancer genome atlas colon adenocarci- noma [TCGA-COAD] collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='7937/K9/TCIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' HJJHBOXZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kirk, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sadow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Levine, Radiology data from the cancer genome atlas rectum adenocarcinoma [TCGA-READ] collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='7937/K9/TCIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='F7PPNPNU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man, and Cybernetics 9 (1) (1979) 62–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1109/TSMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='4310076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Boˇziˇc, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tabernik, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Skoˇcaj, Mixed supervision for surface-defect detection: From weakly to fully supervised learning, Computers in In- dustry 129 (2021) 103459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Raschka, Model evaluation, model selection, and algorithm selection in machine learning, arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='12808 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [40] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sun, Deep residual learning for image recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='1109/CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 33 [41] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kingma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ba, Adam: A method for stochastic optimization, in: ICLR (Poster), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [42] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' of Tumours Editorial Board, WHO classification of tumours of the digestive system, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 5, World Health Organization, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' [43] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Ruamviboonsuk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Tiwari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Sayres, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Nganthavee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Hemarat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Kongprayoon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Raman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Levinstein, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' Schaekermann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=', Real-time diabetic retinopathy screening by deep learning in a multisite national screening programme: a prospective interventional cohort study, The Lancet Digital Health 4 (4) (2022) e235–e244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE0T4oBgHgl3EQfuwEX/content/2301.02608v1.pdf'} diff --git a/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf b/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..993c1ff8abc48ae304a5dadac76376f660c7eec5 --- /dev/null +++ b/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:44edec5b3f481e72d9b56d0fc5b31803c065aaf3faee3765a64d63b089c0879a +size 347933 diff --git a/3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss b/3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ebf49e8588e982bab689c043f6e1817dbed26ce5 --- /dev/null +++ b/3NE0T4oBgHgl3EQfeABh/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:14624e64cdd30113541af8f70d07edd0a0dcaf0abbd95296d05afe130a9907ef +size 2293805 diff --git a/3NE0T4oBgHgl3EQfeABh/vector_store/index.pkl b/3NE0T4oBgHgl3EQfeABh/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..23cc0311044c334179b95ab191afd3d28118bece --- /dev/null +++ b/3NE0T4oBgHgl3EQfeABh/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f0a5d69744d79c569f32eb517c2aa5a10f794e2ad3cb7bc38d2acf6ab3bbd618 +size 77683 diff --git a/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf b/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..00951ce12e7e77f1eaf6156420c26661a09a9d11 --- /dev/null +++ b/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:53cc2003fd204352489201d57931cf871fa65b07c427461100a5b6593e39fbdc +size 2425722 diff --git a/3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss b/3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ebdc3b93406a36734088cf24a908a14c5460ef09 --- /dev/null +++ b/3tFQT4oBgHgl3EQfHDWI/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9966a9144a046a56e8921f8b39f978c527a73306f6a7954da3e990304edc070f +size 5832749 diff --git a/3tFQT4oBgHgl3EQfHDWI/vector_store/index.pkl b/3tFQT4oBgHgl3EQfHDWI/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e075aeb7939d9f330e196bd877fddac2a772faf9 --- /dev/null +++ b/3tFQT4oBgHgl3EQfHDWI/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76a088ee89284b4c4181a14d348a3ea15bd0f08245dd35547f80f79082762b1b +size 191278 diff --git a/4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss b/4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8fe59ed250cd4772b82ea515bcb0437b3259b114 --- /dev/null +++ b/4dAyT4oBgHgl3EQfpPgc/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a98d90550a166ad0ed06d6d8060718d7c46b7fccf6c97867ce92a56d57487482 +size 5308461 diff --git a/4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss b/4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8acb41fe2f47d04bdc540f18010cf3fe077c4354 --- /dev/null +++ b/4dE0T4oBgHgl3EQfvQGV/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f730d428de41eded28e91ac7c54e034fa2a3ff133a9fec074f17dbb59a882f4 +size 458797 diff --git a/4dE0T4oBgHgl3EQfvQGV/vector_store/index.pkl b/4dE0T4oBgHgl3EQfvQGV/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7e1360c64cc36d40815173f8a9e5df7d129ab83b --- /dev/null +++ b/4dE0T4oBgHgl3EQfvQGV/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cb0ef2120cc7ce0b2d91dad1a8bec2a41a884c9775898dd663aa34929dcafc92 +size 17939 diff --git a/7NE0T4oBgHgl3EQffQAA/content/tmp_files/2301.02400v1.pdf.txt b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/2301.02400v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fae845f7df485fe30e3f65096257e6194bfa579b --- /dev/null +++ b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/2301.02400v1.pdf.txt @@ -0,0 +1,2236 @@ +arXiv:2301.02400v1 [cs.IT] 6 Jan 2023 +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +with Flexible Array Size and Large Set Size +Gobinda Ghosh1, Sudhan Majhi2* and Shubhabrata Paul1 +1Mathematics, IIT Patna, Bihta, Patna, 801103, Bihar, India. +2*Electrical Communication Engineering, IISc Bangalore, CV +Raman Rd, Bengaluru, 560012, Karnataka, India. +*Corresponding author(s). E-mail(s): smajhi@iisc.ac.in; +Contributing authors: gobinda 1921ma06@iitp.ac.in; +shubhabrata@iitp.ac.in; +Abstract +In this paper, we propose a direct construction of optimal two- +dimensional +Z-complementary +array +code +sets +(2D-ZCACS) +using +multivariable functions (MVFs). In contrast to earlier works, the +proposed construction allows for a flexible array size and a large +set size. Additionally, the proposed design can be transformed into +a one-dimensional Z-complementary code set (1D-ZCCS). Many of +the 1D-ZCCS described in the literature appeared to be special +cases of this proposed construction. At last, we compare our work +with the current state of the art and then draw our conclusions. +Keywords: Two dimensional complete complementary codes (2D-CCC), +multivariable function (MVF), two dimensional Z- complementary array code +set (2D-ZCACS). +1 Introduction +For an asynchronous two dimensional multi-carrier code-division multiple +access (2D-MC-CDMA) system, the ideal 2D correlation properties of two +dimensional complete complementary codes (2D-CCCs)[1] can be properly uti- +lized to obtain interference-free performance [2]. Similar to one dimensional +complete complementary code (1D-CCC)[3–5], one of the most significant +1 + +Springer Nature 2021 LATEX template +2 +A Direct Construction of Optimal 2D-ZCACS +drawbacks of 2D-CCC is that the set size is restricted [6]. Motivated by +the scarcity of 2D-CCC with flexible set sizes, Zeng et al. proposed 2D Z- +complementary array code sets (2D-ZCACSs) in [6, 7]. For a 2D − (K, Z1 × +Z2)−ZCACSL1×L2 +M +, K, Z1×Z2, L1×L2 and M denote the set size, two dimen- +sional zero-correlation zone (2D-ZCZ) width, array size and the number of +constituent arrays, respectively. In [6, 7], authors obtained ternary 2D-ZCACSs +by inserting some zeros into the existing binary 2D-ZCACSs. In 2021, Pai +et al. presented a new construction method of 2D binary Z-complementary +array pairs (2D-ZCAP) [8]. Recently, Das et al. in [9] proposed a construction +of 2D-ZCACS by using Z-paraunitary (ZPU) matrices. All these construc- +tions of 2D-ZCACS depend heavily on initial sequences and matrices which +increase hardware storage. For the first time in the literature, Roy et al. in +[10] proposed a direct construction of 2D-ZCACS based on MVF. The array +size of the proposed 2D-ZCACS is of the form L1 × L2, where L1 = 2m, +L2 = 2pm1 +1 pm2 +2 +. . . pmk +k , m ≥ 1, mi ≥ 2 and the set size is of the form 2p2 +1p2 +2 . . . p2 +k +where pi is a prime number. Therefore the array size and the set size is +restricted to some even numbers. +Existing array and set size limitations through direct construction in the +literature motivates us to search multivariable function (MVF) for more flex- +ible array and set sizes. Our proposed construction provides 2D-ZCACS with +parameter 2D − (R1R2M1M2, N1 × N2) − ZCACSR1N1×R2N2 +M1M2 +where M1 = +�a +i=1 pki +i , M2 = �b +j=1 qtj +j , pi is any prime or 1, qj is prime, a, b, ki, tj ≥ 1, R1 +and R2 are positive integer, such that R1 ≥ 1 and R2 ≥ 2, N1 = �a +i=1 pmi +i , +N2 = �b +j=1 qnj +j , mi, nj ≥ 1. The set size in our proposed 2D-ZCACS construc- +tion, R1R2M1M2, is more adaptable than the set size of 2D-ZCACS given in +[10]. Unlike [10], the proposed 2D-ZCACS can be reduced to 1D-ZCCS [11–18] +also. As a result, many existing optimal 1D-ZCCSs have become special cases +of the proposed construction [16–18]. The proposed construction also derived a +new set of optimal 1D-ZCCS that had not previously been presented by direct +method. +The rest of the paper is organized as follows. Section 2 discusses construc- +tion related definitions and lemmas. Section 3 contains the construction of +2D-ZCACS and the comparison with the existing state-of-the-art. Finally, in +Section 4, the conclusions are drawn. +2 Notations and definitions +The following notations will be followed throughout this paper: ωn += +exp +� +2π√−1/n +� +, An = {0, 1, . . ., n− 1} ⊂ Z, where n is a positive integer and +Z is the ring of integer. +2.1 Two Dimensional Array +Definition 1 ([9]) Let A = +� +ag,i +� +and B = +� +bg,i +� +be complex-valued arrays of size +l1 × l2 where 0 ≤ g < l1, 0 ≤ i < l2. The two dimensional aperiodic cross correlation + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +3 +function (2D-ACCF) of arrays A and B at shift (τ1, τ2) is defined as +C (A, B) (τ1, τ2) = + + + + + + + + + + + + + + + + + + + + + + + + + +�l1−1−τ1 +g=0 +�l2−1−τ2 +i=0 +ag,ib∗ +g+τ1,i+τ2, if +0 ≤ τ1 < l1, +0 ≤ τ2 < l2; +�l1−1−τ1 +g=0 +�l2−1+τ2 +i=0 +ag,i−τ2b∗ +g+τ1,i, if +0 ≤ τ1 < l1, +−l2 < τ2 < 0; +�l1−1+τ1 +g=0 +�l2−1−τ2 +i=0 +ag−τ1,ib∗ +g,i+τ2, if −l1 < τ1 < 0, +0 ≤ τ2 < l2; +�l1−1+τ1 +g=0 +�l2−1+τ2 +i=0 +ag−τ1,i−τ2b∗ +g,i, if −l1 < τ1 < 0, +−l2 < τ2 < 0. +Here, (.)∗ denotes the complex conjugate. If A = B, then C (A, B) (τ1, τ2) +is called the two dimensional aperiodic auto correlation function (2D-AACF) +of A and referred to as C (A) (τ1, τ2). +When l1 = 1, the complex-valued arrays A and B are reduced to one +dimensional complex-valued sequences A = (aj)l2−1 +j=0 and B = (bj)l2−1 +j=0 with +the corresponding one dimensional aperiodic cross correlation function (1D- +ACCF) given by +C(A, B)(τ2) = + + + + + + + +�l2−1−τ2 +i=0 +aib∗ +i+τ2, +0 ≤ τ2 < l2, +�l2+τ2−1 +i=0 +ai−τ2b∗ +i , +−l2 < τ2 < 0, +0, +otherwise. +(1) +Definition 2 [19],[9] For a set of s sets of arrays A = +� +Ak | k = 0, 1, . . . , s − 1}, +each set Ak = +� +Ak +0, Ak +1, . . . , Ak +s−1 +� +is composed of s arrays of size is l1 × l2. The +set A is said to be 2D-CCC with parameters (s, s, l1, l2) if the following holds +C +� +Ak, Ak′� +(τ1, τ2) = +s−1 +� +i=0 +C +� +Ak +i , Ak′ +i +� +(τ1, τ2) += + + + + + + + +sl1l2, +(τ1, τ2) = (0, 0), k = k′; +0, +(τ1, τ2) ̸= (0, 0), k = k′; +0, +k ̸= k′. +(2) +Definition 3 [10],[9] Let z1, z2, l1, l2 are positive integers and z1 ≤ l1, z2 ≤ l2. +Consider the sets of ˆs set of arrays A = +� +Ak | k = 0, 1, . . . , ˆs − 1}, where each set +Ak = +� +Ak +0, . . . , Ak +s−1 +� +is composed of s arrays of size l1 × l2. The set A is said to + +Springer Nature 2021 LATEX template +4 +A Direct Construction of Optimal 2D-ZCACS +be 2D − (ˆs, z1 × z2) − ZCACSl1×l2 +s +if the following holds +C +� +Ak, Ak′� +(τ1, τ2) = +s−1 +� +i=0 +C +� +Ak +i , Ak′ +i +� +(τ1, τ2) += + + + + + + + +sl1l2, +(τ1, τ2) = (0, 0), k = k′; +0, +(τ1, τ2) ̸= (0, 0),|τ1| < z1,|τ2| < z2, k = k′; +0, +|τ1| < z1,|τ2| < z2, k ̸= k′. +(3) +When z1 = l1, z2 = l2, ˆs = s the 2D-ZCACS becomes 2D-CCC[19, 20] with +parameter (s, l1, l2). It should be noted that for l1 = 1, each array Ak +i becomes +l2-length sequence. Therefore, 2D-ZCACS can be reduced to a conventional +1D-(ˆs, z2) − ZCCSl2 +s [21], [22],[23], where, ˆs, s, z2, l2 represents no. of set, set +size, ZCZ width and sequence length respectively. +Lemma 1 [9] For a 2D − (ˆs, z1 × z2) − ZCACSl1×l2 +s +, the following inequality holds +ˆsz1z2 ≤ s (l1 + z1 − 1) (l2 + z2 − 1) . +(4) +We called 2D-ZCACS is optimal if the following equality holds +ˆs = s +� l1 +z1 +�� l2 +z2 +� +, +(5) +where ⌊.⌋ denotes the floor function. +2.2 Multivariable Function +Let a, b, mi, and nj be positive integers for 1 ≤ i ≤ a and 1 ≤ j ≤ b. Let pi be +any prime or 1, and qj be a prime number. A multivariable function (MVF) +can be defined as +f : Am1 +p1 × Am2 +p2 × · · · × Ama +pa × An1 +q1 × An2 +q2 × · · · × Anb +qb → Z. +Let c, d ≥ 0 be integers such that 0 ≤ c < r and 0 ≤ d < s where r = +pm1 +1 pm2 +2 +. . . pma +a +and s = qn1 +1 qn2 +2 . . . qnb +b . Then c and d can be written as +c = c1 + c2pm1 +1 ++ · · · + capm1 +1 pm2 +2 +. . . pma−1 +a−1 , +d = d1 + d2qn1 +1 + · · · + dbqn1 +1 qn2 +2 . . . qnb−1 +b−1 , +(6) +where, 0 ≤ ci < pmi +i +and 0 ≤ dj < qnj +j . Let Ci = (ci,1, ci,2, . . . , ci,mi) ∈ Ami +pi , +be the vector representation of ci with base pi, i.e., ci = �mi +k=1 ci,kpk−1 +i +and +Dj = (dj,1, dj,2, . . . , dj,nj) ∈ Anj +qj be the vector representation of dj with base +qj, i.e., dj = �nj +l=1 dj,lql−1 +j +where 0 ≤ ci,k < pi, and 0 ≤ dj,l < qj. We define +vectors associated with c and d as +φ(c) = (C1, C2, . . . , Ca) ∈ Am1 +p1 × Am2 +p2 × · · · × Ama +pa , +φ(d) = (D1, D2, . . . , Db) ∈ An1 +q1 × An2 +q2 × · · · × Anb +qb , + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +5 +respectively. We also define an array associated with f as +ψλ(f) = + + + + + + + +ωf0,0 +λ +ωf0,1 +λ +· · · +ωf0,r−1 +λ +ωf1,0 +λ +ωf1,1 +λ +· · · +ωf1,r−1 +λ +... +... +... +... +ωfs−1,0 +λ +ωfs−1,1 +λ +· · · ωfs−1,r−1 +λ + + + + + + + +, +(7) +where fc,d = f +� +φ(c), φ(d) +� +and λ is a positive integer. +Lemma 2 ([24]) Let t and t′ be two non-negative integers, where t ̸= t′, and p is a +prime number. Then +p−1 +� +j=0 +ω(t−t′)j +p += 0. +(8) +Let us consider the set C as +C = +� +Am1 +p1 × Am2 +p2 × · · · × Ama +pa +� +× +� +An1 +q1 × An2 +q2 × · · · × Anb +qb +� +. +(9) +Let 0 ≤ γ < pm1 +1 pm2 +2 +. . . pma +a +and 0 ≤ µ < qn1 +1 qn2 +2 . . . qnb +b +be positive integers +such that +γ = γ1 + +a +� +i=2 +γi + + +i−1 +� +i1=1 +p +mi1 +i1 + + , +µ = µ1 + +b +� +j=2 +µj + + +j−1 +� +j1=1 +q +nj1 +j1 + + , +(10) +where 0 ≤ γi < pmi +i +and 0 ≤ µj < qnj +j . Let γi = (γi,1, γi,2, . . . , γi,mi) ∈ +Ami +pi be the vector representation of γi with base pi, i.e., γi = �mi +k=1 γi,kpk−1 +i +, +where 0 ≤ γi,k < pi. Similarly µj = (µj,1, µj,2, . . . , µj,nj) ∈ Anj +qj be the vector +representation of µj with base qj i.e., µj = �nj +l=1 µj,lql−1 +j +where 0 ≤ µj,l < qj. +Let +φ(γ) = (γ1, γ2, . . . , γa) ∈ Am1 +p1 ×Am2 +p2 × · · · × Ama +pa , +(11) +be the vector associated with γ and +φ(µ) = (µ1, µ2, . . . , µb) ∈ An1 +q1×An2 +q2 × · · · × Anb +qb , +(12) +be the vector associated with µ. Let πi and σj be any permutations of the +set {1, 2, . . ., mi} and {1, 2, . . ., nj}, respectively. Let us also define the MVF + +Springer Nature 2021 LATEX template +6 +A Direct Construction of Optimal 2D-ZCACS +f : C → Z, as +f(φ(γ), φ(µ)) += f (γ1, γ2, . . . , γa, µ1, µ2, . . . , µb) += +a +� +i=1 +λ +pi +mi−1 +� +e=1 +γi,πi(e)γi,πi(e+1) + +a +� +i=1 +mi +� +e=1 +di,eγi,e + +b +� +j=1 +λ +qj +nj−1 +� +o=1 +µj,σj(o)µj,σj(o+1) ++ +b +� +j=1 +nj +� +o=1 +cj,oµj,o, +(13) +where di,e, cj,o ∈ {0, 1, . . ., λ − 1} and λ = l.c.m.(p1, . . . , pa, q1, . . . , qb). Let us +define the set Θ and T as +Θ = {θ : θ = (r1, r2, . . . , ra, s1, s2, . . . , sb)}, +T = {t : t = (x1, x2, . . . , xa, y1, y2, . . . , yb)}, +where 0 ≤ ri, xi < pki +i +and 0 ≤ sj, yj < qrj +j +and ki, rj are positive integers. +Now, we define a function aθ +t: C →Z, as +aθ +t +� +φ(γ), φ(µ) +� += aθ +t (γ1, γ2, . . . , γa, µ1, µ2, . . . , µb) +=f +� +φ(γ), φ(µ) +� ++ +a +� +i=1 +λ +pi +γi,πi(1)ri + +b +� +j=1 +λ +qj +µj,σj(1)sj + +a +� +i=1 +λ +pi +γi,πi(mi)xi ++ +b +� +j=1 +λ +qj +µj,σj(nj)yj + dθ, +(14) +where 0 ≤ dθ < λ, γi,πi(1), γi,πi(mi) denote πi(1)−th and πi(mi)−th element +of γi respectively. Similarly, µj,σj(1), µj,σj(nj) denote σj(1)−th and σj(nj)−th +element of µj respectively. For simplicity, we denote aθ +t +� +φ(γ), φ(µ) +� +by (aθ +t)γ,µ +and f +� +φ(γ), φ(µ) +� +by fγ,µ. +Lemma 3 ([20]) We define the ordered set of arrays At = {ψλ +� +aθ +t +� +: θ ∈ Θ}. +Then the set {At : t ∈ T } forms a 2D-CCC with parameter (α, α, m, n), where, α = +�a +i=1 pki +i +�b +j=1 qrj +j , m = �a +i=1 pmi +i +, n = �b +j=1 qnj +j +and ki, mi, nj, rj are non-negative +integers. + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +7 +3 Proposed construction of 2D-ZCACS +Let a′, b′ be positive integers for 1 ≤ i′ ≤ a′ and 1 ≤ j′ ≤ b′, p′ +i′ be any +prime or 1, and q′ +j′ be prime number. Let γ′, µ′ are positive integers such that +0 ≤ γ′ < +��a +i=1 pmi +i +� ��a′ +i′=1 p′ +i′ +� +and 0 ≤ µ′ < +��b +j=1 qnj +j +� ��b′ +j′=1 q′ +j′ +� +. Then +γ′, µ′ can be written as +γ′ =γ1+ +a +� +i=2 +γi + + +i−1 +� +i1=1 +p +mi1 +i1 + ++ + + +γ′ +1 + +a′ +� +i′=2 +γ′ +i′ + + +i′−1 +� +i1=1 +p′ +i1 + + + + + m, +µ′ =µ1+ +b +� +j=2 +µj + + +j−1 +� +j1=1 +q +nj1 +j1 + ++ + + +µ′ +1 + +b′ +� +j′=2 +µ′ +j′ + + +j′−1 +� +j1=1 +q′ +j1 + + + + + n, +(15) +where m = �a +i=1 pmi +i , n = �b +j=1 qnj +j , 0 ≤ γi < pmi +i , 0 ≤ µj < qnj +j , 0 ≤ γ′ +i′ < p′ +i′ +and 0 ≤ µ′ +j′ < q′ +j′. We denote the vectors associated with γ′ and µ′ are +φ(γ′) = +� +γ1, . . . , γa, γ′ +1, . . . , γ′ +a +� +∈ Am1 +p1 × . . . × Ama +pa × Ap′ +1 × . . . × Ap′ +a′ , +φ(µ′) = +� +µ1, . . . , µb, µ′ +1, . . . , µ′ +b +� +∈ An1 +q1 × . . . × Anb +qb × Aq′ +1 × . . . × Aq′ +b′ , +(16) +respectively, where γi ∈ Ami +pi , µj ∈ Anj +qj +are the vectors associated with +γi and µj +respectively i.e., γi += +(γi,1, γi,2, . . . , γi,mi) +∈ +Ami +pi , µj += +(µj,1, µj,2, . . . , µj,nj) ∈ Anj +qj , γi = �mi +k=1 γi,kpk−1 +i +, µj = �nj +l=1 µi,lql−1 +j +, 0 ≤ +γi,k < pi and 0 ≤ µj,l < qj. Let us consider the set D as +D = Am1 +p1 ×. . .×Ama +pa ×Ap′ +1 ×. . .×Ap′ +a′ ×An1 +q1 ×. . .×Anb +qb ×Aq′ +1 ×. . .×Aq′ +b′ . (17) +Let f be the function as defined (13). We define the MVF M c,d : D → Z as +M c,d � +φ(γ′), φ(µ′) +� += M c,d � +γ1, . . . , γa, γ′ +1, . . . , γ′ +a′, µ1, . . . , µb, µ′ +1, . . . , µ′ +b′ +� += δ +λf (γ1, . . . , γa, µ1, . . . , µb)+ +a′ +� +i′=1 +ci′ δ +p′ +i′ γ′ +i′ + +b′ +� +j′=1 +dj′ δ +q′ +j′ µ′ +j′, +(18) +where +0 +≤ +ci′ +< +p′ +i′, +0 +≤ +dj′ +< +q′ +j′, +c += +(c1, c2, . . . , ca′) +and +d += +(d1, d2, . . . , db′). +For +simplicity, +now +on-wards +we +denote +M c,d(γ1, . . . , γa, γ′ +1, . . . , γ′ +a′, µ1, . . . , µb, µ′ +1, . . . , µ′ +b′) by M c,d. Consider the set +Θ and T as +Θ = {θ : θ = (r1, r2, . . . , ra, s1, s2, . . . , sb)}, +T = {t : t = (x1, x2, . . . , xa, y1, y2, . . . , yb)}, + +Springer Nature 2021 LATEX template +8 +A Direct Construction of Optimal 2D-ZCACS +where 0 ≤ ri, xi < pki +i +and 0 ≤ sj, yj < qrj +j +and ki, rj are positive integers. Let +us define MVF, bθ,c,d +t +: D → Z, as +bθ,c,d +t +=M c,d + +a +� +i=1 +δ +pi +γi,πi(1)ri + +b +� +j=1 +δ +qj +µj,σj(1)sj + +a +� +i=1 +δ +pi +γi,πi(mi)xi ++ +b +� +j=1 +δ +qj +µj,σj(nj)yj + δ +λdθ, +(19) +where 0 ≤ dθ < λ. By (14), (18) and (19) we have +bθ,c,d +t += δ +λaθ +t + +a′ +� +i′=1 +ci′ δ +p′ +i′ γ′ +i′ + +b′ +� +j′=1 +dj′ δ +q′ +j′ µ′ +j′. +(20) +We define the ordered set of arrays as +Ωc,d +t += {ψδ(bθ,c,d +t +) : θ ∈ Θ}. +(21) +where δ = l.c.m(λ, p′ +1, p′ +2, . . . , p′ +a′, q′ +1, q′ +2, . . . , q′ +b′). +Theorem 1 Let m = �a +i=1 pmi +i +, n = �b +j=1 qnj +j , c = (c1, . . . , ca′), d = (d1, . . . , db′). +Then the set S = {Ωc,d +t +: t ∈ T, 0 ≤ ci′ < p′ +i′, 0 ≤ dj′ < q′ +j′} forms a 2D − (α1, z1 × +z2) − ZCACSl1×l2 +α +, where, α1 = +��a′ +i′=1 p′ +i′ +� ��b′ +j′=1 q′ +j′ +� +α, l1 = m +��a′ +i′=1 p′ +i′ +� +, +l2 = n +��b′ +j′=1 q′ +j′ +� +, z1 = m ,z2 = n, α = (�a +i=1 pki +i )(�b +j=1 qrj +j ), ki, rj, mi, nj ≥ 1. +Proof Let ˆγ, ˆµ are positive integers such that 0 ≤ ˆγ < l1 and 0 ≤ ˆµ < l2. Then ˆγ, ˆµ +can be written as +ˆγ = γ1+ +a +� +i=2 +γi + + +i−1 +� +i1=1 +p +mi1 +i1 + ++ + + +γ′ +1 + +a′ +� +i′=2 +γ′ +i′ + + +i′−1 +� +i1=1 +p′ +i1 + + + + + m, +ˆµ = µ1+ +b +� +j=2 +µj + + +j−1 +� +j1=1 +qnj1 +j1 + ++ + + + +µ′ +1 + +b′ +� +j′=2 +µ′ +j′ + + + +j′−1 +� +j1=1 +q′ +j1 + + + + + + + n, +where 0 ≤ γi < pmi +i +, 0 ≤ µj < qnj +j , 0 ≤ γ′ +i′ < p′ +i′ and 0 ≤ µ′ +j′ < q′ +j′. The proof will +be split into following cases +Case 1. (τ1 = 0, τ2 = 0) + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +9 +The ACCF between Ωc,d +t +and Ωc′,d′ +t′ +at τ1 = 0 and τ2 = 0 can be expressed as +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, 0) += +� +θ∈Θ +C(ψδ((bθ,c,d +t +)), ψδ((bθ,c′,d′ +t′ +)))(0, 0) += +� +θ∈Θ +l1−1 +� +ˆγ=0 +l2−1 +� +ˆµ=0 +ω +(bθ,c,d +t +)ˆγ,ˆ +µ−(bθ,c′,d′ +t′ +)ˆγ,ˆ +µ +δ += +� +θ∈Θ +m−1 +� +γ=0 +n−1 +� +µ=0 +p′ +1−1 +� +γ′ +1=0 +. . . +p′ +a′ −1 +� +γ′ +a′=0 +q′ +1−1 +� +µ1=0 +. . . +q′ +b′ −1 +� +µ′ +b′ =0 +ωD +δ , +(22) +where D = δ +λ +� +(aθ +t )γ,µ − (aθ +t′)γ,µ +� ++�a′ +i′=1 +δ +p′ +i′ (ci′ −c′ +i′)γi′ +�b′ +j′=1 +δ +q′ +j′ (dj′ −d′ +j′)µj′. +After splitting (22), we get +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, 0) += + +� +θ∈Θ +m−1 +� +γ=0 +n−1 +� +µ=0 +ω +δ +λ +� +(aθ +t )γ,µ−(aθ +t′ )γ,µ +� +δ + + EF += + +� +θ∈Θ +m−1 +� +γ=0 +n−1 +� +µ=0 +ω +� +(aθ +t )γ,µ−(aθ +t′ )γ,µ +� +λ + + EF += C(At, At′ +)(0, 0)EF, +(23) +where +E = +a′ +� +i′=1 + + + +p′ +i′ −1 +� +γ′ +i′ =0 +ω +(ci′−c′ +i′)γ′ +i′ +p′ +i′ + + + , +F = +b′ +� +j′=1 + + + + +q′ +j′ −1 +� +µ′ +j′ =0 +ω +(dj′ −d′ +j′ )µ′ +j′ +q′ +j′ + + + + . +(24) +Subcase (i): (t ̸= t′) +By lemma 2 we know, the set {At : t ∈ T } forms a 2D-CCC. Hence By lemma 2, we +have +C(At, At′ +)(0, 0) = 0. +(25) +Hence by (23) and (25) we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, 0) = 0. +(26) +Subcase (ii): (t = t′) +By lemma 2, we know +C(At, At′ +)(0, 0) = + + +a +� +i=1 +pmi+ki +i + + + + +b +� +j=1 +qnj+rj +j + + . +(27) + +Springer Nature 2021 LATEX template +10 +A Direct Construction of Optimal 2D-ZCACS +Let M = +��a +i=1 pmi+ki +i +� ��b +j=1 qnj+rj +j +� +hence by Lemma 2, (23), (24), (27), we +have the following +C(Ωc,d +t +, Ωc′,d′ +t +)(0, 0) = + + + + + + + + + + + + + +M +��a′ +i′=1 p′ +i′ +� ��b′ +j′=1 q′ +j′ +� +c = c′, d = d′ +0, +c ̸= c′, d = d′ +0, +c = c′, d ̸= d′ +0, +c ̸= c′, d ̸= d′. +(28) +Case 2. (0 < τ1 < �a +i=1 pmi +i +, 0 < τ2 < �b +j=1 qnj +j ) +Let σ, ρ are positive integers such that 0 ≤ σ < m′ and 0 ≤ ρ < n′ where m′ = +�a′ +i′=1 p′ +i′, n′ = �b′ +j′=1 q′ +j′. Then σ and ρ can be written as +σ = σ1 + σ2p′ +1 + . . . + σa′ + + +a′−1 +� +i′=1 +p′ +i′ + + , +ρ = ρ1 + ρ2q′ +1 + . . . + ρb′ + + +b′−1 +� +j′=1 +q′ +j′ + + , +(29) +respectively where 0 ≤ σi′ < p′ +i′ and 0 ≤ ρj′ < q′ +j′ . We define vectors associated +with σ and ρ to be +φ(σ) = (σ1, . . . , σa′) ∈ Ap′ +1 × . . . × Ap′ +a′ , +φ(ρ) = (ρ1, . . . , ρb′) ∈ Aq′ +1 × . . . × Aq′ +b′ , +(30) +respectively. The ACCF between Ωc,d +t +and Ωc′,d′ +t′ +for 0 < τ1 < �a +i=1 pmi +i +and 0 < +τ2 < �b +j=1 qnj +j , can be derived as +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, τ2) =C(At, At′ +)(τ1, τ2)DE+C(At, At′ +)(τ1− +a +� +i=1 +pmi +i +, τ2)D′E+ +C(At, At′ +)(τ1, τ2 − +b +� +j=1 +qnj +j )DE′ + C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, τ2 − +b +� +j=1 +qnj +j )D′E′, +(31) +where +D = +m′−1 +� +σ=0 + + +a′ +� +i′=1 +ω +(ci′−c′ +i′ )(σi′ ) +p′ +i′ + + , +(32) +E = +n′−1 +� +ρ=0 + + +b′ +� +j′=1 +ω +(dj′ −d′ +j′)(ρj′ ) +q′ +j′ + + , +(33) +D′ = +m′−2 +� +σ=0 + + +a′ +� +i′=1 +ω(ci′(σi′ )−c′ +i′ (σ+1)i′) +p′ +i′ + + , +(34) +E′ = +n′−2 +� +ρ=0 + + +b′ +� +j′=1 +ω +� +dj′ (ρj′ )−d′ +j′(ρ+1)j′ +� +q′ +j′ + + , +(35) + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +11 +and (σ + 1)i′ , (ρ + 1)j′ denotes the i′-th and j′-th components of φ (σ + 1) and +φ (ρ + 1) respectively. By Lemma 2, for 0 < τ1 < �a +i=1 pmi +i +and 0 < τ2 < �b +j=1 qnj +j , +we have +C(At, At′ +)(τ1, τ2) = 0, +(36) +C(At, At′ +)(τ1− +a +� +i=1 +pmi +i +, τ2) = 0, +(37) +C(At, At′ +)(τ1, τ2 − +b +� +j=1 +qnj +j ) = 0, +(38) +C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, τ2 − +b +� +j=1 +qnj +j ) = 0. +(39) +By (31), (36), (37), (38), (39) we have +C(Ωc,d +t +, Ωc +′ ,d +′ +t′ +)(τ1, τ2) = 0. +(40) +Case 3. (0 < τ1 < �a +i=1 pmi +i +, − �b +j=1 qnj +j +< τ2 < 0) +The ACCF between Ωc,d +t +and Ωc′,d′ +t′ +for 0 < τ1 < �a +i=1 pmi +i +and − �b +j=1 qnj +j +< τ2 < +0, can be derived as +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, τ2) +=C(At, At′ +)(τ1, τ2)DE+C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, τ2)D′E ++ C(At, At′ +)(τ1, +b +� +j=1 +qnj +j ++ τ2)DE′′ + C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, +b +� +j=1 +qnj +j ++ τ2)D′E′′, +(41) +where +E′′ = +n′−2 +� +ρ=0 + + +b′ +� +j′=1 +ω +� +dj′ (ρ+1)j′ −d′ +j′ (ρj′ ) +� +q′ +j′ + + . +(42) +By Lemma 2, for 0 < τ1 < �a +i=1 pmi +i +and − �b +j=1 qnj +j +< τ2 < 0, we have +C(At, At′ +)(τ1, +b +� +j=1 +qnj +j ++ τ2) = 0, +(43) +C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, +b +� +j=1 +qnj +j ++ τ2) = 0. +(44) +By (41) , (43) and (44) we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, τ2) = 0. +(45) +Case 4. (0 < τ1 < �a +i=1 pmi +i +, τ2 = 0) + +Springer Nature 2021 LATEX template +12 +A Direct Construction of Optimal 2D-ZCACS +The ACCF between Ωc,d +t +and Ωc′,d′ +t′ +for 0 < τ1 < �a +i=1 pmi +i +and τ2 = 0 , can be +derived as +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, 0) =C(At, At′ +)(τ1, 0)DE+ C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, 0)D′E. +(46) +By Lemma 2, for 0 < τ1 < �a +i=1 pmi +i +, we have +C(At, At′ +)(τ1, 0) = 0. +C(At, At′ +)(τ1 − +a +� +i=1 +pmi +i +, 0) = 0, +(47) +by (46) and (47) we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, 0) = 0. +(48) +Case 5. +(τ1 = 0, 0 < τ2 < �b +j=1 qnj +j ) +The ACCF between Ωc,d +t +and Ωc′,d′ +t′ +for τ1 = 0 and 0 < τ2 < �b +j=1 qnj +j , can be +derived as +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, τ2) =C(At, At′ +)(0, τ2)DE+ C(At, At′ +)(0, τ2 − +b +� +j=1 +qnj +j )DE′. +(49) +By Lemma 2, for 0 < τ2 < �b +j=1 qnj +j , we have +C(At, At′ +)(0, τ2) = 0, +C(At, At′ +)(0, τ2 − +b +� +j=1 +qnj +j ) = 0. +(50) +By (49) and (50) we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, τ2) = 0. +(51) +Case 6. (τ1 = 0, − �b +j=1 qnj +j +< τ2 < 0) +Similarly the ACCF between Ωc,d +t +and Ωc′,d′ +t′ +for τ1 = 0 and − �b +j=1 qnj +j +< τ2 < 0 is +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, τ2) =C(At, At′ +)(0, τ2)DE+ C(At, At′ +)(0, τ2 + +b +� +j=1 +qnj +j )DE′′. +(52) +By Lemma 2, for − �b +j=1 qnj +j +< τ2 < 0, we have +C(At, At′ +)(0, τ2 + +b +� +j=1 +qnj +j ) = 0. +(53) +Hence by (50), (52) and (53) we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(0, τ2) = 0. +(54) + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +13 +Combining all the cases we have +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, τ2) = + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +M +��a′ +i′=1 p′ +i′ +� ��b′ +j′=1 q′ +j′ +� +, +(c, d, t) = (c′, d′, t′) +(τ1, τ2) = (0, 0), +0, +(c, d, t) ̸= (c′, d′, t′) +(τ1, τ2) = (0, 0), +0, +0 ≤ τ1 < �a +i=1 pmi +i +, +(τ1, τ2) ̸= (0, 0). +(55) +Similarly it can be shown +C(Ωc,d +t +, Ωc′,d′ +t′ +)(τ1, τ2) = 0, − +a +� +i=1 +pmi +i +< τ1 < 0. +(56) +Hence from (55), (56) we derive our conclusion. +□ +Example 1 Suppose that a = 1, b = 1, a′ = 1, b′ = 1, p1 = 2, m1 = 2, k1 = 1, q1 = 3, +n1 = 2, r1 = 1, p′ +1 = 3, q′ +1 = 2. Let δ = 6, λ = 6, γ1 = (γ11, γ12) ∈ A2 +2 = {0, 1}2 +be the vector associated with γ1 where 0 ≤ γ1 ≤ 3, i.e., γ1 = γ11 + 2γ12 and +µ1 = (µ11, µ12) ∈ A2 +3 = {0, 1, 2}2 be the vector associated with µ1 where 0 ≤ µ1 ≤ 8, +i.e., µ1 = µ11+3µ12 and 0 ≤ γ′ +1 ≤ 2, 0 ≤ µ′ +1 ≤ 1. We define the MVF f : A2 +2×A2 +3 → Z +as +f (γ1, µ1)=3γ1,2γ1,1+γ1,1+2γ1,2+2µ1,2µ1,1+2µ1,1+µ1,2. +Consider the MVF, Mc,d : A2 +2 × A3 × A2 +3 × A2 → Z as +Mc,d � +γ1, γ′ +1, µ1, µ′ +1 +� += f(γ1, µ1) + 2c1γ′ +1 + 3d1µ′ +1 += 3γ1,2γ1,1 + γ1,1 + 2γ1,2 + 2µ1,2µ1,1 + 2µ1,1 + µ1,2 + 2c1γ′ +1 + 3d1µ′ +1, +(57) +where 0 ≤ c1 < p′ +1 = 2, 0 ≤ d1 < q′ +1 = 3, c = c1 ∈ {0, 1}, and d = d1 ∈ {0, 1, 2}. We +have +Θ = {θ : θ = (r1, s1) : 0 ≤ r1 ≤ 1, 0 ≤ s1 ≤ 2}, +T = {t : t = (x1, y1) : 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 2}. +(58) +Let dθ = 0, now from (19) we have +bθ,c,d +t += Mc,d + 3γ1,2r1 + 2µ1,2s1 + 3γ1,1x1 + 2µ1,2y1, +(59) +and +Ωc,d +t += +� +ψ6(bθ,c,d +t +) : θ = (r1, s1) ∈ {0, 1} × {0, 1, 2} +� +. +(60) +Therefore, the set +S = {Ωc,d +t +: t ∈ T, 0 ≤ c1 ≤ 1, 0 ≤ d1 ≤ 2}, +(61) +forms an optimal 2D − (36, 4 × 9) − ZCACS12×18 +6 +over Z6. + +Springer Nature 2021 LATEX template +14 +A Direct Construction of Optimal 2D-ZCACS +Table 1 Comparison with Previous Works +Source +No. of set +Array Size +Condition +Based on +[7] +K = K′r +L′ +1×(L′ +2 + r + 1) +r ≥ 0 +2D − ZCACS of +set size K′ and +array size L′ +1×L′ +2 +[8] +1 +2m × 2nL +m, n ≥ 0 +ZCP of length L +[9] +K +K × K +K divides set size +BH matrices +[10] +2 �ki +i=1 p2 +i +2m × �ki +i=1 pmi +i +ki, mi ≥ 1, pi’s are prime +MVF +Thm 2 +rsα +rm × sn +α = (�a +i=1 pki +i )(�b +j=1 q +rj +j ), +m=�a +i=1pmi +i +, n=�b +j=1q +nj +j , +r, s, α ≥ 1, pi, qjareprimes +MVF +Remark 1 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′ +1 = 1, b = 1, q1 = +2, b′ = l, r1 ≥ 2, we have optimal 1D-ZCCS with parameter (�l +i=1 q′ +i2r1, 2n1) − +ZCCS +�l +i=1 q′ +i2n1 +2r1 +, which is exactly the same result as in [18]. Also if we take l = 1, then +we have optimal 1D-ZCCS of the form (q′ +12r1, 2n1) − ZCCSq′ +12n1 +2r1 +, which is exactly +the same result in [17]. Therefore the optimal 1D-ZCCS given by [17, 18] appears as +a special case of the proposed construction +Remark 2 In Theorem 1, if a = 1, p1 = 1, a′ = 1, p′ +1 = 1, b = 1, q1 = 2, b′ = l, r1 = 1, +we have 1d-ZCCS with parameter (2 �l +i=1 q′ +i, 2n1) − ZCCS +�l +i=1 q′ +i2n1 +2 +, which is just +a collection of 2 �l +i=1 q′ +i ZCPs with sequence length �l +i=1 q′ +i2n1 and ZCZ width 2n1. +Hence our work produces collections of ZCPs[15] as well. +Remark 3 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′ +1 = 1, b = 1, q1 = 2, +b′ = r, q′ +1 = q′ +2 = . . . = q′r = 2, n1 = m − r and r1 = s + 1 then we have 1D-ZCCS +with parameter (2s+r+1, 2m−r)−ZCCS2m +2s+1, which is exactly the same result in [16]. +Hence, the ZCCS in [16] appears as a special case of our proposed construction. +Remark 4 The 2D-ZCACS given by the proposed construction satisfies the equality +given in (5). Therefore the 2D-ZCACS obtained by the proposed construction is +optimal. +Remark 5 If we take a = 1, a′ = 1, p1 = 1 and p′ +1 = 1, in Theorem 1, we have optimal +1D-ZCCS with parameter +���b′ +j′=1 q′ +j′ +� �b +j=1 qrj +j , n +� +− ZCCS +n +��b′ +j′=1 q′ +j′ +� +�b +j=1 q +rj +j +where, +n = �b +j=1 qnj +j . Hence, we have optimal 1D-ZCCS of length nm where, n, m > 1 and +m = �b′ +j′=1 q′ +j′. Therefore our construction produces optimal 1D-ZCCS with a new +length which is not present in the literature by direct method. + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +15 +Remark +6 The +set +size +of +our +proposed +2D-ZCACS +is +��a′ +i′=1 p′ +i′ +� ��b′ +j′=1 q′ +j′ +� �a +i=1 pki +i +�b +j=1 qrj +j +where, +ki, tj +≥ +1. +If +we +take +a = 1, p1 = 1, a′ = 1, p′ +1 = 1, r1 = r2 = . . . = rb = 2, b′ = 1, and q′ +1 = 2 then we +have set size 2 �b +j=1 q2 +j which is the set size of the 2D-ZCACS in [10]. Therefore, we +have flexible number of set sizes compared to [10]. +3.1 Comparison with Previous Works +Table I compares the proposed work with indirect constructions from [7–9] and +direct construction from [10]. The constructions in [7–9] heavily rely on initial +sequences, increasing hardware storage. The construction in [10] is direct, but +set size and array sizes are limited to some even numbers. Our construction +doesn’t require initial matrices or sequences and produces flexible parameters. +4 Conclusion +In this paper, 2D-ZCACSs are designed by using MVF. The proposed design +does not depend on initial sequences or matrices, so it is direct. Our proposed +design produces flexible array size and set size compared to existing works. +Also, our proposed construction can be reduced to 1D-ZCCS. As a result, +many 1D-ZCCSs become special cases of our work. Finally, we compare our +work to the existing state-of-the-art and show that it’s more versatile. +References +[1] Farkas, P., Turcs´any, M.: Two-dimensional orthogonal complete com- +plementary codes. In: Joint IEEE 1st Workshop on Mobile Future and +Symposium on Trends in Communications (sympoTIC), pp. 21–24 (2003) +[2] Turcs´any, M., Farkaˇs, P.: New 2d-mc-ds-ss-cdma techniques based on two- +dimensional orthogonal complete complementary codes. in Multi-Carrier +Spread-Spectrum, Berlin, Germany: Springer, 49–56 (2004) +[3] Chen, C.-Y., Wang, C.-H., Chao, C.-C.: Complete complementary codes +and generalized reed-muller codes. IEEE Commun. Lett. 12(11), 849–851 +(2008) +[4] Das, S., Majhi, S., Liu, Z.: A novel class of complete complementary codes +and their applications for apu matrices. IEEE Sig. Process. Lett. 25(9), +1300–1304 (2018) +[5] Liu, Z., Guan, Y.L., Parampalli, U.: New complete complementary codes +for peak-to-mean power control in multi-carrier cdma. IEEE Trans. +Commun. 62(3), 1105–1113 (2014) + +Springer Nature 2021 LATEX template +16 +A Direct Construction of Optimal 2D-ZCACS +[6] Xeng, F., Zhang, Z., Ge, L.: Theoretical limit on two dimensional gener- +alized complementary orthogonal sequence set with zero correlation zone +in ultra wideband communications. International Workshop on UWBST +& IWUWBS, 197–201 (2004) +[7] Zeng, F., Zhang, Z., Ge, L.: Construction of two-dimensional comple- +mentary orthogonal sequences with ZCZ and their lower bound. IET +(2005) +[8] Pai, +C.-Y., +Ni, +Y.-T., +Chen, +C.-Y.: +Two-dimensional +binary +Z- +complementary array pairs. IEEE Trans. Inf. Theory 67(6), 3892–3904 +(2021) +[9] Das, S., Majhi, S.: Two-dimensional Z-complementary array code sets +based on matrices of generating polynomials. IEEE Trans. Signal Process. +68, 5519–5532 (2020) +[10] Roy, A., Majhi, S.: Construction of inter-group complementary code set +and 2D Z-complementary array code set based on multivariable functions +(2021). https://doi.org/10.48550/arXiv.2109.00970 +[11] Shen, B., Meng, H., Yang, Y., Zhou, Z.: New constructions of z- +complementary code sets and mutually orthogonal complementary +sequence sets. Des. Codes Cryptogr., 1–19 (2022) +[12] Sarkar, P., Roy, A., Majhi, S.: Construction of z-complementary code sets +with non-power-of-two lengths based on generalized boolean functions. +IEEE Commun. Lett. 24(8), 1607–1611 (2020) +[13] Sarkar, P., Majhi, S.: A direct construction of optimal zccs with maximum +column sequence pmepr two for mc-cdma system. IEEE Commun. Lett +25(2), 337–341 (2020) +[14] Wu, S.-W., S¸ahin, A., Huang, Z.-M., Chen, C.-Y.: Z-complementary code +sets with flexible lengths from generalized boolean functions. IEEE Access +9, 4642–4652 (2020) +[15] Kumar, P., Sarkar, P., Majhi, S., Paul, S.: A direct construction of even +length zcps with large zcz ratio. Cryptogr. Commun., 1–10 (2022) +[16] Sarkar, P., Majhi, S., Liu, Z.: Optimal Z-complementary code set from +generalized Reed-Muller codes. IEEE Trans. Commun. 67(3), 1783–1796 +(2018) +[17] Sarkar, P., Majhi, S., Liu, Z.: Pseudo-Boolean functions for optimal Z- +complementary code sets with flexible lengths. IEEE Signal Process. Lett. +28, 1350–1354 (2021) + +Springer Nature 2021 LATEX template +A Direct Construction of Optimal 2D-ZCACS +17 +[18] Ghosh, G., Majhi, S., Sarkar, P., Upadhaya, A.K.: Direct construction of +optimal Z-complementary code sets with even lengths by using generalized +boolean functions. IEEE Signal Process. Lett. 29, 872–876 (2022) +[19] Pai, C.-Y., Liu, Z., Zhao, Y.-Q., Huang, Z.-M., Chen, C.-Y.: Design- +ing two-dimensional complete complementary codes for omnidirectional +transmission in massive mimo systems. In: International Symposium on +Information Theory (ISIT), pp. 2285–2290 (2022). IEEE +[20] Ghosh, G., Majhi, S., Upadhyay, A.K.: A direct construction of 2D-CCC +with arbitrary array size and flexible set size using multivariable function +(2022). https://doi.org/10.48550/arXiv.2207.13395 +[21] Wu, S.-W., Chen, C.-Y.: Optimal Z-complementary sequence sets with +good peak-to-average power-ratio property. IEEE Signal Process. Lett. +25(10), 1500–1504 (2018) +[22] Yu, T., Adhikary, A.R., Wang, Y., Yang, Y.: New class of optimal Z- +complementary code sets. IEEE Trans. Signal Process. 29, 1477–1481 +(2022) +[23] Shen, +B., +Yang, +Y., +Fan, +P., +Zhou, +Z.: +New +z- +complementary/complementary sequence sets with non-power-of-two +length and low papr. Cryptogr. Commun. 14(4), 817–832 (2022) +[24] Vaidyanathan, P.: Ramanujan sums in the context of signal process- +ing—part i: Fundamentals. IEEE Trans. Signal Process. 62(16), 4145– +4157 (2014) + + diff --git a/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0f0f814d1c3ff2c349657aa4a8cffec5faac1a6 --- /dev/null +++ b/7NE0T4oBgHgl3EQffQAA/content/tmp_files/load_file.txt @@ -0,0 +1,660 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf,len=659 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='02400v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='IT] 6 Jan 2023 Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS with Flexible Array Size and Large Set Size Gobinda Ghosh1, Sudhan Majhi2* and Shubhabrata Paul1 1Mathematics, IIT Patna, Bihta, Patna, 801103, Bihar, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2*Electrical Communication Engineering, IISc Bangalore, CV Raman Rd, Bengaluru, 560012, Karnataka, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' E-mail(s): smajhi@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Contributing authors: gobinda 1921ma06@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' shubhabrata@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Abstract In this paper, we propose a direct construction of optimal two- dimensional Z-complementary array code sets (2D-ZCACS) using multivariable functions (MVFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In contrast to earlier works, the proposed construction allows for a flexible array size and a large set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Additionally, the proposed design can be transformed into a one-dimensional Z-complementary code set (1D-ZCCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Many of the 1D-ZCCS described in the literature appeared to be special cases of this proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At last, we compare our work with the current state of the art and then draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Keywords: Two dimensional complete complementary codes (2D-CCC), multivariable function (MVF), two dimensional Z- complementary array code set (2D-ZCACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 1 Introduction For an asynchronous two dimensional multi-carrier code-division multiple access (2D-MC-CDMA) system, the ideal 2D correlation properties of two dimensional complete complementary codes (2D-CCCs)[1] can be properly uti- lized to obtain interference-free performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similar to one dimensional complete complementary code (1D-CCC)[3–5], one of the most significant 1 Springer Nature 2021 LATEX template 2 A Direct Construction of Optimal 2D-ZCACS drawbacks of 2D-CCC is that the set size is restricted [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Motivated by the scarcity of 2D-CCC with flexible set sizes, Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' proposed 2D Z- complementary array code sets (2D-ZCACSs) in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For a 2D − (K, Z1 × Z2)−ZCACSL1×L2 M , K, Z1×Z2, L1×L2 and M denote the set size, two dimen- sional zero-correlation zone (2D-ZCZ) width, array size and the number of constituent arrays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In [6, 7], authors obtained ternary 2D-ZCACSs by inserting some zeros into the existing binary 2D-ZCACSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In 2021, Pai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' presented a new construction method of 2D binary Z-complementary array pairs (2D-ZCAP) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Recently, Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' in [9] proposed a construction of 2D-ZCACS by using Z-paraunitary (ZPU) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' All these construc- tions of 2D-ZCACS depend heavily on initial sequences and matrices which increase hardware storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For the first time in the literature, Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' in [10] proposed a direct construction of 2D-ZCACS based on MVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The array size of the proposed 2D-ZCACS is of the form L1 × L2, where L1 = 2m, L2 = 2pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pmk k , m ≥ 1, mi ≥ 2 and the set size is of the form 2p2 1p2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p2 k where pi is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the array size and the set size is restricted to some even numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Existing array and set size limitations through direct construction in the literature motivates us to search multivariable function (MVF) for more flex- ible array and set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our proposed construction provides 2D-ZCACS with parameter 2D − (R1R2M1M2, N1 × N2) − ZCACSR1N1×R2N2 M1M2 where M1 = �a i=1 pki i , M2 = �b j=1 qtj j , pi is any prime or 1, qj is prime, a, b, ki, tj ≥ 1, R1 and R2 are positive integer, such that R1 ≥ 1 and R2 ≥ 2, N1 = �a i=1 pmi i , N2 = �b j=1 qnj j , mi, nj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set size in our proposed 2D-ZCACS construc- tion, R1R2M1M2, is more adaptable than the set size of 2D-ZCACS given in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Unlike [10], the proposed 2D-ZCACS can be reduced to 1D-ZCCS [11–18] also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' As a result, many existing optimal 1D-ZCCSs have become special cases of the proposed construction [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proposed construction also derived a new set of optimal 1D-ZCCS that had not previously been presented by direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Section 2 discusses construc- tion related definitions and lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Section 3 contains the construction of 2D-ZCACS and the comparison with the existing state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Finally, in Section 4, the conclusions are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2 Notations and definitions The following notations will be followed throughout this paper: ωn = exp � 2π√−1/n � , An = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', n− 1} ⊂ Z, where n is a positive integer and Z is the ring of integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='1 Two Dimensional Array Definition 1 ([9]) Let A = � ag,i � and B = � bg,i � be complex-valued arrays of size l1 × l2 where 0 ≤ g < l1, 0 ≤ i < l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The two dimensional aperiodic cross correlation Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 3 function (2D-ACCF) of arrays A and B at shift (τ1, τ2) is defined as C (A, B) (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 �l1−1−τ1 g=0 �l2−1−τ2 i=0 ag,ib∗ g+τ1,i+τ2, if 0 ≤ τ1 < l1, 0 ≤ τ2 < l2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1−τ1 g=0 �l2−1+τ2 i=0 ag,i−τ2b∗ g+τ1,i, if 0 ≤ τ1 < l1, −l2 < τ2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1+τ1 g=0 �l2−1−τ2 i=0 ag−τ1,ib∗ g,i+τ2, if −l1 < τ1 < 0, 0 ≤ τ2 < l2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' �l1−1+τ1 g=0 �l2−1+τ2 i=0 ag−τ1,i−τ2b∗ g,i, if −l1 < τ1 < 0, −l2 < τ2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Here, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' )∗ denotes the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' If A = B, then C (A, B) (τ1, τ2) is called the two dimensional aperiodic auto correlation function (2D-AACF) of A and referred to as C (A) (τ1, τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' When l1 = 1, the complex-valued arrays A and B are reduced to one dimensional complex-valued sequences A = (aj)l2−1 j=0 and B = (bj)l2−1 j=0 with the corresponding one dimensional aperiodic cross correlation function (1D- ACCF) given by C(A, B)(τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 �l2−1−τ2 i=0 aib∗ i+τ2, 0 ≤ τ2 < l2, �l2+τ2−1 i=0 ai−τ2b∗ i , −l2 < τ2 < 0, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (1) Definition 2 [19],[9] For a set of s sets of arrays A = � Ak | k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , s − 1}, each set Ak = � Ak 0, Ak 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ak s−1 � is composed of s arrays of size is l1 × l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set A is said to be 2D-CCC with parameters (s, s, l1, l2) if the following holds C � Ak, Ak′� (τ1, τ2) = s−1 � i=0 C � Ak i , Ak′ i � (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 sl1l2, (τ1, τ2) = (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, (τ1, τ2) ̸= (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (2) Definition 3 [10],[9] Let z1, z2, l1, l2 are positive integers and z1 ≤ l1, z2 ≤ l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the sets of ˆs set of arrays A = � Ak | k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ˆs − 1}, where each set Ak = � Ak 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ak s−1 � is composed of s arrays of size l1 × l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The set A is said to Springer Nature 2021 LATEX template 4 A Direct Construction of Optimal 2D-ZCACS be 2D − (ˆs, z1 × z2) − ZCACSl1×l2 s if the following holds C � Ak, Ak′� (τ1, τ2) = s−1 � i=0 C � Ak i , Ak′ i � (τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 sl1l2, (τ1, τ2) = (0, 0), k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, (τ1, τ2) ̸= (0, 0),|τ1| < z1,|τ2| < z2, k = k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 0, |τ1| < z1,|τ2| < z2, k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (3) When z1 = l1, z2 = l2, ˆs = s the 2D-ZCACS becomes 2D-CCC[19, 20] with parameter (s, l1, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' It should be noted that for l1 = 1, each array Ak i becomes l2-length sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore, 2D-ZCACS can be reduced to a conventional 1D-(ˆs, z2) − ZCCSl2 s [21], [22],[23], where, ˆs, s, z2, l2 represents no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' of set, set size, ZCZ width and sequence length respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 1 [9] For a 2D − (ˆs, z1 × z2) − ZCACSl1×l2 s , the following inequality holds ˆsz1z2 ≤ s (l1 + z1 − 1) (l2 + z2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (4) We called 2D-ZCACS is optimal if the following equality holds ˆs = s � l1 z1 �� l2 z2 � , (5) where ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='⌋ denotes the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='2 Multivariable Function Let a, b, mi, and nj be positive integers for 1 ≤ i ≤ a and 1 ≤ j ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let pi be any prime or 1, and qj be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' A multivariable function (MVF) can be defined as f : Am1 p1 × Am2 p2 × · · · × Ama pa × An1 q1 × An2 q2 × · · · × Anb qb → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let c, d ≥ 0 be integers such that 0 ≤ c < r and 0 ≤ d < s where r = pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma a and s = qn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then c and d can be written as c = c1 + c2pm1 1 + · · · + capm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma−1 a−1 , d = d1 + d2qn1 1 + · · · + dbqn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb−1 b−1 , (6) where, 0 ≤ ci < pmi i and 0 ≤ dj < qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let Ci = (ci,1, ci,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ci,mi) ∈ Ami pi , be the vector representation of ci with base pi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', ci = �mi k=1 ci,kpk−1 i and Dj = (dj,1, dj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , dj,nj) ∈ Anj qj be the vector representation of dj with base qj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', dj = �nj l=1 dj,lql−1 j where 0 ≤ ci,k < pi, and 0 ≤ dj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define vectors associated with c and d as φ(c) = (C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Ca) ∈ Am1 p1 × Am2 p2 × · · · × Ama pa , φ(d) = (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , Db) ∈ An1 q1 × An2 q2 × · · · × Anb qb , Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We also define an array associated with f as ψλ(f) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ωf0,0 λ ωf0,1 λ · · ωf0,r−1 λ ωf1,0 λ ωf1,1 λ · · ωf1,r−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' ωfs−1,0 λ ωfs−1,1 λ · · ωfs−1,r−1 λ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (7) where fc,d = f � φ(c), φ(d) � and λ is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 2 ([24]) Let t and t′ be two non-negative integers, where t ̸= t′, and p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then p−1 � j=0 ω(t−t′)j p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (8) Let us consider the set C as C = � Am1 p1 × Am2 p2 × · · · × Ama pa � × � An1 q1 × An2 q2 × · · · × Anb qb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (9) Let 0 ≤ γ < pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pma a and 0 ≤ µ < qn1 1 qn2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qnb b be positive integers such that γ = γ1 + a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8 , µ = µ1 + b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 q nj1 j1 \uf8f6 \uf8f8 , (10) where 0 ≤ γi < pmi i and 0 ≤ µj < qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let γi = (γi,1, γi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γi,mi) ∈ Ami pi be the vector representation of γi with base pi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γi = �mi k=1 γi,kpk−1 i , where 0 ≤ γi,k < pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similarly µj = (µj,1, µj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µj,nj) ∈ Anj qj be the vector representation of µj with base qj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', µj = �nj l=1 µj,lql−1 j where 0 ≤ µj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let φ(γ) = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa) ∈ Am1 p1 ×Am2 p2 × · · · × Ama pa , (11) be the vector associated with γ and φ(µ) = (µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) ∈ An1 q1×An2 q2 × · · · × Anb qb , (12) be the vector associated with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let πi and σj be any permutations of the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', mi} and {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', nj}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us also define the MVF Springer Nature 2021 LATEX template 6 A Direct Construction of Optimal 2D-ZCACS f : C → Z, as f(φ(γ), φ(µ)) = f (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) = a � i=1 λ pi mi−1 � e=1 γi,πi(e)γi,πi(e+1) + a � i=1 mi � e=1 di,eγi,e + b � j=1 λ qj nj−1 � o=1 µj,σj(o)µj,σj(o+1) + b � j=1 nj � o=1 cj,oµj,o, (13) where di,e, cj,o ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', λ − 1} and λ = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , pa, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , qb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us define the set Θ and T as Θ = {θ : θ = (r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ra, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , sb)}, T = {t : t = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , yb)}, where 0 ≤ ri, xi < pki i and 0 ≤ sj, yj < qrj j and ki, rj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Now, we define a function aθ t: C →Z, as aθ t � φ(γ), φ(µ) � = aθ t (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb) =f � φ(γ), φ(µ) � + a � i=1 λ pi γi,πi(1)ri + b � j=1 λ qj µj,σj(1)sj + a � i=1 λ pi γi,πi(mi)xi + b � j=1 λ qj µj,σj(nj)yj + dθ, (14) where 0 ≤ dθ < λ, γi,πi(1), γi,πi(mi) denote πi(1)−th and πi(mi)−th element of γi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Similarly, µj,σj(1), µj,σj(nj) denote σj(1)−th and σj(nj)−th element of µj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For simplicity, we denote aθ t � φ(γ), φ(µ) � by (aθ t)γ,µ and f � φ(γ), φ(µ) � by fγ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lemma 3 ([20]) We define the ordered set of arrays At = {ψλ � aθ t � : θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then the set {At : t ∈ T } forms a 2D-CCC with parameter (α, α, m, n), where, α = �a i=1 pki i �b j=1 qrj j , m = �a i=1 pmi i , n = �b j=1 qnj j and ki, mi, nj, rj are non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 7 3 Proposed construction of 2D-ZCACS Let a′, b′ be positive integers for 1 ≤ i′ ≤ a′ and 1 ≤ j′ ≤ b′, p′ i′ be any prime or 1, and q′ j′ be prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let γ′, µ′ are positive integers such that 0 ≤ γ′ < ��a i=1 pmi i � ��a′ i′=1 p′ i′ � and 0 ≤ µ′ < ��b j=1 qnj j � ��b′ j′=1 q′ j′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then γ′, µ′ can be written as γ′ =γ1+ a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edγ′ 1 + a′ � i′=2 γ′ i′ \uf8eb \uf8ed i′−1 � i1=1 p′ i1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 m, µ′ =µ1+ b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 q nj1 j1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edµ′ 1 + b′ � j′=2 µ′ j′ \uf8eb \uf8ed j′−1 � j1=1 q′ j1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 n, (15) where m = �a i=1 pmi i , n = �b j=1 qnj j , 0 ≤ γi < pmi i , 0 ≤ µj < qnj j , 0 ≤ γ′ i′ < p′ i′ and 0 ≤ µ′ j′ < q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We denote the vectors associated with γ′ and µ′ are φ(γ′) = � γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a � ∈ Am1 p1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ama pa × Ap′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ap′ a′ , φ(µ′) = � µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b � ∈ An1 q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Anb qb × Aq′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Aq′ b′ , (16) respectively, where γi ∈ Ami pi , µj ∈ Anj qj are the vectors associated with γi and µj respectively i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γi = (γi,1, γi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γi,mi) ∈ Ami pi , µj = (µj,1, µj,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µj,nj) ∈ Anj qj , γi = �mi k=1 γi,kpk−1 i , µj = �nj l=1 µi,lql−1 j , 0 ≤ γi,k < pi and 0 ≤ µj,l < qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us consider the set D as D = Am1 p1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Ama pa ×Ap′ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Ap′ a′ ×An1 q1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Anb qb ×Aq′ 1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='×Aq′ b′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (17) Let f be the function as defined (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define the MVF M c,d : D → Z as M c,d � φ(γ′), φ(µ′) � = M c,d � γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a′, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b′ � = δ λf (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb)+ a′ � i′=1 ci′ δ p′ i′ γ′ i′ + b′ � j′=1 dj′ δ q′ j′ µ′ j′, (18) where 0 ≤ ci′ < p′ i′, 0 ≤ dj′ < q′ j′, c = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ca′) and d = (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , db′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' For simplicity, now on-wards we denote M c,d(γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γa, γ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , γ′ a′, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µb, µ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , µ′ b′) by M c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the set Θ and T as Θ = {θ : θ = (r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ra, s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , sb)}, T = {t : t = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , yb)}, Springer Nature 2021 LATEX template 8 A Direct Construction of Optimal 2D-ZCACS where 0 ≤ ri, xi < pki i and 0 ≤ sj, yj < qrj j and ki, rj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let us define MVF, bθ,c,d t : D → Z, as bθ,c,d t =M c,d + a � i=1 δ pi γi,πi(1)ri + b � j=1 δ qj µj,σj(1)sj + a � i=1 δ pi γi,πi(mi)xi + b � j=1 δ qj µj,σj(nj)yj + δ λdθ, (19) where 0 ≤ dθ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' By (14), (18) and (19) we have bθ,c,d t = δ λaθ t + a′ � i′=1 ci′ δ p′ i′ γ′ i′ + b′ � j′=1 dj′ δ q′ j′ µ′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (20) We define the ordered set of arrays as Ωc,d t = {ψδ(bθ,c,d t ) : θ ∈ Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (21) where δ = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='m(λ, p′ 1, p′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , p′ a′, q′ 1, q′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , q′ b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Theorem 1 Let m = �a i=1 pmi i , n = �b j=1 qnj j , c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ca′), d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , db′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then the set S = {Ωc,d t : t ∈ T, 0 ≤ ci′ < p′ i′, 0 ≤ dj′ < q′ j′} forms a 2D − (α1, z1 × z2) − ZCACSl1×l2 α , where, α1 = ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � α, l1 = m ��a′ i′=1 p′ i′ � , l2 = n ��b′ j′=1 q′ j′ � , z1 = m ,z2 = n, α = (�a i=1 pki i )(�b j=1 qrj j ), ki, rj, mi, nj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Proof Let ˆγ, ˆµ are positive integers such that 0 ≤ ˆγ < l1 and 0 ≤ ˆµ < l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then ˆγ, ˆµ can be written as ˆγ = γ1+ a � i=2 γi \uf8eb \uf8ed i−1 � i1=1 p mi1 i1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8edγ′ 1 + a′ � i′=2 γ′ i′ \uf8eb \uf8ed i′−1 � i1=1 p′ i1 \uf8f6 \uf8f8 \uf8f6 \uf8f7 \uf8f8 m, ˆµ = µ1+ b � j=2 µj \uf8eb \uf8ed j−1 � j1=1 qnj1 j1 \uf8f6 \uf8f8+ \uf8eb \uf8ec \uf8ec \uf8edµ′ 1 + b′ � j′=2 µ′ j′ \uf8eb \uf8ec \uf8ed j′−1 � j1=1 q′ j1 \uf8f6 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f7 \uf8f8 n, where 0 ≤ γi < pmi i , 0 ≤ µj < qnj j , 0 ≤ γ′ i′ < p′ i′ and 0 ≤ µ′ j′ < q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proof will be split into following cases Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, τ2 = 0) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 9 The ACCF between Ωc,d t and Ωc′,d′ t′ at τ1 = 0 and τ2 = 0 can be expressed as C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = � θ∈Θ C(ψδ((bθ,c,d t )), ψδ((bθ,c′,d′ t′ )))(0, 0) = � θ∈Θ l1−1 � ˆγ=0 l2−1 � ˆµ=0 ω (bθ,c,d t )ˆγ,ˆ µ−(bθ,c′,d′ t′ )ˆγ,ˆ µ δ = � θ∈Θ m−1 � γ=0 n−1 � µ=0 p′ 1−1 � γ′ 1=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p′ a′ −1 � γ′ a′=0 q′ 1−1 � µ1=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' q′ b′ −1 � µ′ b′ =0 ωD δ , (22) where D = δ λ � (aθ t )γ,µ − (aθ t′)γ,µ � +�a′ i′=1 δ p′ i′ (ci′ −c′ i′)γi′ +�b′ j′=1 δ q′ j′ (dj′ −d′ j′)µj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' After splitting (22), we get C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = \uf8eb \uf8ed� θ∈Θ m−1 � γ=0 n−1 � µ=0 ω δ λ � (aθ t )γ,µ−(aθ t′ )γ,µ � δ \uf8f6 \uf8f8 EF = \uf8eb \uf8ed� θ∈Θ m−1 � γ=0 n−1 � µ=0 ω � (aθ t )γ,µ−(aθ t′ )γ,µ � λ \uf8f6 \uf8f8 EF = C(At, At′ )(0, 0)EF, (23) where E = a′ � i′=1 \uf8eb \uf8ec \uf8ed p′ i′ −1 � γ′ i′ =0 ω (ci′−c′ i′)γ′ i′ p′ i′ \uf8f6 \uf8f7 \uf8f8 , F = b′ � j′=1 \uf8eb \uf8ec \uf8ec \uf8ed q′ j′ −1 � µ′ j′ =0 ω (dj′ −d′ j′ )µ′ j′ q′ j′ \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (24) Subcase (i): (t ̸= t′) By lemma 2 we know, the set {At : t ∈ T } forms a 2D-CCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence By lemma 2, we have C(At, At′ )(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (25) Hence by (23) and (25) we have C(Ωc,d t , Ωc′,d′ t′ )(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (26) Subcase (ii): (t = t′) By lemma 2, we know C(At, At′ )(0, 0) = \uf8eb \uf8ed a � i=1 pmi+ki i \uf8f6 \uf8f8 \uf8eb \uf8ed b � j=1 qnj+rj j \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (27) Springer Nature 2021 LATEX template 10 A Direct Construction of Optimal 2D-ZCACS Let M = ��a i=1 pmi+ki i � ��b j=1 qnj+rj j � hence by Lemma 2, (23), (24), (27), we have the following C(Ωc,d t , Ωc′,d′ t )(0, 0) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 M ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � c = c′, d = d′ 0, c ̸= c′, d = d′ 0, c = c′, d ̸= d′ 0, c ̸= c′, d ̸= d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (28) Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i , 0 < τ2 < �b j=1 qnj j ) Let σ, ρ are positive integers such that 0 ≤ σ < m′ and 0 ≤ ρ < n′ where m′ = �a′ i′=1 p′ i′, n′ = �b′ j′=1 q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Then σ and ρ can be written as σ = σ1 + σ2p′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' + σa′ \uf8eb \uf8ed a′−1 � i′=1 p′ i′ \uf8f6 \uf8f8 , ρ = ρ1 + ρ2q′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' + ρb′ \uf8eb \uf8ed b′−1 � j′=1 q′ j′ \uf8f6 \uf8f8 , (29) respectively where 0 ≤ σi′ < p′ i′ and 0 ≤ ρj′ < q′ j′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define vectors associated with σ and ρ to be φ(σ) = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , σa′) ∈ Ap′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Ap′ a′ , φ(ρ) = (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' , ρb′) ∈ Aq′ 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' × Aq′ b′ , (30) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The ACCF between Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t and Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ for 0 < τ1 < �a i=1 pmi i and 0 < τ2 < �b j=1 qnj j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' can be derived as C(Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2) =C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)DE+C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1− a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)D′E+ C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2 − b � j=1 qnj j )DE′ + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2 − b � j=1 qnj j )D′E′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (31) where D = m′−1 � σ=0 \uf8eb \uf8ed a′ � i′=1 ω (ci′−c′ i′ )(σi′ ) p′ i′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (32) E = n′−1 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω (dj′ −d′ j′)(ρj′ ) q′ j′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (33) D′ = m′−2 � σ=0 \uf8eb \uf8ed a′ � i′=1 ω(ci′(σi′ )−c′ i′ (σ+1)i′) p′ i′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (34) E′ = n′−2 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω � dj′ (ρj′ )−d′ j′(ρ+1)j′ � q′ j′ \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (35) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 11 and (σ + 1)i′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (ρ + 1)j′ denotes the i′-th and j′-th components of φ (σ + 1) and φ (ρ + 1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' By Lemma 2, for 0 < τ1 < �a i=1 pmi i and 0 < τ2 < �b j=1 qnj j , we have C(At, At′ )(τ1, τ2) = 0, (36) C(At, At′ )(τ1− a � i=1 pmi i , τ2) = 0, (37) C(At, At′ )(τ1, τ2 − b � j=1 qnj j ) = 0, (38) C(At, At′ )(τ1 − a � i=1 pmi i , τ2 − b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (39) By (31), (36), (37), (38), (39) we have C(Ωc,d t , Ωc ′ ,d ′ t′ )(τ1, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (40) Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' − �b j=1 qnj j < τ2 < 0) The ACCF between Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t and Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ for 0 < τ1 < �a i=1 pmi i and − �b j=1 qnj j < τ2 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' can be derived as C(Ωc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Ωc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='d′ t′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2) =C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)DE+C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' τ2)D′E + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b � j=1 qnj j + τ2)DE′′ + C(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' At′ )(τ1 − a � i=1 pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b � j=1 qnj j + τ2)D′E′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (41) where E′′ = n′−2 � ρ=0 \uf8eb \uf8ed b′ � j′=1 ω � dj′ (ρ+1)j′ −d′ j′ (ρj′ ) � q′ j′ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (42) By Lemma 2, for 0 < τ1 < �a i=1 pmi i and − �b j=1 qnj j < τ2 < 0, we have C(At, At′ )(τ1, b � j=1 qnj j + τ2) = 0, (43) C(At, At′ )(τ1 − a � i=1 pmi i , b � j=1 qnj j + τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (44) By (41) , (43) and (44) we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (45) Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (0 < τ1 < �a i=1 pmi i , τ2 = 0) Springer Nature 2021 LATEX template 12 A Direct Construction of Optimal 2D-ZCACS The ACCF between Ωc,d t and Ωc′,d′ t′ for 0 < τ1 < �a i=1 pmi i and τ2 = 0 , can be derived as C(Ωc,d t , Ωc′,d′ t′ )(τ1, 0) =C(At, At′ )(τ1, 0)DE+ C(At, At′ )(τ1 − a � i=1 pmi i , 0)D′E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (46) By Lemma 2, for 0 < τ1 < �a i=1 pmi i , we have C(At, At′ )(τ1, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' C(At, At′ )(τ1 − a � i=1 pmi i , 0) = 0, (47) by (46) and (47) we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (48) Case 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, 0 < τ2 < �b j=1 qnj j ) The ACCF between Ωc,d t and Ωc′,d′ t′ for τ1 = 0 and 0 < τ2 < �b j=1 qnj j , can be derived as C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) =C(At, At′ )(0, τ2)DE+ C(At, At′ )(0, τ2 − b � j=1 qnj j )DE′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (49) By Lemma 2, for 0 < τ2 < �b j=1 qnj j , we have C(At, At′ )(0, τ2) = 0, C(At, At′ )(0, τ2 − b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (50) By (49) and (50) we have C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (51) Case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (τ1 = 0, − �b j=1 qnj j < τ2 < 0) Similarly the ACCF between Ωc,d t and Ωc′,d′ t′ for τ1 = 0 and − �b j=1 qnj j < τ2 < 0 is C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) =C(At, At′ )(0, τ2)DE+ C(At, At′ )(0, τ2 + b � j=1 qnj j )DE′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (52) By Lemma 2, for − �b j=1 qnj j < τ2 < 0, we have C(At, At′ )(0, τ2 + b � j=1 qnj j ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (53) Hence by (50), (52) and (53) we have C(Ωc,d t , Ωc′,d′ t′ )(0, τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (54) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 13 Combining all the cases we have C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 M ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � , (c, d, t) = (c′, d′, t′) (τ1, τ2) = (0, 0), 0, (c, d, t) ̸= (c′, d′, t′) (τ1, τ2) = (0, 0), 0, 0 ≤ τ1 < �a i=1 pmi i , (τ1, τ2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (55) Similarly it can be shown C(Ωc,d t , Ωc′,d′ t′ )(τ1, τ2) = 0, − a � i=1 pmi i < τ1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (56) Hence from (55), (56) we derive our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' □ Example 1 Suppose that a = 1, b = 1, a′ = 1, b′ = 1, p1 = 2, m1 = 2, k1 = 1, q1 = 3, n1 = 2, r1 = 1, p′ 1 = 3, q′ 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Let δ = 6, λ = 6, γ1 = (γ11, γ12) ∈ A2 2 = {0, 1}2 be the vector associated with γ1 where 0 ≤ γ1 ≤ 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', γ1 = γ11 + 2γ12 and µ1 = (µ11, µ12) ∈ A2 3 = {0, 1, 2}2 be the vector associated with µ1 where 0 ≤ µ1 ≤ 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', µ1 = µ11+3µ12 and 0 ≤ γ′ 1 ≤ 2, 0 ≤ µ′ 1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We define the MVF f : A2 2×A2 3 → Z as f (γ1, µ1)=3γ1,2γ1,1+γ1,1+2γ1,2+2µ1,2µ1,1+2µ1,1+µ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Consider the MVF, Mc,d : A2 2 × A3 × A2 3 × A2 → Z as Mc,d � γ1, γ′ 1, µ1, µ′ 1 � = f(γ1, µ1) + 2c1γ′ 1 + 3d1µ′ 1 = 3γ1,2γ1,1 + γ1,1 + 2γ1,2 + 2µ1,2µ1,1 + 2µ1,1 + µ1,2 + 2c1γ′ 1 + 3d1µ′ 1, (57) where 0 ≤ c1 < p′ 1 = 2, 0 ≤ d1 < q′ 1 = 3, c = c1 ∈ {0, 1}, and d = d1 ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' We have Θ = {θ : θ = (r1, s1) : 0 ≤ r1 ≤ 1, 0 ≤ s1 ≤ 2}, T = {t : t = (x1, y1) : 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (58) Let dθ = 0, now from (19) we have bθ,c,d t = Mc,d + 3γ1,2r1 + 2µ1,2s1 + 3γ1,1x1 + 2µ1,2y1, (59) and Ωc,d t = � ψ6(bθ,c,d t ) : θ = (r1, s1) ∈ {0, 1} × {0, 1, 2} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' (60) Therefore, the set S = {Ωc,d t : t ∈ T, 0 ≤ c1 ≤ 1, 0 ≤ d1 ≤ 2}, (61) forms an optimal 2D − (36, 4 × 9) − ZCACS12×18 6 over Z6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 A Direct Construction of Optimal 2D-ZCACS Table 1 Comparison with Previous Works Source No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' of set Array Size Condition Based on [7] K = K′r L′ 1×(L′ 2 + r + 1) r ≥ 0 2D − ZCACS of set size K′ and array size L′ 1×L′ 2 [8] 1 2m × 2nL m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' n ≥ 0 ZCP of length L [9] K K × K K divides set size BH matrices [10] 2 �ki i=1 p2 i 2m × �ki i=1 pmi i ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' mi ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pi’s are prime MVF Thm 2 rsα rm × sn α = (�a i=1 pki i )(�b j=1 q rj j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' m=�a i=1pmi i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' n=�b j=1q nj j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' α ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' qjareprimes MVF Remark 1 In Theorem 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' if we take a = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' a′ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' p′ 1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' q1 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' b′ = l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' r1 ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' we have optimal 1D-ZCCS with parameter (�l i=1 q′ i2r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2n1) − ZCCS �l i=1 q′ i2n1 2r1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' which is exactly the same result as in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Also if we take l = 1, then we have optimal 1D-ZCCS of the form (q′ 12r1, 2n1) − ZCCSq′ 