Discrete Applied Mathematics最新文献

{"title":"Arc-disjoint in- and out-branchings in semicomplete split digraphs","authors":"Jiangdong Ai ,&nbsp;Yiming Hao ,&nbsp;Zhaoxiang Li ,&nbsp;Qi Shao","doi":"10.1016/j.dam.2025.05.037","DOIUrl":"10.1016/j.dam.2025.05.037","url":null,"abstract":"<div><div>An <em>out-tree (in-tree)</em> is an oriented tree where every vertex except one, called the <em>root</em>, has in-degree (out-degree) one. An <em>out-branching</em> <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> <em>(in-branching</em> <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span><em>)</em> of a digraph <span><math><mi>D</mi></math></span> is a spanning out-tree (in-tree) rooted at <span><math><mi>u</mi></math></span>. A <em>good</em> <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span><em>-pair</em> in <span><math><mi>D</mi></math></span> is a pair of branchings <span><math><mrow><msubsup><mrow><mi>B</mi></mrow><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>−</mo></mrow></msubsup></mrow></math></span> which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A <em>semicomplete split digraph</em> is a digraph where the vertex set is the disjoint union of two non-empty sets, <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, such that <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is an independent set, the subdigraph induced by <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is semicomplete, and every vertex in <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is adjacent to every vertex in <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. In this paper, we prove that every 2-arc-strong semicomplete split digraph <span><math><mi>D</mi></math></span> contains a good <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math></span>-pair for any choice of vertices <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></math></span> of <span><math><mi>D</mi></math></span>, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024].</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 259-268"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144298868","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}

{"title":"Laplacian pair state transfer in Q-graph","authors":"Ming Jiang ,&nbsp;Xiaogang Liu ,&nbsp;Jing Wang","doi":"10.1016/j.dam.2025.05.046","DOIUrl":"10.1016/j.dam.2025.05.046","url":null,"abstract":"<div><div>In 2018, Chen and Godsil proposed the concept of Laplacian perfect pair state transfer which is a brilliant generalization of Laplacian perfect state transfer. Studying Laplacian pair state transfer will provide a theoretical foundation for constructing quantum communication networks capable of quantum state transfer. In this paper, we study the existence of Laplacian perfect pair state transfer in the Q-graph of an <span><math><mi>r</mi></math></span>-regular graph for <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. By combining the spectral decomposition of the graph with the Laplacian eigenvalue support of pair state, we prove that the Q-graph of an <span><math><mi>r</mi></math></span>-regular graph does not have Laplacian perfect pair state transfer when <span><math><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></math></span> is prime or a power of 2. By contrast, we also give sufficient conditions for Q-graph to have Laplacian pretty good pair state transfer. The approach used in this paper can effectively verify the existence of Laplacian perfect (or pretty good) pair state transfer in other families of graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 239-258"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144298867","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}

{"title":"On open-separating dominating codes in graphs","authors":"Dipayan Chakraborty ,&nbsp;Annegret K. Wagler","doi":"10.1016/j.dam.2025.05.045","DOIUrl":"10.1016/j.dam.2025.05.045","url":null,"abstract":"<div><div>Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is to choose a suitable dominating set <span><math><mi>C</mi></math></span> of a graph <span><math><mi>G</mi></math></span> which is also separating in the sense that the neighborhoods of any two distinct vertices of <span><math><mi>G</mi></math></span> have distinct intersections with <span><math><mi>C</mi></math></span>. Such a dominating and separating set <span><math><mi>C</mi></math></span> of a graph is often referred to as a <em>code</em> in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called <em>open-separating dominating code</em>, or <em><span>OD</span>-code</em> for short, is a dominating set and uses open neighborhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of <span>OD</span>-codes. Due to the emergence of a close and yet difficult to establish relation of the <span>OD</span>-code with another well-studied code in the literature called open (neighborhood)-locating dominating code (referred to as the <em>open-separating total-dominating code</em> and abbreviated as <em><span>OTD</span>-code</em> in this paper), we compare the two codes on various graph families. Finally, we also provide an equivalent reformulation of the problem of finding <span>OD</span>-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with <span>OD</span>-codes, again in relation to <span>OTD</span>-codes of some graph families already studied in this context.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 215-238"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144298869","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}

