Journal of Symbolic Computation最新文献

{"title":"Logarithmic Voronoi cells for Gaussian models","authors":"Yulia Alexandr ,&nbsp;Serkan Hoşten","doi":"10.1016/j.jsc.2023.102256","DOIUrl":"10.1016/j.jsc.2023.102256","url":null,"abstract":"<div><p>We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42943658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nash conditional independence curve","authors":"Irem Portakal ,&nbsp;Javier Sendra–Arranz","doi":"10.1016/j.jsc.2023.102255","DOIUrl":"10.1016/j.jsc.2023.102255","url":null,"abstract":"<div><p><span>We study the Spohn conditional independence (CI) variety </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> of an <em>n</em>-player game <em>X</em><span> for undirected graphical models on </span><em>n</em><span> binary random variables consisting of one edge. For a generic game, we show that </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span> is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety </span><span><math><msup><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and we give an explicit formula for its degree and genus. We prove two universality theorems for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span>: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> defined by at most <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span> polynomials is isomorphic to an affine open subset of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> for some game <em>X</em>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44443031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rational dual certificates for weighted sums-of-squares polynomials with boundable bit size","authors":"Maria M. Davis,&nbsp;Dávid Papp","doi":"10.1016/j.jsc.2023.102254","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102254","url":null,"abstract":"<div><p>In <span>Davis and Papp (2022)</span><span><span><span>, the authors introduced the concept of dual certificates of (weighted) sum-of-squares polynomials, which are vectors from the dual cone of weighted sums of squares (WSOS) polynomials that can be interpreted as </span>nonnegativity<span> certificates. This initial theoretical work showed that for every polynomial in the interior of a WSOS cone, there exists a rational dual certificate proving that the polynomial is WSOS. In this article, we analyze the complexity of rational dual certificates of WSOS polynomials by bounding the bit sizes of integer dual certificates as a function of parameters such as the degree and the number of variables of the polynomials, or their distance from the boundary of the cone. After providing a general bound, we explore several special cases, such as univariate polynomials nonnegative over the real line or a </span></span>bounded interval, represented in different commonly used bases. We also provide an algorithm which runs in rational arithmetic and computes a rational certificate with boundable bit size for a WSOS lower bound of the input polynomial.</span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49727690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the computation of rational solutions of linear integro-differential equations with polynomial coefficients","authors":"Moulay Barkatou,&nbsp;Thomas Cluzeau","doi":"10.1016/j.jsc.2023.102252","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102252","url":null,"abstract":"<div><p><span><span>We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using </span>partial fraction decomposition<span>, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our </span></span><span>Maple</span> implementation.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49727689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isolating all the real roots of a mixed trigonometric-polynomial","authors":"Rizeng Chen ,&nbsp;Haokun Li ,&nbsp;Bican Xia ,&nbsp;Tianqi Zhao ,&nbsp;Tao Zheng","doi":"10.1016/j.jsc.2023.102250","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102250","url":null,"abstract":"<div><p>Mixed trigonometric-polynomials (MTPs) are functions of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>,</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo>)</mo></math></span> where <em>f</em> is a trivariate polynomial with rational coefficients, and the argument <em>x</em> ranges over the reals. In this paper, an algorithm “isolating” all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval <span><math><mo>[</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>]</mo></math></span> while the other consists of countably many roots in <span><math><mi>R</mi><mo>﹨</mo><mo>[</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>]</mo></math></span>. For the roots in <span><math><mo>[</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>]</mo></math></span>, the algorithm returns isolating intervals and corresponding multiplicities while for those greater than <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, it returns finitely many mutually disjoint small intervals <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊂</mo><mo>[</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></math></span>, integers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span> and multisets of root multiplicity <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup></math></span> such that any root <span><math><mo>&gt;</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> is in the set <span><math><mo>(</mo><msub><mrow><mo>∪</mo></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mo>∪</mo></mrow><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></msub><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>)</mo><mo>)</mo></math></span> and any interval <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mn>2</mn><mi>k</mi><mi>π</mi><mo>⊂</mo><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> contains exactly <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> distinct roots with multiplicities <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mr","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49767672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deformations of half-canonical Gorenstein curves in codimension four","authors":"Patience Ablett ,&nbsp;Stephen Coughlan","doi":"10.1016/j.jsc.2023.102251","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102251","url":null,"abstract":"<div><p>Recent work of Ablett (<span>2021</span>) and Kapustka, Kapustka, Ranestad, Schenck, Stillman and Yuan (<span>2021</span>) outlines a number of constructions for singular Gorenstein codimension four varieties. Earlier work of Coughlan, Gołȩbiowski, Kapustka and Kapustka (<span>2016</span>) details a series of nonsingular Gorenstein codimension four constructions with different Betti tables. In this paper we exhibit a number of flat deformations between Gorenstein codimension four varieties in the same Hilbert scheme, realising many of the singular varieties as specialisations of the earlier nonsingular varieties.