12n1 2r1 , which is exactly the same result in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the optimal 1D-ZCCS given by [17, 18] appears as a special case of the proposed construction Remark 2 In Theorem 1, if a = 1, p1 = 1, a′ = 1, p′ 1 = 1, b = 1, q1 = 2, b′ = l, r1 = 1, we have 1d-ZCCS with parameter (2 �l i=1 q′ i, 2n1) − ZCCS �l i=1 q′ i2n1 2 , which is just a collection of 2 �l i=1 q′ i ZCPs with sequence length �l i=1 q′ i2n1 and ZCZ width 2n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence our work produces collections of ZCPs[15] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 3 In Theorem 1, if we take a = 1, p1 = 1, a′ = 1, p′ 1 = 1, b = 1, q1 = 2, b′ = r, q′ 1 = q′ 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' = q′r = 2, n1 = m − r and r1 = s + 1 then we have 1D-ZCCS with parameter (2s+r+1, 2m−r)−ZCCS2m 2s+1, which is exactly the same result in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence, the ZCCS in [16] appears as a special case of our proposed construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 4 The 2D-ZCACS given by the proposed construction satisfies the equality given in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore the 2D-ZCACS obtained by the proposed construction is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Remark 5 If we take a = 1, a′ = 1, p1 = 1 and p′ 1 = 1, in Theorem 1, we have optimal 1D-ZCCS with parameter ���b′ j′=1 q′ j′ � �b j=1 qrj j , n � − ZCCS n ��b′ j′=1 q′ j′ � �b j=1 q rj j where, n = �b j=1 qnj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Hence, we have optimal 1D-ZCCS of length nm where, n, m > 1 and m = �b′ j′=1 q′ j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore our construction produces optimal 1D-ZCCS with a new length which is not present in the literature by direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 15 Remark 6 The set size of our proposed 2D-ZCACS is ��a′ i′=1 p′ i′ � ��b′ j′=1 q′ j′ � �a i=1 pki i �b j=1 qrj j where, ki, tj ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' If we take a = 1, p1 = 1, a′ = 1, p′ 1 = 1, r1 = r2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' = rb = 2, b′ = 1, and q′ 1 = 2 then we have set size 2 �b j=1 q2 j which is the set size of the 2D-ZCACS in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Therefore, we have flexible number of set sizes compared to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='1 Comparison with Previous Works Table I compares the proposed work with indirect constructions from [7–9] and direct construction from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The constructions in [7–9] heavily rely on initial sequences, increasing hardware storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The construction in [10] is direct, but set size and array sizes are limited to some even numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our construction doesn’t require initial matrices or sequences and produces flexible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 4 Conclusion In this paper, 2D-ZCACSs are designed by using MVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' The proposed design does not depend on initial sequences or matrices, so it is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Our proposed design produces flexible array size and set size compared to existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Also, our proposed construction can be reduced to 1D-ZCCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' As a result, many 1D-ZCCSs become special cases of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Finally, we compare our work to the existing state-of-the-art and show that it’s more versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' References [1] Farkas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Turcs´any, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Two-dimensional orthogonal complete com- plementary codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In: Joint IEEE 1st Workshop on Mobile Future and Symposium on Trends in Communications (sympoTIC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 21–24 (2003) [2] Turcs´any, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Farkaˇs, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': New 2d-mc-ds-ss-cdma techniques based on two- dimensional orthogonal complete complementary codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' in Multi-Carrier Spread-Spectrum, Berlin, Germany: Springer, 49–56 (2004) [3] Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Chao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Complete complementary codes and generalized reed-muller codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 12(11), 849–851 (2008) [4] Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': A novel class of complete complementary codes and their applications for apu matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 25(9), 1300–1304 (2018) [5] Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Guan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Parampalli, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': New complete complementary codes for peak-to-mean power control in multi-carrier cdma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 62(3), 1105–1113 (2014) Springer Nature 2021 LATEX template 16 A Direct Construction of Optimal 2D-ZCACS [6] Xeng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Ge, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Theoretical limit on two dimensional gener- alized complementary orthogonal sequence set with zero correlation zone in ultra wideband communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' International Workshop on UWBST & IWUWBS, 197–201 (2004) [7] Zeng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Ge, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Construction of two-dimensional comple- mentary orthogonal sequences with ZCZ and their lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IET (2005) [8] Pai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Ni, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : Two-dimensional binary Z- complementary array pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Theory 67(6), 3892–3904 (2021) [9] Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Two-dimensional Z-complementary array code sets based on matrices of generating polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 68, 5519–5532 (2020) [10] Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Construction of inter-group complementary code set and 2D Z-complementary array code set based on multivariable functions (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='00970 [11] Shen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Meng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': New constructions of z- complementary code sets and mutually orthogonal complementary sequence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Codes Cryptogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', 1–19 (2022) [12] Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Construction of z-complementary code sets with non-power-of-two lengths based on generalized boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 24(8), 1607–1611 (2020) [13] Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': A direct construction of optimal zccs with maximum column sequence pmepr two for mc-cdma system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett 25(2), 337–341 (2020) [14] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', S¸ahin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : Z-complementary code sets with flexible lengths from generalized boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Access 9, 4642–4652 (2020) [15] Kumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Paul, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': A direct construction of even length zcps with large zcz ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Cryptogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', 1–10 (2022) [16] Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Optimal Z-complementary code set from generalized Reed-Muller codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 67(3), 1783–1796 (2018) [17] Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Pseudo-Boolean functions for optimal Z- complementary code sets with flexible lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 28, 1350–1354 (2021) Springer Nature 2021 LATEX template A Direct Construction of Optimal 2D-ZCACS 17 [18] Ghosh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Sarkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Upadhaya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : Direct construction of optimal Z-complementary code sets with even lengths by using generalized boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 29, 872–876 (2022) [19] Pai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : Design- ing two-dimensional complete complementary codes for omnidirectional transmission in massive mimo systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' In: International Symposium on Information Theory (ISIT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 2285–2290 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE [20] Ghosh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Majhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Upadhyay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : A direct construction of 2D-CCC with arbitrary array size and flexible set size using multivariable function (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='13395 [21] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' : Optimal Z-complementary sequence sets with good peak-to-average power-ratio property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 25(10), 1500–1504 (2018) [22] Yu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Adhikary, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': New class of optimal Z- complementary code sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 29, 1477–1481 (2022) [23] Shen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Fan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=', Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': New z- complementary/complementary sequence sets with non-power-of-two length and low papr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Cryptogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 14(4), 817–832 (2022) [24] Vaidyanathan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=': Ramanujan sums in the context of signal process- ing—part i: Fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} +page_content=' 62(16), 4145– 4157 (2014)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE0T4oBgHgl3EQffQAA/content/2301.02400v1.pdf'} diff --git a/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf b/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0434209df212839d43523518b39dafdd849165c0 --- /dev/null +++ b/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:25195e416ca02bf6e54fdf64a8da957ecf23ce821b1d3c4b7ec9d4960e77e789 +size 193776 diff --git a/7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss b/7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..594bf03b7c857dc3c51a42a8c54e0a44e51132ac --- /dev/null +++ b/7NE1T4oBgHgl3EQfTgOa/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9677a5744679db229a3c6f6e8feb3484fd0d953696f509d7c23fc33eb302c6a9 +size 1769517 diff --git a/7NE1T4oBgHgl3EQfTgOa/vector_store/index.pkl b/7NE1T4oBgHgl3EQfTgOa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..30b37e4760837239eb6678fecfbe80610789ff97 --- /dev/null +++ b/7NE1T4oBgHgl3EQfTgOa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3c99fa220eec1b6ec02f5a6f546cc39307755d2479d1ea8fe19b63440cb1d04f +size 73770 diff --git a/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf b/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2a21c78ea76bd7121a2fc88d3ca07482c753b3ff --- /dev/null +++ b/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e4bbf14b29cb6c69027850cd8177ed73b3c75f57cde996b953503a483a34d42 +size 1038570 diff --git a/7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss b/7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e4531ef393b9576a767e085402aa2a7950d02a91 --- /dev/null +++ b/7dAyT4oBgHgl3EQfcvf_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:46aaedc39643a3935c0acbe5ec93bc600f5303db66a769bb42254451f51c58a5 +size 3538989 diff --git a/7dAyT4oBgHgl3EQfcvf_/vector_store/index.pkl b/7dAyT4oBgHgl3EQfcvf_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..acf2236a0204fa12855c71172d7eb5056032136a --- /dev/null +++ b/7dAyT4oBgHgl3EQfcvf_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4e5caf710b94e7e717b4d6a7ef39d03cc389ec24d3473e774f6e305890a3f305 +size 154934 diff --git a/7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss b/7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4ef8fc41b9752512361beec7acb0bca616029ff5 --- /dev/null +++ b/7tE2T4oBgHgl3EQf7whk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1840ff0cf247ec59d7b8a13afe8394a523525caab92bb4304d62c7581ae87041 +size 2031661 diff --git a/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf b/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..828be9fef0af0d223401661d11d01745b72ee489 --- /dev/null +++ b/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fc3143438268239655c5c080276647dbb1a530bfdc8730e9e5edf5cacec39a79 +size 2542622 diff --git a/8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss b/8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1aab1c2c1b775e94834f7fc1ed83b80be8aa232f --- /dev/null +++ b/8tFST4oBgHgl3EQfaTh3/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7b08492e11b4988f3eb45ab26329827f09ed99bf887748c0ab40c2635f76a430 +size 3866669 diff --git a/8tFST4oBgHgl3EQfaTh3/vector_store/index.pkl b/8tFST4oBgHgl3EQfaTh3/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2cff232babec18607f05688f2941ab8a8749beb6 --- /dev/null +++ b/8tFST4oBgHgl3EQfaTh3/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:751a8fe33ac92cc1ab56bb7843b2a31f7164559bf9124ce78d1f5918fe2406a7 +size 135947 diff --git a/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf b/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..262e38ca7d4c95cf772b679bb7126a98b70cec53 --- /dev/null +++ b/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:49e50df0b4cfc15f7f2a7745aed106a68343f2dbab2e2485e072c148e3f4cfdd +size 1761809 diff --git a/99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss b/99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6b4ca653f9067122abd97984bc35a2e6515c7d5b --- /dev/null +++ b/99AyT4oBgHgl3EQfqfgS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:85b57d5419628f66ceb2db578324133dfd812826880e670d17e32ee003142c29 +size 4587565 diff --git a/99AyT4oBgHgl3EQfqfgS/vector_store/index.pkl b/99AyT4oBgHgl3EQfqfgS/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9070df841f18efd89604abb86f1d2630c2ca2461 --- /dev/null +++ b/99AyT4oBgHgl3EQfqfgS/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76221a79ea8ddbc14c684225b2459b6ef2f3a01e8b28edbb32c8e86d5d7bf970 +size 169574 diff --git a/9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf b/9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..52ac95bd1a16440da45e0da7980641912aa1f621 --- /dev/null +++ b/9NAyT4oBgHgl3EQfQ_Zv/content/2301.00057v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9a52f23f1d936ba35c19df5fe1ad3d8d6be76eba9afcd095abf52bafbef53407 +size 476678 diff --git a/9NAyT4oBgHgl3EQfQ_Zv/vector_store/index.pkl b/9NAyT4oBgHgl3EQfQ_Zv/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..bc7bd248b43c76e1294c019e207b4d6bffc1ef51 --- /dev/null +++ b/9NAyT4oBgHgl3EQfQ_Zv/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f2ff7b4e5eb4686090a6d867f5aea9ac39ba1d0e695b1183f8a53ddf81717724 +size 140386 diff --git a/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf b/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d9b090dcab520f5fb1bd7d6b2a1f20f48d9f252c --- /dev/null +++ b/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:68db25aff4d2427f42d5d67b4d172d2a0621d24e2c1e11361ea801facdeff276 +size 208947 diff --git a/9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss b/9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..40360742a924622db548ebd4f9dbfd989328c2b0 --- /dev/null +++ b/9dFQT4oBgHgl3EQf5zZD/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ba63630d9875ff436065035bfd43fa331c51e276671a2455fbf1420e9ccbb197 +size 3211309 diff --git a/9dFQT4oBgHgl3EQf5zZD/vector_store/index.pkl b/9dFQT4oBgHgl3EQf5zZD/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8c179ca446c9bc93b5a1d804630b7db409996ce0 --- /dev/null +++ b/9dFQT4oBgHgl3EQf5zZD/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c61314c21670e5f567e50cc19fe70e26f10349a17d3fadb8001bb49af7b6dd47 +size 119903 diff --git a/BNE2T4oBgHgl3EQf8gli/content/tmp_files/2301.04219v1.pdf.txt b/BNE2T4oBgHgl3EQf8gli/content/tmp_files/2301.04219v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45c26787a874ad0bc3b6d19d69ae3cb9f4be6984 --- /dev/null +++ b/BNE2T4oBgHgl3EQf8gli/content/tmp_files/2301.04219v1.pdf.txt @@ -0,0 +1,741 @@ +arXiv:2301.04219v1 [math.CO] 10 Jan 2023 +EXTENSIONS OF A FAMILY FOR SUNFLOWERS +JUNICHIRO FUKUYAMA +Abstract. This paper refines the original construction of the recent proof +of the sunflower conjecture to prove the same general bound [ck log(k + 1)]m +on the cardinality of a family of m-cardinality sets without a sunflower of k +elements. Our proof uses a structural claim on an extension of a family that +has been previously developed. +1. Motivation, Terminology and Related Facts +The sunflower conjecture states that a family F of sets each of cardinality at +most m includes a k-sunflower if |F| > cm +k for some ck ∈ R>0 depending only on +k, where k-sunflower stands for a family of k different sets with common pair-wise +intersections. It had been open since the sunflower lemma was presented in 1960 +[1], until it was recently proven [2] with the following statement confirmed. +Theorem 1.1. There exists c ∈ R>0 such that for every k, m ∈ Z>0, a family F of +sets each of cardinality at most m includes a k-sunflower if |F| > [ck log(k + 1)]m. +□ +The base of the obtained bound [ck log(k + 1)]m is asymptotically close to the +lower bound k − 1. Our investigation on finding such a near-optimal bound had +continued from the previous work [3]. The paper attempts to explore some combi- +natorial structure involving sunflowers, to prove that a uniform family F includes +three mutually disjoint sets, not just a 3-sunflower, if it satisfies the Γ +� +cm +1 +2 +ǫ� +- +condition for any given ǫ ∈ (0, 1) and c depending on ǫ only. Here the Γ (b)-condition +of F (b ∈ R>0) means |{U : U ∈ F, S ⊂ U}| < b−|S||F| for all nonempty sets S. +The original construction [4] of the work [2] proves the most noted three-petal +case of the conjecture, referring to Theorem 1.2 given below that derives the exten- +sion generator theorem presented in [3]. The goal of this paper is to further refine1 +the original construction to prove the same [ck log(k + 1)]m bound. We will find +such proof at the end of the next section. +The rest of this section describes the similar terminology and related facts. De- +note the universal set by X, its cardinality by n, and a sufficiently small positive +2010 Mathematics Subject Classification. 05D05: Extremal Set Theory (Primary). +Key words and phrases. sunflower lemma, sunflower conjecture, ∆-system. +1Extra information on this paper and the references [2, 3, 4] is available at Penn State Sites. +Web address: https://sites.psu.edu/sunflowerconjecture/2023/01/10/index-page/ +1 + +2 +JUNICHIRO FUKUYAMA +number depending on no other variables by ǫ ∈ (0, 1). In addition, +i, j, m, p, r ∈ Z≥0, +[b] = [1, b] ∩ Z, +F ⊂ 2X, +�X′ +m +� += {U : U ⊂ X′, |U| = m} , +for X′ ⊂ X, +and +F[S] = {U : U ∈ F, S ⊂ U} , +for S ⊂ X, +A set means a subset of X, and one in +�X +m +� +is called m-set. +Weight X by some w : 2X → R≥0, which induces the norm ∥ · ∥ of a family +defined by ∥F∥ = � +U∈F w(U) for any F. Denote set/family subtraction by −, +while we use the symbol \ for the different notion described below. +Use ← to +express substitution into a variable. For simplicity, a real interval may denote the +integral interval of the same range, e.g., use (1, t] instead of (1, t] ∩ Z if it is clear +by context. Obvious floor/ceiling functions will be ommited throughout the paper. +Now let F be a family of m-sets, i.e., F ⊂ +�X +m +� +. We say +κ (F) = +�n +m +� +− ln |F|. +is the sparsity of F. The family satisfies the Γ (b)-condition on ∥ · ∥ (b ∈ R>0) if +∥G∥ = ∥G ∩ F∥, +for all G ⊂ 2X, +and +∥F[S]∥ < b−|S|∥F∥, +for every nonempty set S ⊂ X. +As used above, the norm ∥ · ∥ can be omitted if it is induced by the unit weight, +i.e., +w : V �→ +� 1, +if V ∈ F, +0, +otherwise. +The following theorem is proven2 in [3]. +Theorem 1.2. Let X be weighted to induce the norm ∥ · ∥. For every sufficiently +small ǫ ∈ (0, 1), and F ⊂ +�X +m +� +satisfying the Γ +� 4γn +l +� +-condition on ∥ · ∥ for some +l ∈ [n], m ∈ [l], and γ ∈ +� +ǫ−2, lm−1� +, there are +��n +l +� +(1 − ǫ) +� +sets Y ∈ +�X +l +� +such +that +� +1 − +� 2 +ǫγ +� � l +m +� +� n +m +� ∥F∥ < +���� +�Y +m +����� < +� +1 + +� 2 +ǫγ +� � l +m +� +� n +m +� ∥F∥ . +□ +With Theorem 1.2, we can prove the aforementioned extention generator theorem +that is about the l-extension of F, i.e., +Ext (F, l) = +� +T : T ∈ +�X +l +� +, and ∃U ∈ F, U ⊂ T +� +, +for l ∈ [n] − [m]. +It is not difficult to see +(1.1) +κ [Ext (F, l)] ≤ κ (F) , +as in [3], where it is also shown: +Lemma 1.3. For F ⊂ +�X +m +� +such that m ≤ n/2, +κ +�� X +2m +� +− Ext (F, 2m) +� +≥ 2κ +��X +m +� +− F +� +. +□ +2In [3], the theorem uses so called Γ2 (b, 1)-condition on ∥ · ∥. It is straightforward to check it +means the Γ (b)-condition on ∥ · ∥ here. + +EXTENSIONS OF A FAMILY FOR SUNFLOWERS +3 +Further denote +Gp = G × G × · · · × G +� +�� +� +p +, +X = (X1, X2, . . . , Xp) ∈ +� +2X�p , +Rank (X) = p, +and +Union (X) = +p� +j=1 +Xj. +Suppose m divides n and p = m. If Union (X) = X, and all Xi are mutually +disjoint n/m-sets, then such an X is an m-split of X with Xi called strips. Its +subsplit X′ of rank r ∈ [m], or r-subsplit of X, is the tuple of some r strips of X +preserving the order. +A set S is on X′ if S ⊂ Union +� +X′� +, and |Xi ∩ S| ∈ {0, 1} for every strip Xi of +X′. Denote +- by 2X′ the family of all sets on X′, +- by +�X′ +p +� +the family of p-sets on X′, +- by X \ X′ the subsplit of rank m − m′ consisting of the strips in X but not in +X′, +- and by X′ \ B for a set B, abusing the symbol \, the subsplit of X consisting of +the strips each disjoint with B. +For notational convenience, allow Rank +� +X′� += 0 for which X′ = (∅), Union +� +X′� += +∅, and +�X′ +p +� += {∅}. We have: +Lemma 1.4. For any nonempty family F ⊂ +�X +m +� +such that m divides n = |X|, +there exists an m-split X of X such that +����F ∩ +�X +m +����� ≥ +� n +m +�m |F| +� n +m +� > |F| exp (−m) . +□ +The lemma proven in [2] poses a special case of the general statement presented in +[3]. +2. Proof of Theorem 1.1 +We prove Theorem 1.1 with Theorem 1.2 in the two subsections below. Given F +and k, we will find a subfamily ˆF ⊂ F with a property that implies the existence +of a k-sunflower in itself. +2.1. Formulation and Construction. Letting +h = exp +�1 +ǫ +� +, +and +c = exp (h) , +assume WLOG that +- k ≥ 3, +- n = |X| is larger than ckm and divisible by m. Otherwise add some extra elements +to X. +- F ⊂ +�X +m +� +for an m-split X of X by Lemma 1.4, satisfying the Γ (cck ln k)-condition +and |F| > (cck ln k)m. +- |F| < (km)m and m > cc ln k, otherwise F includes a k-sunflower by the sunflower +lemma. + +4 +JUNICHIRO FUKUYAMA +FindCores +Input: +i) the family F ⊂ +�X +m +� +. +Outputs: +i) C ⊂ +�X +r0 +� +for some r0 ∈ [0, m]. +ii) ˆF ⊂ � +C∈C F[C] such that | ˆF| ≥ 3−m−1|F|. +1. F′ ← F; +ˆF ← ∅; +C ← ∅; +2. for r = m down to 0 do: +2-1. repeat: +a) find an r-set C such that |F′[C]| ≥ f(r) putting TC ← F′[C]; +b) if found then: +F′ ← F′ − TC; +ˆF ← ˆF ∪ TC; +C ← C ∪ {C}; +else exit Loop 2-1; +2-2. if | ˆF| ≥ 3−m+r−1|F| then return +� +r, C, ˆF +� +; +Figure 1. Algorithm FindCores +Let +i ∈ [k], +r ∈ [0, m], +b = ck ln k, +δ = +ǫ +k ln k, +F′, Fi ⊂ F, +C ∈ 2X, +and +Yi ∈ 2X. +A tuple Z = (C, Y1; F1, Y2; F2, · · · , Yk; Fk) is said to be a partial sunflower of +rank r over F′ if there exists an r-subsplit X∗ of X satisfying the four conditions: +Z-i) C ∈ �m/c +u=0 +�X∗ +u +� +. +Z-ii) Yi are mutually disjoint k subsets of Union (X∗ \ C) such that +|Yi ∩ X†| = δ|X†| for each strip X† of X∗ \ C. +Z-iii) The k families Fi are each nonempty included in +F′[C] ∩ +�X − Union (X∗ \ C) ∪ Yi +m +� +, +and are identical if Rank (X∗ \ C) = 0. +Z-iv) |Fi| < 2|Fi′| for i ∈ [k] and i′ ∈ [k] − {i}. +We say that such an Fi occurs on Z and in Z with the core C. Also Z and +Fi are on X∗. A family Z of Z on one or more X∗ is a partial sunflower family +(PSF) of rank r over F′, if each two Fi occurring on two different Z are mutually +disjoint, i.e., the universal disjoint property of Z is met. Denote +F (Z) := +� +Z∈Z +i∈[k] +Fi of Z, +for any PSF Z abusing the symbol F. +In the rest of our proof, we construct a nonempty PSF of rank m over F. This +means a k-sunflower in F proving Theorem 1.1. +With +f : Z≥0 → R≥0, +x �→ ǫ3m(chk)−x +k +|F|, + +EXTENSIONS OF A FAMILY FOR SUNFLOWERS +5 +obtain the families C and ˆF by the algorithm FindCores described in Fig. 1. It +is straightforward to see that the two outputs correctly satisfy the properties i)-ii). +In addition: +A) | ˆF[U]| < f(|U|) for all U ∈ �m +r′=r0+1 +�X +r′ +� +. +B) We will construct partial sunflowers over ˆF with cores C in C. The families TC +Step 2-1 finds for r = r0 are mutually disjoint each with |TC| ≥ f(r0). By the +Γ (cck ln k)-condition of F and cc ln k < m, +k−1ǫ3m(chk ln k)−|C||F| = f (r0) ≤ |TC| ≤ |F[C]| < (cck ln k)−|C||F|, +⇒ +r0 = |C| < ln k − 3m ln ǫ +(c − h) ln c +< m +2c +�ln k +m + 1 +� +< m +c . +Define a statement on the obtained objects. +Proposition Πr for r ∈ [r0, m]: there exists a PSF Z of rank r over ˆF such that +|F(Z)| > ǫ2r−2r0| ˆF|. +□ +Such a Z is said to be r-normal. By definition, Z is the union of PSFs ZX∗ on +r-subsplits X∗ satisfying the universal disjoint property. +Our final goal of finding a nonempty PSF of rank m over F would be met if Πm. +The proposition Πr0 holds since +Z = + + + + +C, ∅; TC, ∅; TC, · · · , ∅; TC +� +�� +� +k + + : C ∈ ˆC + + + +is an r0-normal PSF such that F (Z) = ˆF by B) where TC are the ones mentioned +there. So it suffices to show +(2.1) +Πr ⇒ Πr+1, +for every r ∈ [r0, m), +to have proof of a k-sunflower in F. +2.2. Proof of (2.1). We start showing (2.1) as the only remaining task. Assume +Πr for a particular r ∈ [r0, m), so we have an r-normal PSF Z that is the union of +ZX∗ on some r-subsplits X∗ by definition. We confirm Πr+1 in four steps. +Step 1. Reconstruct Z into another PSF Z′. Obtain such a Z′ by the algorithm +Reconstruct described in Fig. 2. It is a PSF of rank r over F (Z) satisfying the +two conditions: +C) |F (Z′) | > 2−1ǫ|F (Z) |. +D) For each Z ∈ Z′ on an r-subsplit X∗, there exists an r+1-subsplit X′ containing +X∗ such that each Fi on Z meets +|Fi[S]| < 1 +b |Fi|, +∀S ∈ +�X′ \ X∗ +1 +� +. +□ +We see their truth by the notes below. Such a Z ∈ Z′ is said to be on the split +pair +� +X∗, X′� +. In Steps 2 and 3, we will construct our desired r + 1-normal PSF +Z′′ from Z′ confirming Πr+1. +Justification of Z′ being a PSF with C) and D). +- F (ZX′) of an X′ disregarded by Step 2-3 is negligible as their union will be +smaller than 2−m� m +r+1 +� +< +� 3 +2 +�−m < +� 3 +2 +�−cc ln k < ǫ3/k of F (Z). + +6 +JUNICHIRO FUKUYAMA +Reconstruct +Input: +an r-normal PSF Z for some r ∈ [r0, m). +Output: +a PSF Z′ of rank r over F (Z) satisfying C) and D). +1. X ← the family of all r-subsplits X∗; +/* The given Z is the union of PSFs ZX∗ on some X∗ by definition. */ +2. for each r + 1-subsplit X′ do: +2-1. X∗ ← the family of all X∗ ∈ X that are r-subsplits of X′; +2-2. ZX′ ← � +X∗∈X∗ ZX∗; +2-3. if |F (ZX′) | < 3−m|F (Z) | then go to Step 2 for the next X′ else X ← X − X∗; +2-4. for each Fi in ZX′ do F′ +i ← Fi; +2-5. for each Fi in ZX′ and on X∗, and sets B ∈ +�X∗ +r +� +and U ∈ +� X′ +r+1 +� +[B] such that +|Fi[U]| > +� +c +√ +hk ln k +�−r−1 +|Fi[B]| do F′ +i ← F′ +i − Fi[U]; +2-6. for each Z ∈ ZX′ do: +a) if |F′ +i| < ǫ|Fi| for some Fi on Z then delete Z from ZX′; +b) else normalize Z for the condition Z-iv) as follows: +b)-i) for each Fi on Z do: +γi ← |F′ +i|−1 mini′∈[k] |F′ +i′|; +F′ +i ← any subfamily of F′ +i of cardinality min (|F′ +i|, ⌊2γi|F′ +i|⌋); +b)-ii) replace all Fi by the F′ +i to reconstruct Z; +3. return the union of all ZX′ found in Loop 2 as Z′; +Figure 2. Algorithm Reconstruct +- |F(ZX′)[U]| ≤ | ˆF[U]| < f (r + 1) for all U ∈ +� X +r+1 +� +by A). So, +|F (ZX′) [U]| +|F (ZX′) | +< +f (r + 1) +3−mǫ2r−2r0| ˆF| +< +� +chk ln k +�−r−1 +k +, +before Step 2-4. +- Fi[B] are mutually disjoint for all different Fi occurring in ZX′ each on an X∗, +and B ∈ +�X∗ +r +� +right before Step 2-5, by the universal disjoint property of Z. (By +the rule Z-iii), Fi on a single Z are identified when r = r0.) In addition, Fi[U] +are mutually disjoint for all Fi and U ∈ +� X′ +r+1 +� +meeting +� +X∗∈X∗, Fi in ZX′ and on X∗ +B∈(X∗ +r ), U∈(X′ +m′)[B] +Fi[U] = F(ZX′). +- By the above two, Step 2-5 may reduce +V = +� +Fi in ZX′ +F′ +i +by less than its ǫ3/k, leaving only Fi, B, and U such that +|F′ +i[U]| < (cb)−r−1|F′ +i[B]|, +⇒ +|F′ +i[B]| > (cb)r+1. +- By Z-iv) of Z, Step 2-6-a) may only reduce less than 2ǫ3 of V. +- The process of normalization is well-defined by Step 2-6-b) due to |F′ +i| > (cb)r+1 +before it. It could further reduce V into its ǫ/2 or larger. + +EXTENSIONS OF A FAMILY FOR SUNFLOWERS +7 +- The condition Z-iv) of the obtained Z′ follows the above as well as the two +properties C) and D). +Step 2. For each Fi occurring in Z′, construct a family Yi of Y ∈ +� +X† +δ|X†| +� +such +that Fi ∩ +�X−X†∪Y +m +� +is sufficiently large, where X† = Union(X′ \ X∗). Consider +each Z ∈ Z′ on (X∗, X′) with the unique strip X† of X′ \ X∗, and an Fi on Z. +Weight X† by 2X† → Z≥0, W �→ |Fi[W]| inducing the norm ∥ · ∥ as in Section 1. +The family H = +�X† +1 +� +satisfies the Γ(b)-condition on ∥ ·∥ by D). Apply Theorem 1.2 +to H. There exists Yi ⊂ +� X† +δ|X†| +� +such that +|Yi| > +� |X†| +δ|X†| +� +[1 − exp (−h)] , +(2.2) +δ [1 − exp (−h)] |Fi| < +��FY +i +�� < δ [1 + exp (−h)] |Fi|, +for every Y ∈ Yi, +where +FY +i ⊂ Fi ∩ +�X − X† ∪ Y +m +� +. +Step 3. Find Z′′ by (2.2). Now consider the same Z with the k families Fi and +sets Yi. Put +δ′ = 2δ ln k, +and +Y′ +i = Ext (Yi, δ′|X†|) , +for each Fi to see +|Y′ +i| > +� |X†| +δ′|X†| +� +[1 − exp (−h ln k)] > +� |X†| +δ′|X†| +� � +1 − ǫ +k +� +, +by Lemma 1.3, (1.1) and (2.2): to the Yi, repeatedly apply the lemma ⌈log2 ln k⌉ +times doubling the second parameter of Ext. Then κ +�� +X† +δ′|X†| +� +− Y′ +i +� +> h ln k. +Hence, there exist more than +� |X†| +δ′|X†| +��|X†| − δ′|X†| +δ′|X†| +� +· · · +�|X†| − (k − 1)δ′|X†| +δ′|X†| +� +(1 − ǫ) +tuples (Y ′ +1, Y ′ +2, . . . , Y ′ +k) ∈ +� +X† +δ′|X†| +�k such that each Y ′ +i is in Y′ +i, disjoint with the other +k − 1. +For such a (Y ′ +1, Y ′ +2, . . . , Y ′ +k), find a δ|X†|-set Y † +i ∈ Yi included in each Y ′ +i . Add +the tuple +� +C, Y1 ∪ Y † +1 ; F +Y † +1 +1 , Y2 ∪ Y † +2 ; F +Y † +2 +2 , . . . , Yk ∪ Y † +k ; F +Y † +k +k +� +to Z′′, where the set C is that of Z. By construction, it satisfies the conditions Z-i) +to iii) with F′ ← F (Z′). +Subtract �k +i=1 F +Y † +1 +i +from Fi. Repeat the above ǫ−1/2 times including Step 2 for +the current Z. Then denote an element of Z′′ by Z′, and family F +Y † +i +i +by F′ +i. Such +a Z′ and F′ +i are produced from Z and Fi, and we assume it for the four objects +anywhere below. +Finally, normalize each Z′ for Z-iv) the same way as Step 2-6-b)-i) of Recon- +struct with the γi given there. It possibly reduces F′ +i into its 1 − exp (−h/2) or +larger. +Perform the process for all Z ∈ Z′ to complete our construction of Z′′. + +8 +JUNICHIRO FUKUYAMA +Step 4. Confirm Πr+1 to finish the proof. For the r + 1-normality of Z′′, it can +be checked by straightforward recursive arguments with (2.2) that +1 +2ǫ1/2|Fi| < ∆|Fi| < 2ǫ1/2|Fi|, +where |Fi| expresses the value after Step 1, and ∆|Fi| the difference between |Fi| +and its final value. This means Step 2 can use the Γ +� +b +� +1 − 2ǫ1/2�� +-condition on ∥ · +∥B instead of the Γ (b)-condition throughout the construction, constantly achieving +(2.2) for a Z. In addition, the normalization of each Z′ always keeps more than +half of its �k +i=1 F′ +i. +Hence, the recursive loop for every Z terminates without an exception defining +our Z′′ with |F (Z′′) | > ǫ2/3|F (Z′) | > ǫ2|F (Z) | by C). As it is a PSF with the +universal disjoint property by construction, we confirm the proposition Πr+1 to +complete our proof of (2.1). +We now have Theorem 1.1. +References +1. Erd¨os, P., Rado, R. : Intersection theorems for systems of sets. Journal of the London Math- +ematical Society, Second Series, 35 (1), pp. 85 - 90 (1960) +2. Fukuyama, J. : The sunflower conjecture proven. arXiv:2212.13609 [math.CO] (2022) +3. Fukuyama, +J.: +Improved +bound +on +sets +including +no +sunflower +with +three +petals. +arXiv:1809.10318v3 [math.CO] (2021) +4. Fukuyama, J. : The case k = 3 of the sunflower conjecture. Private work available at Penn +State Sites. +Web address: +https://sites.psu.edu/sunflowerconjecture/files/2023/01/Proof-of-the-3-petal- +sunflower-conjecture.pdf (2023) +Department of Computer Science and Engineering, The Pennsylvania State Univer- +sity, PA 16802, USA +E-mail address: jxf140@psu.edu + diff --git a/BNE2T4oBgHgl3EQf8gli/content/tmp_files/load_file.txt b/BNE2T4oBgHgl3EQf8gli/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb733982dd6fedb2d1c6c16726fa7d86c568b1b4 --- /dev/null +++ b/BNE2T4oBgHgl3EQf8gli/content/tmp_files/load_file.txt @@ -0,0 +1,271 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf,len=270 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='04219v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='CO] 10 Jan 2023 EXTENSIONS OF A FAMILY FOR SUNFLOWERS JUNICHIRO FUKUYAMA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' This paper refines the original construction of the recent proof of the sunflower conjecture to prove the same general bound [ck log(k + 1)]m on the cardinality of a family of m-cardinality sets without a sunflower of k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Our proof uses a structural claim on an extension of a family that has been previously developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Motivation, Terminology and Related Facts The sunflower conjecture states that a family F of sets each of cardinality at most m includes a k-sunflower if |F| > cm k for some ck ∈ R>0 depending only on k, where k-sunflower stands for a family of k different sets with common pair-wise intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It had been open since the sunflower lemma was presented in 1960 [1], until it was recently proven [2] with the following statement confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' There exists c ∈ R>0 such that for every k, m ∈ Z>0, a family F of sets each of cardinality at most m includes a k-sunflower if |F| > [ck log(k + 1)]m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ The base of the obtained bound [ck log(k + 1)]m is asymptotically close to the lower bound k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Our investigation on finding such a near-optimal bound had continued from the previous work [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The paper attempts to explore some combi- natorial structure involving sunflowers, to prove that a uniform family F includes three mutually disjoint sets, not just a 3-sunflower, if it satisfies the Γ � cm 1 2 +ǫ� condition for any given ǫ ∈ (0, 1) and c depending on ǫ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Here the Γ (b)-condition of F (b ∈ R>0) means |{U : U ∈ F, S ⊂ U}| < b−|S||F| for all nonempty sets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The original construction [4] of the work [2] proves the most noted three-petal case of the conjecture, referring to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2 given below that derives the exten- sion generator theorem presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The goal of this paper is to further refine1 the original construction to prove the same [ck log(k + 1)]m bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We will find such proof at the end of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The rest of this section describes the similar terminology and related facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' De- note the universal set by X, its cardinality by n, and a sufficiently small positive 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 05D05: Extremal Set Theory (Primary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' sunflower lemma, sunflower conjecture, ∆-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 1Extra information on this paper and the references [2, 3, 4] is available at Penn State Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Web address: https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='edu/sunflowerconjecture/2023/01/10/index-page/ 1 2 JUNICHIRO FUKUYAMA number depending on no other variables by ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' In addition, i, j, m, p, r ∈ Z≥0, [b] = [1, b] ∩ Z, F ⊂ 2X, �X′ m � = {U : U ⊂ X′, |U| = m} , for X′ ⊂ X, and F[S] = {U : U ∈ F, S ⊂ U} , for S ⊂ X, A set means a subset of X, and one in �X m � is called m-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Weight X by some w : 2X → R≥0, which induces the norm ∥ · ∥ of a family defined by ∥F∥ = � U∈F w(U) for any F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Denote set/family subtraction by −, while we use the symbol \\ for the different notion described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Use ← to express substitution into a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For simplicity, a real interval may denote the integral interval of the same range, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', use (1, t] instead of (1, t] ∩ Z if it is clear by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Obvious floor/ceiling functions will be ommited throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Now let F be a family of m-sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', F ⊂ �X m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We say κ (F) = �n m � − ln |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' is the sparsity of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The family satisfies the Γ (b)-condition on ∥ · ∥ (b ∈ R>0) if ∥G∥ = ∥G ∩ F∥, for all G ⊂ 2X, and ∥F[S]∥ < b−|S|∥F∥, for every nonempty set S ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' As used above, the norm ∥ · ∥ can be omitted if it is induced by the unit weight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', w : V �→ � 1, if V ∈ F, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The following theorem is proven2 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Let X be weighted to induce the norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For every sufficiently small ǫ ∈ (0, 1), and F ⊂ �X m � satisfying the Γ � 4γn l � condition on ∥ · ∥ for some l ∈ [n], m ∈ [l], and γ ∈ � ǫ−2, lm−1� , there are ��n l � (1 − ǫ) � sets Y ∈ �X l � such that � 1 − � 2 ǫγ � � l m � � n m � ∥F∥ < ���� �Y m ����� < � 1 + � 2 ǫγ � � l m � � n m � ∥F∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ With Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2, we can prove the aforementioned extention generator theorem that is about the l-extension of F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', Ext (F, l) = � T : T ∈ �X l � , and ∃U ∈ F, U ⊂ T � , for l ∈ [n] − [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It is not difficult to see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1) κ [Ext (F, l)] ≤ κ (F) , as in [3], where it is also shown: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For F ⊂ �X m � such that m ≤ n/2, κ �� X 2m � − Ext (F, 2m) � ≥ 2κ ��X m � − F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ 2In [3], the theorem uses so called Γ2 (b, 1)-condition on ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It is straightforward to check it means the Γ (b)-condition on ∥ · ∥ here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' EXTENSIONS OF A FAMILY FOR SUNFLOWERS 3 Further denote Gp = G × G × · · · × G � �� � p , X = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' , Xp) ∈ � 2X�p , Rank (X) = p, and Union (X) = p� j=1 Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Suppose m divides n and p = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' If Union (X) = X, and all Xi are mutually disjoint n/m-sets, then such an X is an m-split of X with Xi called strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Its subsplit X′ of rank r ∈ [m], or r-subsplit of X, is the tuple of some r strips of X preserving the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' A set S is on X′ if S ⊂ Union � X′� , and |Xi ∩ S| ∈ {0, 1} for every strip Xi of X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Denote by 2X′ the family of all sets on X′, by �X′ p � the family of p-sets on X′, by X \\ X′ the subsplit of rank m − m′ consisting of the strips in X but not in X′, and by X′ \\ B for a set B, abusing the symbol \\, the subsplit of X consisting of the strips each disjoint with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For notational convenience, allow Rank � X′� = 0 for which X′ = (∅), Union � X′� = ∅, and �X′ p � = {∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We have: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For any nonempty family F ⊂ �X m � such that m divides n = |X|, there exists an m-split X of X such that ����F ∩ �X m ����� ≥ � n m �m |F| � n m � > |F| exp (−m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ The lemma proven in [2] poses a special case of the general statement presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1 We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2 in the two subsections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Given F and k, we will find a subfamily ˆF ⊂ F with a property that implies the existence of a k-sunflower in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Formulation and Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Letting h = exp �1 ǫ � , and c = exp (h) , assume WLOG that k ≥ 3, n = |X| is larger than ckm and divisible by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Otherwise add some extra elements to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F ⊂ �X m � for an m-split X of X by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='4, satisfying the Γ (cck ln k)-condition and |F| > (cck ln k)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' |F| < (km)m and m > cc ln k, otherwise F includes a k-sunflower by the sunflower lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 4 JUNICHIRO FUKUYAMA FindCores Input: i) the family F ⊂ �X m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Outputs: i) C ⊂ �X r0 � for some r0 ∈ [0, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' ii) ˆF ⊂ � C∈C F[C] such that | ˆF| ≥ 3−m−1|F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F′ ← F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' ˆF ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' C ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' for r = m down to 0 do: 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' repeat: a) find an r-set C such that |F′[C]| ≥ f(r) putting TC ← F′[C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' b) if found then: F′ ← F′ − TC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' ˆF ← ˆF ∪ TC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' C ← C ∪ {C};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' else exit Loop 2-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' if | ˆF| ≥ 3−m+r−1|F| then return � r, C, ˆF � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Algorithm FindCores Let i ∈ [k], r ∈ [0, m], b = ck ln k, δ = ǫ k ln k, F′, Fi ⊂ F, C ∈ 2X, and Yi ∈ 2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' A tuple Z = (C, Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F1, Y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F2, · · · , Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Fk) is said to be a partial sunflower of rank r over F′ if there exists an r-subsplit X∗ of X satisfying the four conditions: Z-i) C ∈ �m/c u=0 �X∗ u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Z-ii) Yi are mutually disjoint k subsets of Union (X∗ \\ C) such that |Yi ∩ X†| = δ|X†| for each strip X† of X∗ \\ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Z-iii) The k families Fi are each nonempty included in F′[C] ∩ �X − Union (X∗ \\ C) ∪ Yi m � , and are identical if Rank (X∗ \\ C) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Z-iv) |Fi| < 2|Fi′| for i ∈ [k] and i′ ∈ [k] − {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We say that such an Fi occurs on Z and in Z with the core C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Also Z and Fi are on X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' A family Z of Z on one or more X∗ is a partial sunflower family (PSF) of rank r over F′, if each two Fi occurring on two different Z are mutually disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', the universal disjoint property of Z is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Denote F (Z) := � Z∈Z i∈[k] Fi of Z, for any PSF Z abusing the symbol F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' In the rest of our proof, we construct a nonempty PSF of rank m over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' This means a k-sunflower in F proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' With f : Z≥0 → R≥0, x �→ ǫ3m(chk)−x k |F|, EXTENSIONS OF A FAMILY FOR SUNFLOWERS 5 obtain the families C and ˆF by the algorithm FindCores described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It is straightforward to see that the two outputs correctly satisfy the properties i)-ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' In addition: A) | ˆF[U]| < f(|U|) for all U ∈ �m r′=r0+1 �X r′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' B) We will construct partial sunflowers over ˆF with cores C in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The families TC Step 2-1 finds for r = r0 are mutually disjoint each with |TC| ≥ f(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' By the Γ (cck ln k)-condition of F and cc ln k < m, k−1ǫ3m(chk ln k)−|C||F| = f (r0) ≤ |TC| ≤ |F[C]| < (cck ln k)−|C||F|, ⇒ r0 = |C| < ln k − 3m ln ǫ (c − h) ln c < m 2c �ln k m + 1 � < m c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Define a statement on the obtained objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Proposition Πr for r ∈ [r0, m]: there exists a PSF Z of rank r over ˆF such that |F(Z)| > ǫ2r−2r0| ˆF|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ Such a Z is said to be r-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' By definition, Z is the union of PSFs ZX∗ on r-subsplits X∗ satisfying the universal disjoint property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Our final goal of finding a nonempty PSF of rank m over F would be met if Πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The proposition Πr0 holds since Z = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8edC, ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' TC, ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' TC, · · · , ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' TC � �� � k \uf8f6 \uf8f8 : C ∈ ˆC \uf8fc \uf8fd \uf8fe is an r0-normal PSF such that F (Z) = ˆF by B) where TC are the ones mentioned there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' So it suffices to show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1) Πr ⇒ Πr+1, for every r ∈ [r0, m), to have proof of a k-sunflower