Pub Date : 2024-03-01Epub Date: 2023-08-03DOI: 10.1002/rsa.21173
Lucas Aragão, Maurício Collares, João Pedro Marciano, Taísa Martins, Robert Morris
The set-coloring Ramsey number is defined to be the minimum such that if each edge of the complete graph is assigned a set of colors from , then one of the colors contains a monochromatic clique of size . The case is the usual -color Ramsey number, and the case was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that if is bounded away from 0 and 1. In the range , however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine up to polylogarithmic factors in the exponent for essentially all , , and .
集合着色拉姆齐数 Rr,s(k)的定义是：如果完整图 Kn 的每条边都从 {1,...,r}中分配了一组 s 种颜色，则其中一种颜色包含大小为 k 的单色小块，那么最小 n 的集合着色拉姆齐数 Rr,s(k)。康伦、福克斯、何、穆巴伊、苏克和韦斯特拉特直到最近才首次获得关于一般 s 的重要结果，他们证明了如果 s/r 在 0 和 1 之间有界，则 Rr,s(k)=2Θ(kr)。在本说明中，我们引入了一种新的（随机）着色，并用它来确定 Rr,s(k)，基本上所有 r、s 和 k 的指数都可以达到多对数因子。
{"title":"A lower bound for set-coloring Ramsey numbers.","authors":"Lucas Aragão, Maurício Collares, João Pedro Marciano, Taísa Martins, Robert Morris","doi":"10.1002/rsa.21173","DOIUrl":"10.1002/rsa.21173","url":null,"abstract":"<p><p>The set-coloring Ramsey number <math><mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math> is defined to be the minimum <math><mrow><mrow><mi>n</mi></mrow></mrow></math> such that if each edge of the complete graph <math><mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mrow></math> is assigned a set of <math><mrow><mrow><mi>s</mi></mrow></mrow></math> colors from <math><mrow><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>r</mi><mo>}</mo></mrow></mrow></math>, then one of the colors contains a monochromatic clique of size <math><mrow><mrow><mi>k</mi></mrow></mrow></math>. The case <math><mrow><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></mrow></math> is the usual <math><mrow><mrow><mi>r</mi></mrow></mrow></math>-color Ramsey number, and the case <math><mrow><mrow><mi>s</mi><mo>=</mo><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mrow></math> was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general <math><mrow><mrow><mi>s</mi></mrow></mrow></math> were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that <math><mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>Θ</mi><mo>(</mo><mi>k</mi><mi>r</mi><mo>)</mo></mrow></msup></mrow></mrow></math> if <math><mrow><mrow><mi>s</mi><mo>/</mo><mi>r</mi></mrow></mrow></math> is bounded away from 0 and 1. In the range <math><mrow><mrow><mi>s</mi><mo>=</mo><mi>r</mi><mo>-</mo><mi>o</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></math>, however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine <math><mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math> up to polylogarithmic factors in the exponent for essentially all <math><mrow><mrow><mi>r</mi></mrow></mrow></math>, <math><mrow><mrow><mi>s</mi></mrow></mrow></math>, and <math><mrow><mrow><mi>k</mi></mrow></mrow></math>.</p>","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10952192/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76773187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pedro Araújo, Jan Hladký, Eng Keat Hng, Matas Šileikis
Abstract Flip processes, introduced in [ Garbe, Hladký, Šileikis, Skerman: From flip processes to dynamical systems on graphons ], are a class of random graph processes defined using a rule which is just a function from all labelled graphs of a fixed order into itself. The process starts with an arbitrary given ‐vertex graph . In each step, the graph is obtained by sampling random vertices of and replacing the induced graph by . Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, ‘extremist flip processes’ (in which is either a clique or an independent set, depending on whether has less or more than half of all potential edges), and ‘ignorant flip processes’ in which the output does not depend on .
{"title":"Prominent examples of flip processes","authors":"Pedro Araújo, Jan Hladký, Eng Keat Hng, Matas Šileikis","doi":"10.1002/rsa.21192","DOIUrl":"https://doi.org/10.1002/rsa.21192","url":null,"abstract":"Abstract Flip processes, introduced in [ Garbe, Hladký, Šileikis, Skerman: From flip processes to dynamical systems on graphons ], are a class of random graph processes defined using a rule which is just a function from all labelled graphs of a fixed order into itself. The process starts with an arbitrary given ‐vertex graph . In each step, the graph is obtained by sampling random vertices of and replacing the induced graph by . Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, ‘extremist flip processes’ (in which is either a clique or an independent set, depending on whether has less or more than half of all potential edges), and ‘ignorant flip processes’ in which the output does not depend on .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135540100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).
{"title":"Sharp thresholds in adaptive random graph processes","authors":"Calum MacRury, Erlang Surya","doi":"10.1002/rsa.21197","DOIUrl":"https://doi.org/10.1002/rsa.21197","url":null,"abstract":"Abstract The ‐process is a single player game in which the player is initially presented the empty graph on vertices. In each step, a subset of edges is independently sampled according to a distribution . The player then selects one edge from , and adds to its current graph. For a fixed monotone increasing graph property , the objective of the player is to force the graph to satisfy in as few steps as possible. The ‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for in the ‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020).","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marcelo Campos, Maurício Collares, Guilherme Oliveira Mota
Abstract For every , we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length . This solves a conjecture of Kohayakawa, Morris and the last two authors.