{"title":"Infinite grid exploration with synchronous myopic robots without chirality","authors":"Quentin Bramas ,&nbsp;Pascal Lafourcade ,&nbsp;Stéphane Devismes","doi":"10.1016/j.dam.2025.05.031","DOIUrl":"10.1016/j.dam.2025.05.031","url":null,"abstract":"<div><div>In this paper, we consider the exploration of an infinite grid by a swarm of fully-synchronous robots with weak capabilities: they are disoriented, opaque, do not communicate explicitely, have limited visibility, and cannot occupy the same position at the same time.</div><div>Our first result shows that, in this context, minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to our exploration problem that is optimal <em>w.r.t.</em> both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. More precisely, the first algorithm solves the problem using eight oblivious robots under visibility range two (this visibility being optimal when considering oblivious robots), and the second algorithm solves the problem under visibility range one using six robots and two colors (which is optimal under this visibility range). Finally, we also tackle the optimality in terms of number of robots. According to the lower bound given in Bramas et al. (2020), we propose an algorithm working with a minimum number of robots (five) under visibility range one. This latter uses twelve colors and also guarantees that nodes are visited infinitely often.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 193-214"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144298870","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}

{"title":"Fairness in graph-theoretical optimization problems","authors":"Christopher Hojny,&nbsp;Frits Spieksma,&nbsp;Sten Wessel","doi":"10.1016/j.dam.2025.05.026","DOIUrl":"10.1016/j.dam.2025.05.026","url":null,"abstract":"<div><div>There is arbitrariness in optimum solutions of graph-theoretic problems that can give rise to unfairness. Incorporating fairness in such problems, however, can be done in multiple ways. For instance, fairness can be defined on an individual level, for individual vertices or edges of a given graph, or on a group level. In this work, we analyze in detail two individual-fairness measures that are based on finding a probability distribution over the set of solutions. One measure guarantees uniform fairness, i.e., entities have equal chance of being part of the solution when sampling from this probability distribution. The other measure maximizes the minimum probability for every entity of being selected in a solution. In particular, we reveal that computing these individual-fairness measures is in fact equivalent to computing the fractional covering number and the fractional partitioning number of a hypergraph. In addition, we show that for a general class of problems that we classify as independence systems, these two measures coincide. We also analyze group fairness and how this can be combined with the individual-fairness measures. Finally, we establish the computational complexity of determining group-fair solutions for a variant of the matching problem.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290572","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}

{"title":"The resistance distance of a dual number weighted graph","authors":"Yu Li,&nbsp;Lizhu Sun,&nbsp;Changjiang Bu","doi":"10.1016/j.dam.2025.05.041","DOIUrl":"10.1016/j.dam.2025.05.041","url":null,"abstract":"<div><div>For a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, assigning each edge <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math></span> a weight of a dual number <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>e</mi></mrow></msub><mi>ɛ</mi></mrow></math></span>, the weighted graph <span><math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></math></span> is called a dual number weighted graph, where <span><math><mrow><mo>−</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>e</mi></mrow></msub></mrow></math></span> can be regarded as the perturbation of the unit resistor on the edge <span><math><mi>e</mi></math></span> of <span><math><mi>G</mi></math></span>. For a connected dual number weighted graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>, we give some expressions and block representations of generalized inverses of the Laplacian matrix of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>. We give the perturbation bounds for the resistance distance and Kirchhoff index of <span><math><mi>G</mi></math></span>. In particular, when only the edge <span><math><mrow><mi>e</mi><mo>=</mo><mrow><mo>{</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>}</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices <span><math><mi>i</mi></math></span> and <span><math><mi>j</mi></math></span> of <span><math><mi>G</mi></math></span>, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 154-165"},"PeriodicalIF":1.0,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144289075","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}