in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We start showing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1) as the only remaining task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Assume Πr for a particular r ∈ [r0, m), so we have an r-normal PSF Z that is the union of ZX∗ on some r-subsplits X∗ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We confirm Πr+1 in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Reconstruct Z into another PSF Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Obtain such a Z′ by the algorithm Reconstruct described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It is a PSF of rank r over F (Z) satisfying the two conditions: C) |F (Z′) | > 2−1ǫ|F (Z) |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' D) For each Z ∈ Z′ on an r-subsplit X∗, there exists an r+1-subsplit X′ containing X∗ such that each Fi on Z meets |Fi[S]| < 1 b |Fi|, ∀S ∈ �X′ \\ X∗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' □ We see their truth by the notes below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Such a Z ∈ Z′ is said to be on the split pair � X∗, X′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' In Steps 2 and 3, we will construct our desired r + 1-normal PSF Z′′ from Z′ confirming Πr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Justification of Z′ being a PSF with C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F (ZX′) of an X′ disregarded by Step 2-3 is negligible as their union will be smaller than 2−m� m r+1 � < � 3 2 �−m < � 3 2 �−cc ln k < ǫ3/k of F (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 6 JUNICHIRO FUKUYAMA Reconstruct Input: an r-normal PSF Z for some r ∈ [r0, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Output: a PSF Z′ of rank r over F (Z) satisfying C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' X ← the family of all r-subsplits X∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' /* The given Z is the union of PSFs ZX∗ on some X∗ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' */ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' for each r + 1-subsplit X′ do: 2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' X∗ ← the family of all X∗ ∈ X that are r-subsplits of X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' ZX′ ← � X∗∈X∗ ZX∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' if |F (ZX′) | < 3−m|F (Z) | then go to Step 2 for the next X′ else X ← X − X∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' for each Fi in ZX′ do F′ i ← Fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' for each Fi in ZX′ and on X∗, and sets B ∈ �X∗ r � and U ∈ � X′ r+1 � [B] such that |Fi[U]| > � c √ hk ln k �−r−1 |Fi[B]| do F′ i ← F′ i − Fi[U];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' for each Z ∈ ZX′ do: a) if |F′ i| < ǫ|Fi| for some Fi on Z then delete Z from ZX′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' b) else normalize Z for the condition Z-iv) as follows: b)-i) for each Fi on Z do: γi ← |F′ i|−1 mini′∈[k] |F′ i′|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F′ i ← any subfamily of F′ i of cardinality min (|F′ i|, ⌊2γi|F′ i|⌋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' b)-ii) replace all Fi by the F′ i to reconstruct Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' return the union of all ZX′ found in Loop 2 as Z′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Algorithm Reconstruct |F(ZX′)[U]| ≤ | ˆF[U]| < f (r + 1) for all U ∈ � X r+1 � by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' So, |F (ZX′) [U]| |F (ZX′) | < f (r + 1) 3−mǫ2r−2r0| ˆF| < � chk ln k �−r−1 k , before Step 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Fi[B] are mutually disjoint for all different Fi occurring in ZX′ each on an X∗, and B ∈ �X∗ r � right before Step 2-5, by the universal disjoint property of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' (By the rule Z-iii), Fi on a single Z are identified when r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=') In addition, Fi[U] are mutually disjoint for all Fi and U ∈ � X′ r+1 � meeting � X∗∈X∗, Fi in ZX′ and on X∗ B∈(X∗ r ), U∈(X′ m′)[B] Fi[U] = F(ZX′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' By the above two, Step 2-5 may reduce V = � Fi in ZX′ F′ i by less than its ǫ3/k, leaving only Fi, B, and U such that |F′ i[U]| < (cb)−r−1|F′ i[B]|, ⇒ |F′ i[B]| > (cb)r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' By Z-iv) of Z, Step 2-6-a) may only reduce less than 2ǫ3 of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The process of normalization is well-defined by Step 2-6-b) due to |F′ i| > (cb)r+1 before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It could further reduce V into its ǫ/2 or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' EXTENSIONS OF A FAMILY FOR SUNFLOWERS 7 The condition Z-iv) of the obtained Z′ follows the above as well as the two properties C) and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For each Fi occurring in Z′, construct a family Yi of Y ∈ � X† δ|X†| � such that Fi ∩ �X−X†∪Y m � is sufficiently large, where X† = Union(X′ \\ X∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Consider each Z ∈ Z′ on (X∗, X′) with the unique strip X† of X′ \\ X∗, and an Fi on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Weight X† by 2X† → Z≥0, W �→ |Fi[W]| inducing the norm ∥ · ∥ as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' The family H = �X† 1 � satisfies the Γ(b)-condition on ∥ ·∥ by D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2 to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' There exists Yi ⊂ � X† δ|X†| � such that |Yi| > � |X†| δ|X†| � [1 − exp (−h)] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2) δ [1 − exp (−h)] |Fi| < ��FY i �� < δ [1 + exp (−h)] |Fi|, for every Y ∈ Yi, where FY i ⊂ Fi ∩ �X − X† ∪ Y m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Find Z′′ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Now consider the same Z with the k families Fi and sets Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Put δ′ = 2δ ln k, and Y′ i = Ext (Yi, δ′|X†|) , for each Fi to see |Y′ i| > � |X†| δ′|X†| � [1 − exp (−h ln k)] > � |X†| δ′|X†| � � 1 − ǫ k � , by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='3, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2): to the Yi, repeatedly apply the lemma ⌈log2 ln k⌉ times doubling the second parameter of Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Then κ �� X† δ′|X†| � − Y′ i � > h ln k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Hence, there exist more than � |X†| δ′|X†| ��|X†| − δ′|X†| δ′|X†| � · · �|X†| − (k − 1)δ′|X†| δ′|X†| � (1 − ǫ) tuples (Y ′ 1, Y ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' , Y ′ k) ∈ � X† δ′|X†| �k such that each Y ′ i is in Y′ i, disjoint with the other k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For such a (Y ′ 1, Y ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' , Y ′ k), find a δ|X†|-set Y † i ∈ Yi included in each Y ′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Add the tuple � C, Y1 ∪ Y † 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F Y † 1 1 , Y2 ∪ Y † 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F Y † 2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' , Yk ∪ Y † k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' F Y † k k � to Z′′, where the set C is that of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' By construction, it satisfies the conditions Z-i) to iii) with F′ ← F (Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Subtract �k i=1 F Y † 1 i from Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Repeat the above ǫ−1/2 times including Step 2 for the current Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Then denote an element of Z′′ by Z′, and family F Y † i i by F′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Such a Z′ and F′ i are produced from Z and Fi, and we assume it for the four objects anywhere below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Finally, normalize each Z′ for Z-iv) the same way as Step 2-6-b)-i) of Recon- struct with the γi given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' It possibly reduces F′ i into its 1 − exp (−h/2) or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Perform the process for all Z ∈ Z′ to complete our construction of Z′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 8 JUNICHIRO FUKUYAMA Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Confirm Πr+1 to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' For the r + 1-normality of Z′′, it can be checked by straightforward recursive arguments with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2) that 1 2ǫ1/2|Fi| < ∆|Fi| < 2ǫ1/2|Fi|, where |Fi| expresses the value after Step 1, and ∆|Fi| the difference between |Fi| and its final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' This means Step 2 can use the Γ � b � 1 − 2ǫ1/2�� condition on ∥ · ∥B instead of the Γ (b)-condition throughout the construction, constantly achieving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='2) for a Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' In addition, the normalization of each Z′ always keeps more than half of its �k i=1 F′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Hence, the recursive loop for every Z terminates without an exception defining our Z′′ with |F (Z′′) | > ǫ2/3|F (Z′) | > ǫ2|F (Z) | by C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' As it is a PSF with the universal disjoint property by construction, we confirm the proposition Πr+1 to complete our proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' We now have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Erd¨os, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=', Rado, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' : Intersection theorems for systems of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Journal of the London Math- ematical Society, Second Series, 35 (1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' 85 - 90 (1960) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Fukuyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' : The sunflower conjecture proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='13609 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='CO] (2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Fukuyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=': Improved bound on sets including no sunflower with three petals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='10318v3 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='CO] (2021) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Fukuyama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' : The case k = 3 of the sunflower conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Private work available at Penn State Sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content=' Web address: https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='edu/sunflowerconjecture/files/2023/01/Proof-of-the-3-petal- sunflower-conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='pdf (2023) Department of Computer Science and Engineering, The Pennsylvania State Univer- sity, PA 16802, USA E-mail address: jxf140@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE2T4oBgHgl3EQf8gli/content/2301.04219v1.pdf'} diff --git a/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/2301.08656v1.pdf.txt b/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/2301.08656v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4595a8eed9b85218fc0442c938a62220f548cb2 --- /dev/null +++ b/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/2301.08656v1.pdf.txt @@ -0,0 +1,1052 @@ +Quantum Control of Trapped Polyatomic Molecules for eEDM Searches +Lo¨ıc Anderegg,1, 2, ∗ Nathaniel B. Vilas,1, 2 Christian Hallas,1, 2 Paige +Robichaud,1, 2 Arian Jadbabaie,3 John M. Doyle,1, 2 and Nicholas R. Hutzler3, † +1Department of Physics, Harvard University, Cambridge, MA 02138, USA +2Harvard-MIT Center for Ultracold Atoms, Cambridge, MA 02138, USA +3Division of Physics, Mathematics, and Astronomy, +California Institute of Technology, Pasadena, CA 91125, USA +(Dated: January 23, 2023) +Ultracold polyatomic molecules are promising candidates for experiments in quantum science, +quantum sensing, ultracold chemistry, and precision measurements of physics beyond the Standard +Model. A key, yet unrealized, requirement of these experiments is the ability to achieve full quantum +control over the complex internal structure of the molecules. Here, we establish coherent control of +individual quantum states in a polyatomic molecule, calcium monohydroxide (CaOH), and use these +techniques to demonstrate a method for searching for the electron electric dipole moment (eEDM). +Optically trapped, ultracold CaOH molecules are prepared in a single quantum state, polarized in +an electric field, and coherently transferred into an eEDM sensitive state where an electron spin +precession measurement is performed. To extend the coherence time of the measurement, we utilize +eEDM sensitive states with tunable, near-zero magnetic field sensitivity. The spin precession coher- +ence time is limited by AC Stark shifts and uncontrolled magnetic fields. These results establish a +path for eEDM searches with trapped polyatomic molecules, towards orders-of-magnitude improved +experimental sensitivity to time-reversal-violating physics. +The rich structure of polyatomic molecules makes them +an appealing platform for experiments in quantum sci- +ence [1–4], ultracold chemistry [5], and precision mea- +surements [6–10]. Key to this structure is the presence +of near-degenerate states of opposite parity, which allow +the molecules to be easily polarized in the laboratory +frame with the application of a small electric field. Such +states are a novel resource, generic among polyatomic +molecules while rare in diatomics, that may be useful +for applications such as analog simulation of quantum +magnetism models [1, 2] or for realizing switchable inter- +actions and long-lived qubit states for quantum comput- +ing [4]. Additionally, the parity-doublet states in trapped +polyatomic molecules are expected to be an invaluable +tool for systematic error rejection in precision measure- +ments of physics beyond the Standard Model (BSM) [6]. +To date, several species of polyatomic molecules have +been laser cooled and/or trapped at ultracold temper- +atures [11–17]. +One powerful avenue for tabletop BSM searches is +probing for the electric dipole moment of the electron +(eEDM) [18–22], de, which violates time-reversal (T) +symmetry and is predicted by many BSM theories to +be orders of magnitude larger than the Standard Model +prediction [19, 20]. Current state-of-the-art eEDM ex- +periments are broadly sensitive to T-violating physics at +energies much greater than 1 TeV [23–28]. All such ex- +periments use Ramsey spectroscopy to measure an en- +ergy shift due to the interaction of the electron with the +large electric field present inside a polarized molecule [24– +27, 29]. Molecular beam experiments have achieved high +statistical sensitivity by measuring a large number of +molecules over a ≈ 1 ms coherence time [24, 25], while +molecular ion-based experiments have used long Ram- +sey interrogation times (≈ 1 s) though with lower num- +bers [26, 27, 29]. +Measurements with trapped neutral +polyatomic molecules can potentially combine the best +features of each approach to achieve orders-of-magnitude +improved statistical sensitivity [6]. +In this Report, we demonstrate full quantum control +over the internal states of a trapped polyatomic molecule +in a vibrational bending mode with high polarizability +in small electric fields. The method starts with prepar- +ing ultracold, optically trapped molecules in a single hy- +perfine level, after which a static electric field is applied +to polarize the molecules. +The strength of the polar- +izing electric field is tuned to obtain near-zero g-factor +spin states, which have strongly suppressed sensitivity +to magnetic field noise while retaining eEDM sensitivity. +Microwave pulses are applied to create a coherent super- +position of these zero g-factor spin states that precesses +under the influence of an external magnetic field. The +precession phase is then read out by a combination of +microwave pulses and optical cycling. +We observe spin precession over a range of electric and +magnetic fields and characterize the current limitations +to the coherence time of the measurement. With readily +attainable experimental parameters, coherence times on +the order of the state lifetime (>100 ms) could be realisti- +cally achieved. We therefore realize the key components +of an eEDM measurement in this system. Although the +light mass of CaOH precludes a competitive eEDM mea- +surement [30], the protocol demonstrated here is directly +transferable to heavier laser-cooled alkaline earth mono- +hydroxides with identical internal level structures, such +as SrOH, YbOH, and RaOH, which have significantly en- +arXiv:2301.08656v1 [physics.atom-ph] 20 Jan 2023 + +2 +FIG. 1. +(a) A geometric picture of the bending molecule at the zero g-factor crossing, showing the electron spin (⃗S) has a finite +projection on the molecule axis (ˆn), giving eEDM sensitivity. However, the electron spin (⃗S) is orthogonal to the magnetic field +( ⃗B), resulting in suppressed magnetic field sensitivity. (b) The magnetic sensitivity (upper plot) and eEDM sensitivity (lower +plot) for a pair of zero g-factor states (N = 1, J = 1/2+, F = 1, MF = ±1) are shown as a function of the applied electric +field. (c) Experimental sequence to prepare the eEDM sensitive state. First, the molecules are pumped into a single quantum +state (N = 1, J = 1/2−, F = 0) with a combination of microwave drives and optical pumping (I). Next, a microwave π-pulse +drives the molecules into the N = 2, J = 3/2−, F = 2, MF = 0 state (II). Lastly, the eEDM measurement state is prepared as +a coherent superposition of the N = 1, J = 1/2−, F = 1 MF = ±1 states with a microwave π-pulse (III). The states which are +optically detectable with the detection light are shown in black, while those not addressed by the detection light are in grey. +hanced sensitivity to the eEDM [6, 11, 12, 30, 31]. +In eEDM measurements with polarized molecules, the +electron spin ⃗S precesses under the influence of an ex- +ternal magnetic field BZ and the internal electric field of +the molecule, Eeff, which can be large due to relativistic +effects. Time evolution is described by the Hamiltonian +H = gSµBBZ ⃗S · ˆZ − deEeff⃗S · ˆn += gSµBBZMS − deEeffΣ. +(1) +Here, gS ≈ 2 is the electron spin g-factor, µB is the +Bohr magneton, BZ points along the lab ˆZ axis, and +the internal field Eeff points along the molecule’s inter- +nuclear axis ˆn. +We define the quantities MS = ⃗S · ˆZ +and Σ = ⃗S · ˆn to describe the electron’s magnetic sen- +sitivity and EDM sensitivity, respectively. The effect of +the eEDM can be isolated by switching the orientation of +the applied magnetic field or, alternatively, by switching +internal states to change the sign of MS or Σ. Perform- +ing both switches is a powerful technique for suppressing +systematic errors [25, 26]. +Current EDM bounds rely on specific states in di- +atomic molecules that have an unusually small g-factor, +reducing sensitivity to stray magnetic fields [24, 26]. +However, CaOH, like other laser-coolable molecules with +structure amenable to eEDM searches [6, 31–33], has +a single valence electron, which results in large mag- +netic g-factors. In this work, we engineer reduced mag- +netic sensitivity by using an applied electric field EZ to +tune MS to a zero-crossing, while maintaining signifi- +cant eEDM sensitivity Σ. This technique is generic to +polyatomic molecules with parity-doublets. Details of a +specific M = ±1 pair of zero g-factor states are shown +in Figure 1 (a)-(b), with further information in the Sup- +plemental Material. Sensitivity to transverse magnetic +fields is also suppressed in these zero g-factor states (see +Supplemental Material). +The +experiment +begins +with +laser-cooled +CaOH +molecules loaded from a magneto-optical trap [14] into +an optical dipole trap (ODT) formed by a 1064 nm laser +beam with a 25 µm waist size, as described in previous +work [15]. The ODT is linearly polarized and its polar- +ization vector ⃗ϵODT defines the ˆZ axis, along which we +also apply magnetic and electric fields, ⃗B = BZ ˆZ and +⃗E = EZ ˆZ, respectively, as depicted in Figure 1(a). We +first non-destructively image the molecules in the ODT +for 10 ms as normalization against variation in the num- +ber of trapped molecules. The molecules are then opti- +cally pumped into the N = 1− levels of the � +X2Σ+(010) +vibrational bending mode [15] (Figure 1(c)), and the trap +depth is adiabatically lowered by 3.5× to reduce the effect +of AC Stark shifts from the trap light and to lower the +temperature of the molecules to 34 µK. Any molecules +that were not pumped into N = 1− levels of the bending + +(c) +A21(010)2- +(a) +(b) +1/2 +0,1+ +μb(Ms) (MHz/G) +0.3 +0.2 +623 nm +2 +M= +E,B,EoDT + 0. 0 +(Z)= . 