{"title":"Counting orientations of random graphs with no directed <i>k</i>‐cycles","authors":"Marcelo Campos, Maurício Collares, Guilherme Oliveira Mota","doi":"10.1002/rsa.21196","DOIUrl":"https://doi.org/10.1002/rsa.21196","url":null,"abstract":"Abstract For every , we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length . This solves a conjecture of Kohayakawa, Morris and the last two authors.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider a well‐known model of random directed acyclic graphs of order , obtained by recursively adding vertices, where each new vertex has a fixed outdegree and the endpoints of the edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number of vertices that are descendants of . We show that converges in distribution; the limit distribution is, up to a constant factor, given by the th root of a Gamma distributed variable with distribution . When , the limit distribution can also be described as a chi distribution . We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.
{"title":"The number of descendants in a random directed acyclic graph","authors":"Svante Janson","doi":"10.1002/rsa.21195","DOIUrl":"https://doi.org/10.1002/rsa.21195","url":null,"abstract":"Abstract We consider a well‐known model of random directed acyclic graphs of order , obtained by recursively adding vertices, where each new vertex has a fixed outdegree and the endpoints of the edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number of vertices that are descendants of . We show that converges in distribution; the limit distribution is, up to a constant factor, given by the th root of a Gamma distributed variable with distribution . When , the limit distribution can also be described as a chi distribution . We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135539423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
António Girão, Freddie Illingworth, Alex Scott, David R. Wood
Abstract We prove that the vertices of every ‐uniform hypergraph with maximum degree may be colored with colors such that each vertex is in at most monochromatic edges. This result, which is best possible up to the value of the constant , generalizes the classical result of Erdős and Lovász who proved the case.
{"title":"Defective coloring of hypergraphs","authors":"António Girão, Freddie Illingworth, Alex Scott, David R. Wood","doi":"10.1002/rsa.21190","DOIUrl":"https://doi.org/10.1002/rsa.21190","url":null,"abstract":"Abstract We prove that the vertices of every ‐uniform hypergraph with maximum degree may be colored with colors such that each vertex is in at most monochromatic edges. This result, which is best possible up to the value of the constant , generalizes the classical result of Erdős and Lovász who proved the case.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136261739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $mathbb{R}^3$ whose intersection graph is triangle-free of chromatic number $Omega(n^{1/15})$. This improves the previously best known bound $Omega(loglog n)$ by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of $n$ points in $mathbb{R}^d$, whose Delaunay graph with respect to axis-parallel boxes has independence number at most $ncdot (log n)^{-(d-1)/2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.
{"title":"Coloring lines and Delaunay graphs with respect to boxes","authors":"Tomon, István","doi":"10.1002/rsa.21193","DOIUrl":"https://doi.org/10.1002/rsa.21193","url":null,"abstract":"The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $mathbb{R}^3$ whose intersection graph is triangle-free of chromatic number $Omega(n^{1/15})$. This improves the previously best known bound $Omega(loglog n)$ by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number. (ii) Second, we construct a set of $n$ points in $mathbb{R}^d$, whose Delaunay graph with respect to axis-parallel boxes has independence number at most $ncdot (log n)^{-(d-1)/2+o(1)}$. This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tao Jiang, Shoham Letzter, Abhishek Methuku, Liana Yepremyan
Abstract We show that for every integer and large , every properly edge‐colored graph on vertices with at least edges contains a rainbow subdivision of . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.
{"title":"Rainbow subdivisions of cliques","authors":"Tao Jiang, Shoham Letzter, Abhishek Methuku, Liana Yepremyan","doi":"10.1002/rsa.21186","DOIUrl":"https://doi.org/10.1002/rsa.21186","url":null,"abstract":"Abstract We show that for every integer and large , every properly edge‐colored graph on vertices with at least edges contains a rainbow subdivision of . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135113115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if .
本文研究了一种新的离散树及其分支过程，我们称之为厄朗加权树(EWT)。EWT作为La and Kabkab提出的随机图模型的局部弱极限出现，互联网数学，11 (2015)，no。6, 528 - 554。与众所周知的随机图模型的局部弱极限相比，EWT具有相互依赖的结构。特别是，它的顶点编码了一个具有不可数多类型的多类型分支过程。我们推导了EWT的主要性质，如灭绝概率、增长率等。我们证明了消光概率是算子的最小不动点。然后，我们从点过程的角度分析增长率算子。导出了生长算子的Krein-Rutman特征值和相应的特征函数，并证明了消光的概率等于1当且仅当。
{"title":"The Erlang weighted tree, a new branching process","authors":"Mehrdad Moharrami, Vijay Subramanian, Mingyan Liu, Rajesh Sundaresan","doi":"10.1002/rsa.21180","DOIUrl":"https://doi.org/10.1002/rsa.21180","url":null,"abstract":"Abstract In this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .
{"title":"Spatial mixing and the random‐cluster dynamics on lattices","authors":"Reza Gheissari, Alistair Sinclair","doi":"10.1002/rsa.21191","DOIUrl":"https://doi.org/10.1002/rsa.21191","url":null,"abstract":"Abstract An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high‐temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random‐cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this article, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random‐cluster dynamics in finite subsets of for general . These imply a host of new results, including optimal mixing for the random cluster dynamics on torii and boxes on vertices in at all high temperatures and at sufficiently low temperatures, and for large values of quasi‐polynomial (or quasi‐linear when ) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst‐case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on for general .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135616043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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