{"title":"On the size of immune sets in the k-PULL infection model","authors":"Josep Fàbrega ,&nbsp;Xavier Marcote ,&nbsp;Xavier Muñoz","doi":"10.1016/j.dam.2025.05.047","DOIUrl":"10.1016/j.dam.2025.05.047","url":null,"abstract":"<div><div>This paper addresses immune sets in graphs, which are subsets of nodes that remain unaffected during the spread of influence, failure, or infection. The specific propagation model examined is the <span><math><mi>k</mi></math></span>-PULL infection rule, also referred to as bootstrap percolation. Studying immune sets offers important insights into the structural vulnerabilities and defensive capabilities of networks. In particular, we establish upper bounds for the size of minimal <span><math><mi>k</mi></math></span>-immune sets in graphs with a given maximum degree. Additionally, we focus on the <span><math><mi>k</mi></math></span>-immune number of a graph, defined as the minimum number of vertices in a <span><math><mi>k</mi></math></span>-immune set, and we derive bounds for this parameter. Lastly, we investigate the <span><math><mi>k</mi></math></span>-immune number of the Cartesian product of two graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 166-177"},"PeriodicalIF":1.0,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144289074","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}

{"title":"Constant time enumeration of weighted trees","authors":"Mengze Qian,&nbsp;Ryuhei Uehara","doi":"10.1016/j.dam.2025.05.043","DOIUrl":"10.1016/j.dam.2025.05.043","url":null,"abstract":"<div><div>This study specifically addresses the enumeration problem of the class of weighted trees. A weighted tree is a rooted unordered tree, and each vertex in a weighted tree is assigned an integer weight. The task is to enumerate the weighted trees. Given a tree <span><math><mi>T</mi></math></span> and a weight <span><math><mi>w</mi></math></span>, the objective is to enumerate all weighted trees that share the same tree structure as <span><math><mi>T</mi></math></span> and have a total weight of <span><math><mi>w</mi></math></span>. Note that the weight assigned to each vertex must be non-negative. The algorithm utilizes reverse search and enumerates each weighted tree in a constant amortized time. By combining our findings with the enumeration of the class of rooted trees, we can efficiently enumerate every weighted tree with a maximum of <span><math><mi>n</mi></math></span> vertices and a total weight of <span><math><mi>w</mi></math></span> for any given positive inputs <span><math><mi>n</mi></math></span> and <span><math><mi>w</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 129-138"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272534","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}

{"title":"The number of nonisomorphic nonorientable 6-gonal embeddings of complete graphs","authors":"Vladimir P. Korzhik","doi":"10.1016/j.dam.2025.05.040","DOIUrl":"10.1016/j.dam.2025.05.040","url":null,"abstract":"<div><div>We show that the number of nonisomorphic nonorientable 6-gonal embeddings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mrow><mi>n</mi><mo>≡</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, such that the boundary walk of every face passes through some vertices twice is greater than the number of nonisomorphic (both orientable and nonorientable) triangular embeddings of the complete graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 122-128"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272532","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}

{"title":"Constructions of 2-rotation symmetric bent functions based on Maiorana–McFarland’s bent function","authors":"Yu Guan,&nbsp;Sihong Su","doi":"10.1016/j.dam.2025.05.032","DOIUrl":"10.1016/j.dam.2025.05.032","url":null,"abstract":"<div><div>Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for more than five decades. Rotation symmetric Boolean functions, introduced by Pieprzyk and Qu, are those Boolean functions which are invariant under the cyclic shifts of inputs. In this paper, we first give two flexible construction methods of bent functions on <span><math><mi>n</mi></math></span> variables by defining two subsets <span><math><mi>T</mi></math></span>’s of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to modify the support of Maiorana–McFarland’s bent function <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mi>⋅</mi><mi>π</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>⊕</mo><mi>h</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></math></span>, <span><math><mi>π</mi></math></span> is a permutation over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>, and <span><math><mi>h</mi></math></span> is a Boolean function on <span><math><mi>m</mi></math></span> variables. These methods have corresponding constraints on <span><math><mi>π</mi></math></span> and <span><math><mi>h</mi></math></span>. Then, we deduce the dual functions of the newly constructed bent functions. Lastly, we propose the methods of constructing 2-rotation symmetric bent functions by redefining the two subsets <span><math><mi>T</mi></math></span>’s of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and by imposing more restrictive constraints on <span><math><mi>π</mi></math></span> and <span><math><mi>h</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 139-153"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272533","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}