0 +1 +M=+1 +3/2 +-0.2 +2- +n +Ca +40 GHz +(010)+3zX +0.6 +Sensitivity (22) +Relative EDM +0.4 +M=-1 +0.2 +0+ +1/2 + 10 ms), losses from background gas collisions (∼1 sec), +blackbody excitation (∼1 sec), and the spontaneous life- +time of the bending mode (∼0.7 sec) lead to an overall +loss of signal, as characterized in Ref. [15]. This effect +is mitigated with a fixed duration between the first and +second images, making the loss independent of the pre- +cession time. +To map out the location of the zero g-factor cross- +ing, we perform spin precession measurements at a fixed +magnetic field BZ = 110 mG for different electric fields +(Figure 2(b)). The spin precession frequency corresponds +to an effective g-factor at that electric field. +We find +that the zero g-factor crossing within the N = 1, J = +1/2+, F = 1, M = ±1 eEDM manifold occurs at an elec- +tric field of 59.6 V/cm, in agreement with theory cal- +culations described in the Supplemental Material. +We +note that there is another zero g-factor crossing for the +N = 1, J = 3/2+, F = 1 manifold at ≈ 64 V/cm, which + +(a) +(b) +(au) +0.6 +0.5 + remaining +(MHz/G) +0.55 +0 +0.05 +0.5 +eff +Fraction +0 +0.45 +-0.05 +-0.5 +58 +59 +60 +61 +0.4 +40 +50 +60 +70 +500 +1000 +80 +0 +Time (μs) +E- (V/cm)4 +FIG. 3. Coherence time of the spin precession signal. (a) Measured coherence times τ versus BZ at different electric fields +(red and blue markers, corresponding to different magnetic field sensitivity). The coherence time scales as 1/BZ due to AC +Stark shift broadening, then plateaus at a limit set by the magnetic field instability δB. This limit increases as the g-factor +approaches zero. Solid and dashed curves are fit to the data. The ambient magnetic field noise determined from the fit is +δB = 4+2 +−1 mG, while the fitted decoherence time due to light shifts is τ = (1/BZ) × 80+20 +−10 ms×mG. (b) The spin precession +coherence time at BZ = 15 mG is extended by 16× by approching the zero g-factor point. +has a smaller eEDM sensitivity but the opposite slope +of geff vs. +EZ, thereby providing a powerful resource +to reject systematic errors related to imperfect field re- +versals (see Supplemental Material). We emphasize that +while the location of these crossings is dependent on the +structure of a specific molecule, their existence is generic +in polyatomic molecules, which naturally have parity- +doublet structure [6]. +A critical component of the spin precession measure- +ment is the coherence time, which sets the sensitivity +of an eEDM search. +Figure 3(a) shows the measured +coherence time of our system at different applied fields +BZ and EZ. We characterize two dominant limitations +that wash out oscillations at long times. Variations in +the spin precession frequency can be linearly expanded +as δωSP = µeff(δBZ) + (δµeff)BZ. +The first term de- +scribes magnetic field noise and drift of the applied bias +field, given by δBZ. The second term describes noise and +drifts in the g-factor, δgeff, which can arise from insta- +bility in the applied electric field, EZ, or from AC Stark +shifts (described below). Drifts in the bias electric field +EZ are found to be negligible in our apparatus. +Decoherence due to magnetic field noise, δBZ, is inde- +pendent of the applied magnetic field but is proportional +to µeff, and can be mitigated by operating near the zero g- +factor crossing. As shown in Fig. 3(b), at an electric field +of 90 V/cm, corresponding to a large magnetic moment +of µeff = 1.0 MHz/G, we realize a magnetic field noise- +limited coherence time of 0.5 ms at BZ ≈ 15 mG. At an +electric field of 61.5 V/cm, corresponding to µeff = 0.06 +MHz/G, much closer to the zero g-factor location, we +find a coherence time of 4 ms at the same BZ. +At higher magnetic fields, the primary limitation to +the coherence time is due to AC stark shifts from the +optical trapping light (Fig. 4). The intense Z-polarized +ODT light leads to a shift in the electric field at which the +zero g-factor crossing occurs. Due to the finite temper- +ature of the molecules within the trap, they will explore +different intensities of trap light and hence have differ- +ent values of geff. The spread δgeff causes variation of +ωSP, which leads to decoherence. In contrast to the mag- +netic field noise term, this effect is independent of the +electric field EZ but decreases monotonically with BZ, +which scales the frequency sensitivity to g-factor vari- +ations, δωSP = BZδµeff. +The insensitivity of g-factor +broadening to the exact value of geff is demonstrated in +Fig. 4(c). Decoherence due to AC Stark shifts can be +reduced by cooling the molecules to lower temperatures +or by decreasing BZ. +The bias magnetic field can be +reduced arbitrarily far until either transverse magnetic +fields or magnetic field noise become dominant. +From +the decoherence rates measured in this work, it is ex- +pected that AC Stark shift-limited coherence times ∼1 s +could be achieved at bias fields of BZ ∼ 100 µG. +From the above discussion, it is expected that the +longest achievable coherence times will occur for very +small g-factors, geff ≈ 0, and very small bias fields, +BZ ≈ 0. Minimizing BZ requires reducing the effects of + +(a) +(b) +0.55 +0.06 MHz/G +Ambient Magnetic Field Noise +(ne) +10 +1.0 MHz/G +raction +0.5 +5 +Trap Light Shift +0.45 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +(sw +Time (μus) +Abient Magnetic Field Noise +0.5 +0.5 +0.1 +0 +0.06 MHz/G +g +1.0 MHz/G +0.45 +5 +10 +50 +100 +500 +0 +100 +200 +300 +400 +500 +600 +(mG) +Time (μs)5 +FIG. 4. Effect of trap light on coherence time. (a) The trap +light shifts the location of the zero crossing in µeff. As a result, +molecules at a finite temperature explore different magnetic +field sensitivities µeff. (b) Dependence of the spin precession +frequency (scaled by the trap depth U0) on position within +the trap. +At lower magnetic fields, the relative change in +spin precession frequency is reduced. (c) Two spin precession +curves taken at the same magnetic field (BZ = 210 mG) but +at different electric fields, showing that the AC Stark shift +limitation is independent of the effective g-factor, since AC +Stark shifts dominate the coherence time for large bias fields. +both magnetic field noise and transverse magnetic fields +to well below the level of the bias field energy shifts. We +cancel the transverse magnetic fields to below 1 mG by +maximizing the spin precession period under the influ- +ence of transverse B fields only, and actively monitor +and feedback on the magnetic field along each axis to +minimize noise and drifts in BZ. Note that the stainless +steel vacuum chamber has no magnetic shielding, lead- +ing to high levels of magnetic field noise which would not +be present in an apparatus designed for an eEDM search. +Even under these conditions, we achieve a coherence time +of 30 ms at an electric field of 60.3 V/cm (corresponding +to µeff = 0.02 MHz/G) and a bias field of BZ ≈ 2 mG, +(see Supplemental Material). However, at such a low bias +field, the molecules are sensitive to 60 Hz magnetic field +noise present in the unshielded apparatus, which is on +the same order as the bias field. Since the experiment is +phase stable with respect to the AC line frequency, this +60 Hz magnetic field fluctuation causes a time-dependent +spin precession frequency. Nevertheless, our prototype +experiment confirms that long coherence times are possi- +ble. Any future eEDM experiment would have magnetic +shielding that would greatly suppress nefarious magnetic +fields from the environment. Such shielding could readily +enable coherence times exceeding that of the ∼ 0.5 s life- +time of the bending modes of similar linear polyatomic +molecules with larger eEDM sensitivity [15]. +In summary, we have realized coherent control of opti- +cally trapped polyatomic molecules and demonstrated a +realistic experimental roadmap for future eEDM mea- +surements. +By leveraging the unique features of the +quantum levels in polyatomic molecules, we achieve a +coherence time of 30 ms for paramagnetic molecules in a +stainless steel chamber with no magnetic shielding. With +common shielding techniques employed in past EDM ex- +periments, there is a clear path to reducing stray fields +and extending coherence times to > 100 ms. +At such +a level, the dominant limitation becomes the finite life- +time of the bending mode [15]. Even longer coherence +times are possible with the right choice of parity dou- +blet states, as found in symmetric or asymmetric top +molecules [6, 13, 34, 35]. +Following +our +established +roadmap +with +heavier +trapped polyatomic molecules has the potential to +provide orders-of-magnitude improvements to current +bounds on T-violating physics. +Using a recent study +of the � +X(010) state in YbOH [36], we have identified +similar N = 1 zero g-factor states for eEDM measure- +ments with significantly improved sensitivity. In addi- +tion to the g-factor tuning demonstrated in this work, +polyatomic molecules provide the ability to reverse the +sign of Σ without reversing MS - a crucial feature of re- +cent experiments that have greatly improved the limit +on the eEDM [25, 27]. For example, in the N = 1 mani- +fold of CaOH, there is another zero g-factor crossing at a +nearby electric field value, with 69% smaller values of Σ +and opposite sign. Since the ratio of eEDM sensitivity to +g-factor vs. EZ slope differs between these two crossings, +measurements at both points could be used to suppress +systematics due to non-reversing fields coupling to the +electric field dependence of the g-factor [25]. +This work provides a first experimental demonstration +of the advantages of the rich level structure of polyatomic +molecules for precision measurements. +While we have +focused here on spin precession with T-reversed states +(M = ±1), many levels of interest can be favorably en- +gineered for precision measurement experiments. +In a +recent proposal [9], parity-doublets, magnetically tuned +to degeneracy in optically trapped polyatomic molecules, +were shown to be advantageous for searches for parity vi- +olating physics. In another recent work [7], a microwave +clock between rovibrational states in SrOH was proposed +as a sensitive probe of ultra-light dark matter, utilizing +transitions tuned to electric and/or magnetic insensitiv- +ity. In these proposals, and now experimentally demon- +strated in our work, coherent control and state engineer- +ing in polyatomic molecules can mitigate systematic er- +rors and enable robust searches for new physics. + +(a) +(b) +0.04 +10 +μeff (MHz/G) +0.02 +100 mG +I=1/2 +I=0 +0. +5 +50 mG +-0.02 +10 mG +0 +-0.04 +59 +60 +61 +-wo +0 +Wo +Ez (V/cm) +Position in trap +(c) +0.58 +0.01 MHz/G, 210 mG +(a.u.) +0.06 MHz/G, 210 mG +0.55 +Fraction remaining ( +0.52 +0.49 +0.46 +0.43 +0.4 +0 +200 +400 +600 +800 +1000 +Time (μs)6 +This work was supported by the AFOSR and the NSF. +LA acknowledges support from the HQI, NBV from the +DoD NDSEG fellowship program, and PR from the NSF +GRFP. NRH and AJ acknowledge support from NSF CA- +REER (PHY-1847550), The Gordon and Betty Moore +Foundation (GBMF7947), and the Alfred P. Sloan Foun- +dation (G-2019-12502). AJ acknowledges helpful discus- +sions with Chi Zhang and Phelan Yu. +∗ anderegg@g.harvard.edu +† hutzler@caltech.edu +[1] M. L. Wall, K. Maeda, and L. D. Carr, Simulating quan- +tum magnets with symmetric top molecules, Ann. Phys. +(Berlin) 525, 845 (2013). +[2] M. Wall, K. Maeda, and L. D. Carr, Realizing un- +conventional quantum magnetism with symmetric top +molecules, New J. Phys. 17, 025001 (2015). +[3] Q. Wei, S. Kais, B. Friedrich, and D. Herschbach, Entan- +glement of polar symmetric top molecules as candidate +qubits, J. Chem Phys 135, 154102 (2011). +[4] P. Yu, L. W. Cheuk, I. Kozyryev, and J. M. Doyle, A +scalable quantum computing platform using symmetric- +top molecules, New J. Phys. 21, 093049 (2019). +[5] L. D. Augustoviˇcov´a and J. L. Bohn, Ultracold collisions +of polyatomic molecules: CaOH, New J. Phys. 21, 103022 +(2019). +[6] I. Kozyryev and N. R. Hutzler, Precision measurement of +time-reversal symmetry violation with laser-cooled poly- +atomic molecules, Phys. Rev. Lett. 119, 133002 (2017). +[7] I. Kozyryev, Z. Lasner, and J. M. Doyle, Enhanced sen- +sitivity to ultralight bosonic dark matter in the spectra +of the linear radical SrOH, Phys. Rev. A 103, 043313 +(2021). +[8] N. R. Hutzler, Polyatomic molecules as quantum sensors +for fundamental physics, Quantum Science and Technol- +ogy 5, 044011 (2020). +[9] E. B. Norrgard, D. S. Barker, S. Eckel, J. A. Fedchak, +N. N. Klimov, and J. Scherschligt, Nuclear-spin depen- +dent parity violation in optically trapped polyatomic +molecules, Communications Physics 2, 1 (2019). +[10] Y. +Hao, +P. +Navr´atil, +E. +B. +Norrgard, +M. +Iliaˇs, +E. Eliav, R. G. E. Timmermans, V. V. Flambaum, and +A. Borschevsky, Nuclear spin-dependent parity-violating +effects in light polyatomic molecules, Phys. Rev. A 102, +052828 (2020). +[11] I. Kozyryev, L. Baum, K. Matsuda, B. L. Augenbraun, +L. Anderegg, A. P. Sedlack, and J. M. Doyle, Sisyphus +laser cooling of a polyatomic molecule, Phys. Rev. Lett. +118, 173201 (2017). +[12] B. L. Augenbraun, Z. D. Lasner, A. Frenett, H. Sawaoka, +C. Miller, T. C. Steimle, and J. M. Doyle, Laser- +cooled polyatomic molecules for improved electron elec- +tric dipole moment searches, New J. Phys. 22, 022003 +(2020). +[13] D. Mitra, N. B. Vilas, C. Hallas, L. Anderegg, B. L. +Augenbraun, L. Baum, C. Miller, S. Raval, and J. M. +Doyle, Direct laser cooling of a symmetric top molecule, +Science 369, 1366 (2020). +[14] N. B. Vilas, C. Hallas, L. Anderegg, P. Robichaud, +A. Winnicki, D. Mitra, and J. M. Doyle, Magneto- +optical trapping and sub-Doppler cooling of a polyatomic +molecule, Nature 606, 70 (2022). +[15] C. Hallas, N. B. Vilas, L. Anderegg, P. Robichaud, +A. Winnicki, C. Zhang, L. Cheng, and J. M. Doyle, Opti- +cal trapping of a polyatomic molecule in an ℓ-type parity +doublet state, arXiv:2208.13762 (2022). +[16] M. Zeppenfeld, B. G. Englert, R. Gl¨ockner, A. Prehn, +M. Mielenz, C. Sommer, L. D. van Buuren, M. Motsch, +and G. Rempe, Sisyphus cooling of electrically trapped +polyatomic molecules, Nature 491, 570 (2012). +[17] A. Prehn, M. Ibr¨ugger, R. Gl¨ockner, G. Rempe, and +M. Zeppenfeld, Optoelectrical cooling of polar molecules +to submillikelvin temperatures, Phys. Rev. Lett. 116, +063005 (2016). +[18] D. DeMille, J. M. Doyle, and A. O. Sushkov, Probing +the frontiers of particle physics with tabletop-scale ex- +periments, Science 357, 990 (2017). +[19] T. E. Chupp, P. Fierlinger, M. J. Ramsey-Musolf, and +J. T. Singh, Electric dipole moments of atoms, molecules, +nuclei, and particles, Rev. Mod. Phys. 91, 015001 (2019). +[20] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, +A. Derevianko, and C. W. Clark, Search for new physics +with atoms and molecules, Rev. Mod. Phys. 90, 025008 +(2018). +[21] C. Cesarotti, Q. Lu, Y. Nakai, A. Parikh, and M. Reece, +Interpreting the electron EDM constraint, Journal of +High Energy Physics 2019, 59 (2019). +[22] M. Pospelov and A. Ritz, Electric dipole moments as +probes of new physics, Annals of Physics 318, 119 (2005). +[23] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, +M. R. Tarbutt, and E. A. Hinds, Improved measurement +of the shape of the electron, Nature 473, 493 (2011). +[24] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, +G. Gabrielse, Y. V. Gurevich, P. W. Hess, N. R. Hut- +zler, E. Kirilov, I. Kozyryev, et al., Order of magnitude +smaller limit on the electric dipole moment of the elec- +tron, Science 343, 269 (2014). +[25] A. Collaboration et al., Improved limit on the electric +dipole moment of the electron., Nature 562, 355 (2018). +[26] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, +T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell, +Precision measurement of the electron’s electric dipole +moment using trapped molecular ions, Phys. Rev. Lett. +119, 153001 (2017). +[27] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, +Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, +A. Wang, J. Ye, and E. A. Cornell, A new bound on +the electron’s electric dipole moment, arXiv:2212.11841 +(2022). +[28] R. Alarcon, J. Alexander, V. Anastassopoulos, T. Aoki, +R. Baartman, S. Baeßler, L. Bartoszek, D. H. Beck, +F. Bedeschi, R. Berger, et al., Electric dipole moments +and the search for new physics, arXiv:2203.08103 (2022). +[29] Y. Zhou, Y. Shagam, W. B. Cairncross, K. B. Ng, T. S. +Roussy, T. Grogan, K. Boyce, A. Vigil, M. Pettine, +T. Zelevinsky, J. Ye, and E. A. Cornell, Second-Scale +Coherence Measured at the Quantum Projection Noise +Limit with Hundreds of Molecular Ions, Phys. Rev. Lett. +124, 053201 (2020). +[30] K. Gaul and R. Berger, Ab initio study of parity +and time-reversal violation in laser-coolable triatomic +molecules, Phys. Rev. A 101, 012508 (2020). +[31] T. A. Isaev, A. V. Zaitsevskii, and E. Eliav, Laser- + +7 +coolable polyatomic molecules with heavy nuclei, J. Phys. +B 50, 225101 (2017). +[32] T. A. Isaev and R. Berger, Polyatomic candidates for +cooling of molecules with lasers from simple theoretical +concepts, Phys. Rev. Lett. 116, 063006 (2016). +[33] I. Kozyryev, L. Baum, K. Matsuda, and J. M. Doyle, Pro- +posal for laser cooling of complex polyatomic molecules, +ChemPhysChem 17, 3641 (2016). +[34] B. L. Augenbraun, J. M. Doyle, T. Zelevinsky, and +I. Kozyryev, Molecular asymmetry and optical cycling: +Laser cooling asymmetric top molecules, Phys. Rev. X +10, 031022 (2020). +[35] B. L. Augenbraun, Z. D. Lasner, A. Frenett, H. Sawaoka, +A. T. Le, J. M. Doyle, and T. C. Steimle, Observa- +tion and laser spectroscopy of ytterbium monomethoxide, +YbOCH3, Phys. Rev. A 103, 022814 (2021). +[36] A. Jadbabaie, Y. Takahashi, N. H. Pilgram, C. J. +Conn, +C. Zhang, and N. R. Hutzler, Characteriz- +ing the fundamental bending vibration of a linear +polyatomic molecule for symmetry violation searches, +arXiv:2301.04124 (2022). + +8 +Supplemental Material for “Quantum Control of Trapped Polyatomic Molecules for +eEDM Searches” +ZERO g-FACTOR STATES +Origin +In 2Σ electronic states of linear polyatomic molecules, the spin-rotation interaction, γ ⃗N · ⃗S, couples the molecular +rotation N and the electron spin S to form the total angular momentum J. These states are well described in the +Hund’s case (b) coupled basis. An applied electric field EZ will interact with the molecular-frame electric dipole +moment µE, connecting states with opposite parity, ∆MF = 0, and ∆J ≤ 1. When µEEZ ≫ γ, N and S are +uncoupled and well described by their lab frame projections MN and MS. However, in the intermediate field regime +with µEEZ ∼ γ, the molecular eigenstates are mixed in both the Hund’s case (b) coupled basis and the decoupled +basis. MF remains a good quantum number in the absence of transverse fields. In this regime, MF ̸= 0 states with +⟨MS⟩ = 0 can arise at specific field values. These states have no first order electron spin magnetic sensitivity, and, +unlike MF = 0 clock states, have large eEDM sensitivity near BZ = 0. We refer to these states as zero g-factor +states [6]. +Zero g-factor states arise from avoided level crossings as free field states are mixed by the electric field. One of the +crossing states has ⟨MS⟩ < 0, the other state has ⟨MS⟩ > 0, and both have mixed MN. The spin-rotation interaction +couples the states and lifts the crossing degeneracy, resulting in eigenstates that are superpositions of electron spin +up and down with ⟨MS⟩ = 0, while retaining non-zero molecular orientation with ⟨ˆn⟩ = ⟨MNℓ⟩ ̸= 0. The lab frame +projection of ˆn ensures that the eEDM interaction in the molecule frame does not rotationally average away. +Zero g-factor states are generically present in the Stark tuning of polyatomic molecules. The reduction of symmetry +in a polyatomic molecule allows for rotation about the internuclear axis, resulting in closely spaced doublets of opposite +parity. When these doublets are mixed by an applied electric field, they split into 2N +1 groups of levels representing +the values of the molecular orientation ⟨MNℓ⟩. For each N manifold with parity doubling, avoided level crossings +generically occur between an MNℓ = ±1 Stark manifold and an MNℓ = 0 Stark manifold. +In diatomic molecules without parity-doubling, the existence of zero g-factor states requires an inverted spin rotation +structure (γ < 0), such that the two J states are tuned closer to each other by an electric field. For example, the +YbF molecule (γ = −13.4 MHz [37, 38]) has zero g-factor states at E ≈ 866 V/cm in the N = 1 manifold, while +CaF does not. However, since |γ|/B ≪ 1 for most 2Σ diatomic molecules, the electric fields that mix spin-rotation +states are much less than those that polarize the molecule. Therefore, zero g-factor states occur when the molecule +has negligible lab-frame polarization, limiting eEDM sensitivity. For example, the aforementioned states in YbF have +|⟨Σ⟩| ≈ 0.006, which is ∼3% the value of Σ in the zero g-factor states used in this work. +Characterization +To locate zero g-factor crossings and calculate eEDM sensitivities, we model the � +X(010) level structure using an +effective Hamiltonian approach [40–42]: +Heff = HRot + HSR + Hℓ + HHyp + HZeeman + HStark + HODT +(2a) +HRot = B +� +⃗N 2 − ℓ2� +(2b) +HSR = γ +� +⃗N · ⃗S − NzSz +� +(2c) +Hℓ = −qℓ +� +N 2 ++e−i2φ + N 2 +−ei2φ� +(2d) +HHyp = bF ⃗I · ⃗S + c +3 +� +3IzSz − ⃗I · ⃗S +� +(2e) +HZeeman = gSµBBZSZ +(2f) +HStark = −µZEZ +(2g) +HODT = −⃗d · ⃗EODT +(2h) + +9 +Here, we use a similar Hamilton as Ref. [7]. +HRot is the rotational energy; HSR is the spin-rotation interaction +accurate for low-N bending mode levels, with z defined in the molecule frame; Hℓ is the ℓ-type doubling Hamiltonian, +with ± defined in the molecule frame, φ as the nuclear bending coordinate, and using the same phase convention as +Ref. [43]; HHyp is the hyperfine Fermi-contact and dipolar spin interactions, defined in the molecule frame; HZeeman +describes the interaction of the electron spin magnetic moment with the lab-frame magnetic field; HStark is the +interaction of the Z-component of molecule-frame electric dipole moment µE with the lab frame DC electric field, +EZ; and HODT is the interaction of the molecular dipole moment operator ⃗d with the electric field of the ODT laser, +⃗EODT = E0/2(ˆϵODTe−iωt + c.c.). +To evaluate the molecule frame matrix elements, we follow the techniques outlined in Refs. [40, 41] to transform into +the lab frame. The field-free Hamiltonian parameters are taken from Ref. [44], except for the hyperfine parameters, +which were determined by the observed line positions to be bF = 2.45 MHz and c = 2.6 MHz, similar to those of the +� +X(000) state [45]. We use the same dipole moment, |µ| = 1.47 D, as the � +X(000) state, determined in Ref. [46]. Matrix +elements of HODT are calculated following Ref. [47] using the 1064nm dynamic polarizabilities reported in Ref. [15]. +For the calculations discussed below and in the main text, the ODT is polarized along the laboratory Z axis and +the molecules sit at a fixed trap depth of 160 µK (corresponding to the average trap intensity seen by the molecules in +the experiment). As detailed in the main text, when the trapping light is aligned with EZ, it acts like a weak electric +field, shifting the zero g-factor crossing by ∼ 1 V/cm from the field-free value. If the trapping light polarization is +rotated relative to EZ, tensor light shifts can couple states with ∆MF = ±2 or ±1 (the linearity of the light ensures +there are no ∆MF = ±1 vector shifts) [47]. The effects of this coupling are similar to those of transverse magnetic +fields, which we discuss below. +In the current work, we ignore nuclear and rotational Zeeman effects. Specifically, the magnetic sensitivity of CaOH +receives small contributions from nuclear spin of the H atom and the rotational magnetic moment of both the electrons +and the nuclear framework. While they have not yet been fully characterized, all of these effects will contribute at the +10−3µB level or less. These additional g-factors do not depend strongly on the applied electric field, and result in a +small shift of the zero g-factor crossing location. Future work characterizing rotational magnetic moments of � +X(010) +states of laser-coolable metal hydroxides can enable more accurate predictions of zero g-factor field values. +In CaOH, each rotational state N supports multiple M = ±1 pairs of zero g-factor states. The states at finite +electric field can be labeled in terms of their adiabatically correlated zero-field quantum numbers |N, Jp, F, M⟩. In the +presence of trap shifts, the zero g-factor states for N = 1 occur at E = 59.6 V/cm for |J = 1/2+, F = 1, M = ±1⟩ and +at E = 64.1 V/cm for |J = 3/2+, F = 1, M = ±1⟩. The J = 1/2, M = 1 state is a superposition of 47% MNℓ = −1, +50% MNℓ = 0, and 3% MNℓ = 1, while the J = 3/2, M = 1 state is 43% MNℓ = −1, 48% MN = 0, and 9% MNℓ = 1. +Both states are weak-electric-field seekers, yet the opposite molecule frame orientation of the spin results in differences +(a) +(b) +FIG. S1. Electric field tuning of N = 1 zero g-factor states near BZ = 0 in the absence of trap shifts. Blue lines denote +MF = +1 states and red lines MF = −1. Solid traces denote the J = 1/2 state pair and dashed traces denote the J = 3/2 +pair. The dotted vertical lines mark the electric field value of the zero g-factor crossing without trap shifts, ≈60.5 V/cm for +J = 1/2 and ≈64.4 V/cm for J = 3/2. Grayed out traces are other states in the N = 1 manifold. (a) The g-factor gSµB⟨MS⟩ +as a function of the applied electric field. (b) eEDM sensitivity ⟨Σ⟩ as a function of the applied electric field. A consequence of +the Hund’s case (b) coupling scheme is that Σ asymptotes to a maximum magnitude of S/(N(N + 1)) = 1/4 for fields where +the parity doublets are fully mixed but rotational mixing is negligible [39]. For fields where J is not fully mixed, some states +can exhibit |Σ| > 1/4. + +10 +(a) +(b) +(d) +(c) +FIG. S2. Full electric and magnetic characterization of zero g-factor states in the N = 1 manifold of CaOH, without trap shifts. +(a, b) 2D plots of the effective g-factor difference between two M = ±1 states, defined by geff = gSµB (⟨MS⟩M=+1 − ⟨MS⟩M=−1). +The plotted g-factor is normalized by gSµB. The black line represents the contour where the M = ±1 levels are nominally +degenerate. (c, d) 2D plots of the eEDM sensitivity, ⟨Σ⟩M=+1 − ⟨Σ⟩M=−1. The black line represents the geff = 0 contour. +in the value of Σ and the g-factor slope. For CaOH, the magnetic sensitivity and eEDM sensitivity of N = 1 zero +g-factor states are shown in Fig. S1. +By diagonalizing Heff over a grid of (EZ, BZ) values, we can obtain 2D plots of g-factors and eEDM sensitivities +shown in Fig. S2. For generality, we consider the molecular structure in the absence of trap shifts. Using the Z- +symmetry of the Hamiltonian, we separately diagonalize each MF block to avoid degeneracies at BZ = 0. Continuous +2D surfaces for eigenvalues and eigenvectors are obtained by ordering eigenstates at each value of (E, B) according to +their adiabatically correlated free field state. The application of an external magnetic field parallel