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic optimization of sequential decision problems","authors":"Mareike Dressler ,&nbsp;Marina Garrote-López ,&nbsp;Guido Montúfar ,&nbsp;Johannes Müller ,&nbsp;Kemal Rose","doi":"10.1016/j.jsc.2023.102241","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102241","url":null,"abstract":"<div><p><span><span>We study the optimization of the expected long-term reward in finite partially observable Markov decision processes<span> over the set of stationary stochastic policies. In the case of deterministic observations, also known as state aggregation, the problem is equivalent to optimizing a linear objective subject to </span></span>quadratic constraints<span><span>. We characterize the feasible set of this problem as the intersection of a product of affine varieties of rank one matrices and a polytope. Based on this description, we obtain bounds on the number of critical points of the </span>optimization problem. Finally, we conduct experiments in which we solve the KKT equations or the </span></span>Lagrange equations<span> over different boundary components of the feasible set, and we compare the result to the theoretical bounds and to other constrained optimization methods.</span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Skew-polynomial-sparse matrix multiplication","authors":"Qiao-Long Huang ,&nbsp;Ke Ye ,&nbsp;Xiao-Shan Gao","doi":"10.1016/j.jsc.2023.102240","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102240","url":null,"abstract":"<div><p>Based on the observation that <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for <span><math><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> matrix multiplication over <span><math><mi>Q</mi></math></span>, where <em>p</em> is a prime number. The algorithm has complexity <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>ω</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>T</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is a parameter determined by the skew-polynomial-sparsity of input matrices and <em>ω</em><span> is the asymptotic exponent of matrix multiplication<span>. Here a matrix is skew-polynomial-sparse if its corresponding skew polynomial is sparse. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity </span></span><span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>∼</mo></mrow></msup><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>ω</mi><mo>−</mo><mn>2</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ν</mi></mrow></mfrac><mo>)</mo></math></span>, where <span><math><mi>t</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span> is the skew-polynomial-sparsity of the product and <em>ν</em><span> is the probability parameter. The main feature of the algorithms is the acceleration for matrix multiplication if the input matrices or their products are skew-polynomial-sparse.</span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49756347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Standard generators of finite fields and their cyclic subgroups","authors":"Frank Lübeck","doi":"10.1016/j.jsc.2022.11.001","DOIUrl":"https://doi.org/10.1016/j.jsc.2022.11.001","url":null,"abstract":"<div><p>We define standard constructions of finite fields, and standard generators of (multiplicative) cyclic subgroups in these fields.</p><p>The motivation is to provide a substitute for Conway polynomials which can be used by various software packages and in collections of mathematical data which involve finite fields.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49760253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The factorial-basis method for finding definite-sum solutions of linear recurrences with polynomial coefficients","authors":"Antonio Jiménez-Pastor ,&nbsp;Marko Petkovšek","doi":"10.1016/j.jsc.2022.11.002","DOIUrl":"https://doi.org/10.1016/j.jsc.2022.11.002","url":null,"abstract":"<div><p><span>The problem of finding a nonzero solution of a linear recurrence </span><span><math><mi>L</mi><mi>y</mi><mo>=</mo><mn>0</mn></math></span> with polynomial coefficients where <em>y</em><span> has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of </span><span>Chen and Kauers (2017, Sec. 8)</span>, has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a <em>quasi-triangular</em>, <span><em>shift-compatible </em><em>factorial</em><em> basis</em></span> <span><math><mi>B</mi><mo>=</mo><msubsup><mrow><mo>〈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>〉</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span><span> of the polynomial space </span><span><math><mi>K</mi><mo>[</mo><mi>n</mi><mo>]</mo></math></span> over a field <span><math><mi>K</mi></math></span> of characteristic zero, computes a recurrence satisfied by the coefficient sequence <span><math><mi>c</mi><mo>=</mo><msubsup><mrow><mo>〈</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>〉</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> of the solution <span><math><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> (where, thanks to the quasi-triangularity of <span><math><mi>B</mi></math></span>, the sum on the right terminates for each <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>). More generally, if <span><math><mi>B</mi></math></span> is <em>m</em>-sieved for some <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span>, our algorithm computes a system of <em>m</em> recurrences satisfied by the <em>m</em>-sections of the coefficient sequence <em>c</em>. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of <span><math><mi>L</mi><mi>y</mi><mo>=</mo><mn>0</mn></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}