to the electric field +results in ⟨MS⟩ ̸= 0 for an individual zero g-factor state, but the differential value between a zero g-factor pair can +still have ∆⟨MS⟩ = 0. This differential value means the superposition of a zero g-factor pair can maintain magnetic +insensitivity and EDM sensitivity over a range of fields, for example up to ∼5 G for the J = 1/2, N = 1 pair. +The procedure we use here for identifying zero g-factor states can be generically extended to searching for favorable +transitions between states with differing eEDM sensitivities, similar to what has been already demonstrated in a +recent proposal to search for ultra-light dark matter using SrOH [7]. In addition, there are also fields of BZ ≈ 10 − 20 +G and EZ ≈ 0 where opposite parity states are tuned to near degeneracy. This is the field regime that has been +proposed for precision measurements of parity-violation in optically trapped polyatomic molecules [9]. +We note that zero g-factor pairs also occur in N = 2−. The crossings occur around 400 − 500 V/cm for states + +11 +MF = 0- +MF = 0+ +MF = 1 +MF = -1 +SXBX +~540 kHz +~980 kHz +geffBZ +SXBX +SXBX +SXBX +(a) +(b) +FIG. S3. (a) Stark shifts for N = 1 in CaOH. The J = 1/2+ zero g-factor states are shown with a solid green line, while the +J = 3/2+ zero g-factor states are indicated with a dashed green line. All other levels are grayed out. A vertical dotted line +indicates the location of the J = 1/2+ zero g-factor crossing. (b) A zoomed in level diagram of the J = 1/2+ zero g-factor +hyperfine manifold. The bias field splitting geffBZ is not to scale. Transverse field couplings are shown with double sided +arrows, with blue (red) indicating negative (positive) SX matrix element. +correlated with the negative parity manifold. +Since many interactions increase in magnitude with larger N, the +overall electric field scale of the intermediate regime increases. Additionally, the robustness of zero g-factor states +also improves, with some pairs able to maintain ∆⟨MS⟩ = 0 for magnetic fields up to 40 G. These N = 2 pairs also +have non-zero eEDM sensitivity for a wide range of magnetic field values. +TRANSVERSE MAGNETIC FIELDS +Transverse Field Sensitivity +We now expand our discussion to include the effect of transverse magnetic fields. Their effects can by modeled by +adding BXSX and BY SY terms to the effective Hamiltonian, which have the selection rule ∆MF = ±1. For this +discussion, we focus on the level structure of the N = 1, J = 1/2+ manifold in CaOH near the zero g-factor crossing +at 60.5 V/cm in the absence of trap shifts, shown in Figure S3. We note if there were no nuclear spin I, the two zero +g-factor states would be MJ = ±1/2 states separated by ∆M = 1. In such a case these degenerate states would be +directly sensitive to transverse fields at first order, thereby reducing the g-factor suppression. +Due to the hyperfine structure from the nuclear spin of the H atom in CaOH, the degenerate MF = ±1 states in a +zero g-factor pair are coupled by second order transverse field interactions. These interactions are mediated via the +MF = 0± states, where ± denotes the upper or lower states. Using a Schrieffer–Wolff (aka Van-Vleck) transformation, +we can express the effective Hamiltonian matrix for second order coupling between the MF = ±1 states. We write +the states as |MF ⟩, and for convenience we take the transverse field to point along X: + +12 +H+1,−1 = −(gSµBBX)2 +�⟨−1|SX|0+⟩⟨0+|SX| + 1⟩ +∆E0+ ++ ⟨−1|SX|0−⟩⟨0−|SX| + 1⟩ +∆E0− +� +(3) +Here, ∆E0± is the energy difference of the MF = 0± levels from the MF = ±1 levels. Our model provides the following +values: ⟨0−|SX|+1⟩ = ⟨0−|SX|−1⟩ = −0.18, ⟨0+|SX|+1⟩ = −0.16, and ⟨0+|SX|−1⟩ = 0.16. The difference in sign is +a result of Clebsh-Gordon coefficient phases, and only the relative phase is relevant. We also have ∆E0+ = 0.98 MHz +and ∆E0− = −0.54 MHz. The combination of phases precludes the possibility of destructive interference. With these +parameters and defining g⊥ = H+1,−1/BX, then eqn. 3 evaluates to (gSµBBX)2(0.086/MHz) ≈ (0.68 MHz/G2)B2 +X. +Our model estimates the transverse sensitivity at BX ∼ 1 mG to be g⊥µB ∼ 7 × 10−4 MHz/G, of the same order as +the neglected nuclear and rotational Zeeman terms. The suppressed transverse field sensitivity bounds the magnitude +of BZ, which must be large enough to define a quantization axis for the spin, geffBZ ≫ g⊥B⊥. +Cancellation of transverse magnetic fields +When transverse magnetic fields are dominant, the electron will be quantized along the transverse axis and there +is minimal spin precession by the bias BZ field. The transverse coupling results in eigenstates given by (|MF = +1⟩±eiφ|MF = −1⟩)/ +√ +2, where the phase φ is set by the direction of ⃗B in the transverse plane. If φ = 0 or π, only one +of these states is bright to the ˆX-polarized state preparation microwaves, which means the initial state is stationary +under the transverse fields. For all other orientations, the transverse field causes spin precession with varying contrast, +depending on the specific value of φ. +We are able to use transverse spin precesion to measure and zero transverse fields to the mG level. We do so by +operating with minimal bias field BZ ≈ 0 and operating EZ near the zero g-factor crossing, such that geffBZ < g⊥B⊥. +We then apply a small transverse magnetic field to perform transverse spin precession. +Here, the dynamics are +dominated by the transverse fields rather than the Z fields. +We obtain field zeros by iteratively minimizing the +precession frequency by tuning the bias fields BX and BY . +IMPERFECT FIELD REVERSAL +We briefly present a systematic effect involving non-reversing fields in eEDM measurements with zero g-factor +states and discuss methods for its mitigation. The electric field dependence of geff can mimic an eEDM signal when +combined with other systematic effects, very much like in 3∆1 molecules [25, 26]. When the sign of EZ is switched, a +non-reversing electric field ENR will cause a g-factor difference of gNR = (dgeff/dEZ)ENR. This will give an additional +spin precession signal gNRBZ. By perfectly reversing BZ as well, this precession signal can be distinguished from a +true EDM signal. However, if there is also a non-reversing magnetic field BNR, there will still be a residual EDM signal +given by (dg/dE)ENRBNR. Using the measured slope of ∼0.03 (MHz/G)/(V/cm), and using conservative estimates +of ENR ∼ 1 mV/cm and BNR ∼ 1 µG, we obtain an estimate precession frequency of ∼30 µHz. While this is an order +of magnitude smaller than the statistical error for the current best eEDM measurement measurement [48], it is still +desirable to devise methods to reduce the effect further. +Performing eEDM measurements at different zero g-factor states can help suppress systematic errors resulting from +the above mechanism. For example, the N = 1, J = 3/2 zero crossing has a different magnitude for Σ, which can be +used to distinguish a true eEDM from a systematic effect. Both N = 1 crossings are only separated by ∼4 V/cm. +Furthermore, the zero g-factor states in N = 2− can also be used for systematic checks, as they additionally offer +different geff vs EZ slopes as well as different Σ values. +The N = 2− states can be populated directly by the +photon-cycling used to pump into the bending mode. +SPIN PRECESSION NEAR ZERO G-FACTOR +As discussed in the main text, the longest achievable coherence times occur at at combination of low effective g- +factors (which suppress δBZ decoherence) and low magnetic bias fields (which suppress δµeff decoherence). These low +g-factors and bias fields only very weakly enforce a quantization axis along Z, enhancing the potential for transverse +magnetic fields B⊥ to contribute. Such fields have the effect of (a) reducing the spin precession contrast and (b) + +13 +FIG. S4. Spin precession at EZ = 60.3 V/cm and BZ = 2 mG. The fit includes a 60 Hz time-varying magnetic field whose +amplitude and phase are measured with a magnetometer. The coherence time fits to 30 ms. +altering the observed precession frequency. To avoid these effects, the condition geffBZ > g⊥B⊥ must therefore be +satisfied. To achieve this, we zero the transverse magnetic fields by intentionally taking spin precession data at BZ ≈ 0 +and geff ≈ 0 while varying the transverse fields BX and BY . By minimizing the spin precession frequency as a function +of the transverse fields, we reduce B⊥ to approximately 1 mG. In addition, long-term drifts in the dc magnetic field +along all three axes are compensated by actively feeding back on the magnetic field as measured with a fluxgate +magnetometer. Under these conditions, at an electric field of 60.3 V/cm (corresponding to µeff = 0.02 MHz/G) and +a bias field of BZ ≈ 2 mG, we achieve a coherence time of 30 ms (Fig. S4). +At these very low bias fields, the molecules are also sensitive to 60 Hz magnetic field noise present in the unshielded +apparatus, whose amplitude is on the same order as BZ. Since the experiment is phase stable with respect to the AC +line frequency, this 60 Hz magnetic field fluctuation causes a time-dependent spin precession frequency. A fluxgate +magnetometer is used to measure the amplitude and phase of this 60 Hz field, which are then used as fixed parameters +in the fit shown in Figure S4. +[37] B. E. Sauer, J. Wang, and E. A. Hinds, Laser-rf double resonance spectroscopy of 174YbF in the X2Σ+ state: Spin-rotation, +hyperfine interactions, and the electric dipole moment, J. Chem. Phys. 105, 7412 (1996). +[38] C. S. Dickinson, J. A. Coxon, N. R. Walker, and M. C. L. Gerry, Fourier transform microwave spectroscopy of the 2Σ+ +ground states of YbX (X=F, Cl, Br): Characterization of hyperfine effects and determination of the molecular geometries, +J. Chem. Phys. 115, 6979 (2001). +[39] A. Petrov and A. Zakharova, Sensitivity of the YbOH molecule to P,T-odd effects in an external electric field, Phys. Rev. +A 105, L050801 (2022). +[40] J. M. Brown and A. Carrington, Rotational spectroscopy of diatomic molecules (Cambridge University Press, 2003). +[41] E. Hirota, High-Resolution Spectroscopy of Transient Molecules, Springer Series in Chemical Physics, Vol. 40 (Springer +Berlin Heidelberg, Berlin, Heidelberg, 1985). +[42] A. Merer and J. Allegretti, Rotational energies of linear polyatomic molecules in vibrationally degenerate levels of electronic +2Σ and 3Σ states, Canadian Journal of Physics 49, 2859 (1971). +[43] J. M. Brown, The rotational dependence of the Renner-Teller interaction: a new term in the effective Hamiltonian for +linear triatomic molecules in Π electronic states, Mol. Phys. 101, 3419 (2003). +[44] M. Li and J. A. Coxon, High-resolution analysis of the fundamental bending vibrations in the ˜A2Π and ˜ +X2Σ+ states of +caoh and caod: Deperturbation of Renner-Teller, spin-orbit and K-type resonance interactions, J. Chem. Phys. 102, 2663 +(1995). +[45] C. Scurlock, D. Fletcher, and T. Steimle, Hyperfine structure in the (0,0,0) ˜ +X2Σ+ state of CaOH observed by pump/probe +microwave-optical double resonance, J. Mol. Spectrosc. 159, 350 (1993). +[46] T. Steimle, D. Fletcher, K. Jung, and C. Scurlock, A supersonic molecular beam optical stark study of CaOH and SrOH, +J. Chem. Phys. 96, 2556 (1992). +[47] L. Caldwell and M. R. Tarbutt, Sideband cooling of molecules in optical traps, Phys. Rev. Res. 2, 013251 (2020). +[48] Z. Lasner, Order-of-magnitude-tighter bound on the electron electric dipole moment, Ph.D. thesis, Yale University (2019). + +Q +60.3 V/cm +0.44 +Spin Precession with 60 Hz Modulation +Fraction (au) +0.42 +0.4 +0.38 +0 +10 +20 +30 +40 +50 +60 +70 +Time (ms) \ No newline at end of file diff --git a/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/load_file.txt b/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a3b270b96af9dfd3d81d90e7fbad91ac3e49eb0 --- /dev/null +++ b/BNFAT4oBgHgl3EQfsB5-/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf,len=858 +page_content='Quantum Control of Trapped Polyatomic Molecules for eEDM Searches Lo¨ıc Anderegg,1, 2, ∗ Nathaniel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Vilas,1, 2 Christian Hallas,1, 2 Paige Robichaud,1, 2 Arian Jadbabaie,3 John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Doyle,1, 2 and Nicholas R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Hutzler3, † 1Department of Physics, Harvard University, Cambridge, MA 02138, USA 2Harvard-MIT Center for Ultracold Atoms, Cambridge, MA 02138, USA 3Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA (Dated: January 23, 2023) Ultracold polyatomic molecules are promising candidates for experiments in quantum science, quantum sensing, ultracold chemistry, and precision measurements of physics beyond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' A key, yet unrealized, requirement of these experiments is the ability to achieve full quantum control over the complex internal structure of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Here, we establish coherent control of individual quantum states in a polyatomic molecule, calcium monohydroxide (CaOH), and use these techniques to demonstrate a method for searching for the electron electric dipole moment (eEDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Optically trapped, ultracold CaOH molecules are prepared in a single quantum state, polarized in an electric field, and coherently transferred into an eEDM sensitive state where an electron spin precession measurement is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' To extend the coherence time of the measurement, we utilize eEDM sensitive states with tunable, near-zero magnetic field sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The spin precession coher- ence time is limited by AC Stark shifts and uncontrolled magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' These results establish a path for eEDM searches with trapped polyatomic molecules, towards orders-of-magnitude improved experimental sensitivity to time-reversal-violating physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The rich structure of polyatomic molecules makes them an appealing platform for experiments in quantum sci- ence [1–4], ultracold chemistry [5], and precision mea- surements [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Key to this structure is the presence of near-degenerate states of opposite parity, which allow the molecules to be easily polarized in the laboratory frame with the application of a small electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Such states are a novel resource, generic among polyatomic molecules while rare in diatomics, that may be useful for applications such as analog simulation of quantum magnetism models [1, 2] or for realizing switchable inter- actions and long-lived qubit states for quantum comput- ing [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Additionally, the parity-doublet states in trapped polyatomic molecules are expected to be an invaluable tool for systematic error rejection in precision measure- ments of physics beyond the Standard Model (BSM) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' To date, several species of polyatomic molecules have been laser cooled and/or trapped at ultracold temper- atures [11–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' One powerful avenue for tabletop BSM searches is probing for the electric dipole moment of the electron (eEDM) [18–22], de, which violates time-reversal (T) symmetry and is predicted by many BSM theories to be orders of magnitude larger than the Standard Model prediction [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Current state-of-the-art eEDM ex- periments are broadly sensitive to T-violating physics at energies much greater than 1 TeV [23–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' All such ex- periments use Ramsey spectroscopy to measure an en- ergy shift due to the interaction of the electron with the large electric field present inside a polarized molecule [24– 27, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Molecular beam experiments have achieved high statistical sensitivity by measuring a large number of molecules over a ≈ 1 ms coherence time [24, 25], while molecular ion-based experiments have used long Ram- sey interrogation times (≈ 1 s) though with lower num- bers [26, 27, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Measurements with trapped neutral polyatomic molecules can potentially combine the best features of each approach to achieve orders-of-magnitude improved statistical sensitivity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' In this Report, we demonstrate full quantum control over the internal states of a trapped polyatomic molecule in a vibrational bending mode with high polarizability in small electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The method starts with prepar- ing ultracold, optically trapped molecules in a single hy- perfine level, after which a static electric field is applied to polarize the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The strength of the polar- izing electric field is tuned to obtain near-zero g-factor spin states, which have strongly suppressed sensitivity to magnetic field noise while retaining eEDM sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Microwave pulses are applied to create a coherent super- position of these zero g-factor spin states that precesses under the influence of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The precession phase is then read out by a combination of microwave pulses and optical cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' We observe spin precession over a range of electric and magnetic fields and characterize the current limitations to the coherence time of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' With readily attainable experimental parameters, coherence times on the order of the state lifetime (>100 ms) could be realisti- cally achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' We therefore realize the key components of an eEDM measurement in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Although the light mass of CaOH precludes a competitive eEDM mea- surement [30], the protocol demonstrated here is directly transferable to heavier laser-cooled alkaline earth mono- hydroxides with identical internal level structures, such as SrOH, YbOH, and RaOH, which have significantly en- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='08656v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='atom-ph] 20 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' (a) A geometric picture of the bending molecule at the zero g-factor crossing, showing the electron spin (⃗S) has a finite projection on the molecule axis (ˆn), giving eEDM sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' However, the electron spin (⃗S) is orthogonal to the magnetic field ( ⃗B), resulting in suppressed magnetic field sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' (b) The magnetic sensitivity (upper plot) and eEDM sensitivity (lower plot) for a pair of zero g-factor states (N = 1, J = 1/2+, F = 1, MF = ±1) are shown as a function of the applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' (c) Experimental sequence to prepare the eEDM sensitive state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' First, the molecules are pumped into a single quantum state (N = 1, J = 1/2−, F = 0) with a combination of microwave drives and optical pumping (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Next, a microwave π-pulse drives the molecules into the N = 2, J = 3/2−, F = 2, MF = 0 state (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Lastly, the eEDM measurement state is prepared as a coherent superposition of the N = 1, J = 1/2−, F = 1 MF = ±1 states with a microwave π-pulse (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The states which are optically detectable with the detection light are shown in black, while those not addressed by the detection light are in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' hanced sensitivity to the eEDM [6, 11, 12, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' In eEDM measurements with polarized molecules, the electron spin ⃗S precesses under the influence of an ex- ternal magnetic field BZ and the internal electric field of the molecule, Eeff, which can be large due to relativistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Time evolution is described by the Hamiltonian H = gSµBBZ ⃗S · ˆZ − deEeff⃗S · ˆn = gSµBBZMS − deEeffΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' (1) Here, gS ≈ 2 is the electron spin g-factor, µB is the Bohr magneton, BZ points along the lab ˆZ axis, and the internal field Eeff points along the molecule’s inter- nuclear axis ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' We define the quantities MS = ⃗S · ˆZ and Σ = ⃗S · ˆn to describe the electron’s magnetic sen- sitivity and EDM sensitivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The effect of the eEDM can be isolated by switching the orientation of the applied magnetic field or, alternatively, by switching internal states to change the sign of MS or Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Perform- ing both switches is a powerful technique for suppressing systematic errors [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Current EDM bounds rely on specific states in di- atomic molecules that have an unusually small g-factor, reducing sensitivity to stray magnetic fields [24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' However, CaOH, like other laser-coolable molecules with structure amenable to eEDM searches [6, 31–33], has a single valence electron, which results in large mag- netic g-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' In this work, we engineer reduced mag- netic sensitivity by using an applied electric field EZ to tune MS to a zero-crossing, while maintaining signifi- cant eEDM sensitivity Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' This technique is generic to polyatomic molecules with parity-doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Details of a specific M = ±1 pair of zero g-factor states are shown in Figure 1 (a)-(b), with further information in the Sup- plemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Sensitivity to transverse magnetic fields is also suppressed in these zero g-factor states (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The experiment begins with laser-cooled CaOH molecules loaded from a magneto-optical trap [14] into an optical dipole trap (ODT) formed by a 1064 nm laser beam with a 25 µm waist size, as described in previous work [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The ODT is linearly polarized and its polar- ization vector ⃗ϵODT defines the ˆZ axis, along which we also apply magnetic and electric fields, ⃗B = BZ ˆZ and ⃗E = EZ ˆZ, respectively, as depicted in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' We first non-destructively image the molecules in the ODT for 10 ms as normalization against variation in the num- ber of trapped molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' The molecules are then opti- cally pumped into the N = 1− levels of the � X2Σ+(010) vibrational bending mode [15] (Figure 1(c)), and the trap depth is adiabatically lowered by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='5× to reduce the effect of AC Stark shifts from the trap light and to lower the temperature of the molecules to 34 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' Any molecules that were not pumped into N = 1− levels of the bending (c) A21(010)2- (a) (b) 1/2 0,1+ μb(Ms) (MHz/G) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='2 623 nm 2 M= E,B,EoDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' 0 (Z)= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content=' 0 1 M=+1 3/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='2 2- n Ca 40 GHz (010)+3zX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='6 Sensitivity (22) Relative EDM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='4 M=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFAT4oBgHgl3EQfsB5-/content/2301.08656v1.pdf'} +page_content='2 0+ 1/2