{"title":"Classifying one-dimensional discrete models with maximum likelihood degree one","authors":"Arthur Bik , Orlando Marigliano","doi":"10.1016/j.aam.2025.102928","DOIUrl":"10.1016/j.aam.2025.102928","url":null,"abstract":"<div><div>We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of ‘fundamental models’ using a finite number of simple operations. We introduce ‘chipsplitting games’, a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102928"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144330513","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":"Primary decomposition theorem and generalized spectral characterization of graphs","authors":"Songlin Guo , Wei Wang , Wei Wang","doi":"10.1016/j.aam.2025.102927","DOIUrl":"10.1016/j.aam.2025.102927","url":null,"abstract":"<div><div>Suppose <em>G</em> is a controllable graph of order <em>n</em> with adjacency matrix <em>A</em>. Let <span><math><mi>W</mi><mo>=</mo><mo>[</mo><mi>e</mi><mo>,</mo><mi>A</mi><mi>e</mi><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>e</mi><mo>]</mo></math></span> (<em>e</em> is the all-ones vector) and <span><math><mi>Δ</mi><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>></mo><mi>j</mi></mrow></msub><msup><mrow><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> (<span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are eigenvalues of <em>A</em>) be the walk matrix and the discriminant of <em>G</em>, respectively. Wang and Yu (<span><span>arXiv:1608.01144</span><svg><path></path></svg></span>) <span><span>[21]</span></span> showed that if<span><span><span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>gcd</mi><mo></mo><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></msup><mi>det</mi><mo></mo><mi>W</mi><mo>,</mo><mi>Δ</mi><mo>}</mo></math></span></span></span> is odd and squarefree, then <em>G</em> is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph <em>G</em> to be DGS without the squarefreeness assumption on <span><math><mi>θ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102927"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321228","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":"Self-modified difference ascent sequences","authors":"Giulio Cerbai , Anders Claesson , Bruce E. Sagan","doi":"10.1016/j.aam.2025.102929","DOIUrl":"10.1016/j.aam.2025.102929","url":null,"abstract":"<div><div>Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with <em>d</em>-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of <em>d</em>-Fishburn permutations.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102929"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321229","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 average number of cycles in conjugacy class products","authors":"Jesse Campion Loth, Amarpreet Rattan","doi":"10.1016/j.aam.2025.102926","DOIUrl":"10.1016/j.aam.2025.102926","url":null,"abstract":"<div><div>We show that for the product of permutations from two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types <em>α</em> and <em>β</em>, the average number of cycles in their product is between <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>3</mn></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span>, where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the harmonic number.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102926"},"PeriodicalIF":1.0,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144314072","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 nonnegative ranks of matrices in Puiseux series fields","authors":"Yaroslav Shitov","doi":"10.1016/j.aam.2025.102916","DOIUrl":"10.1016/j.aam.2025.102916","url":null,"abstract":"<div><div>The <em>positive part</em> <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> of the field <span><math><mi>C</mi><mo>{</mo><mo>{</mo><mi>t</mi><mo>}</mo><mo>}</mo></math></span> consists of Puiseux series with positive real leading terms. Answering a question of Yu, we show that, if <em>M</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and rank two, then there are an <span><math><mi>m</mi><mo>×</mo><mn>2</mn></math></span> matrix <em>A</em> and <span><math><mn>2</mn><mo>×</mo><mi>n</mi></math></span> matrix <em>B</em> with entries in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> such that <span><math><mi>M</mi><mo>=</mo><mi>A</mi><mi>B</mi></math></span>. We discuss the problem in larger ranks and answer a further question arisen in a work of Brandenburg, Loho, and Sinn.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102916"},"PeriodicalIF":1.0,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239672","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}
Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , Ligong Wang
{"title":"Developments on the Hoffman program of graphs","authors":"Jianfeng Wang , Jing Wang , Maurizio Brunetti , Francesco Belardo , Ligong Wang","doi":"10.1016/j.aam.2025.102915","DOIUrl":"10.1016/j.aam.2025.102915","url":null,"abstract":"<div><div>For each squared graph matrix <em>M</em>, the Hoffman program consists of two aspects: finding all the possible limit points of <em>M</em>-spectral radii of graphs and detecting all the connected graphs whose <em>M</em>-spectral radius does not exceed a fixed limit point. In this survey, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless Laplacian, the Hermitian adjacency and the skew-adjacency matrix of graphs. The correspondent problems related to tensors of hypergraphs are also discussed. Moreover, we obtain new results about the Hoffman program with relation to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrix. In particular, we get two generalized versions of it applicable to nonnegative symmetric matrices with fractional elements. We also retrieve the limit points of spectral radii of (signless) Laplacian matrices of graphs less than <span><math><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><mo>(</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>−</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mn>54</mn><mo>+</mo><mn>6</mn><msqrt><mrow><mn>33</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102915"},"PeriodicalIF":1.0,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190253","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":"Binary sequences meet the Fibonacci sequence","authors":"Piotr Miska , Bartosz Sobolewski , Maciej Ulas","doi":"10.1016/j.aam.2025.102914","DOIUrl":"10.1016/j.aam.2025.102914","url":null,"abstract":"<div><div>We introduce a new family of number sequences <span><math><msub><mrow><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, governed by the recurrence relation<span><span><span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>b</mi><mi>f</mi><mo>(</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>u</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> is a sequence with values <span><math><mn>0</mn><mo>,</mo><mn>1</mn></math></span>. Our study focuses on the properties of the sequence of quotients <span><math><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and its set of values <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> for various <strong>u</strong>. We give a sufficient condition for finiteness of <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and automaticity of <span><math><msub><mrow><mo>(</mo><mi>h</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, which holds in particular when <strong>u</strong> is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software <span>Walnut</span>. On the other hand, we prove that the set <span><math><mi>V</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is infinite for other special binary sequences <strong>u</strong>, and obtain a trichotomy in its topological type when <strong>u</strong> is eventually periodic.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102914"},"PeriodicalIF":1.0,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144169097","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":"Depth and regularity of tableau ideals","authors":"Do Trong Hoang , Thanh Vu","doi":"10.1016/j.aam.2025.102913","DOIUrl":"10.1016/j.aam.2025.102913","url":null,"abstract":"<div><div>We compute the depth and regularity of ideals associated with arbitrary fillings of positive integers to a Young diagram, called the tableau ideals.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102913"},"PeriodicalIF":1.0,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144138028","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":"q-Super Catalan numbers: Combinatorial identities, generating functions, and Narayana refinements","authors":"Arthur Rodelet–Causse , Lenny Tevlin","doi":"10.1016/j.aam.2025.102911","DOIUrl":"10.1016/j.aam.2025.102911","url":null,"abstract":"<div><div>We derive a number of combinatorial identities satisfied by the <em>q</em>-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the <em>q</em>-super Catalan numbers.</div><div>Next, we introduce some <em>q</em>-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for <em>q</em>-Catalan numbers.</div><div>Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the <em>γ</em>-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural <em>q</em>-analogs, prove their <em>q</em>-<em>γ</em>-positivity, and prove some identities they satisfy, generalizing identities of Kreweras <span><span>[17]</span></span> and Le Jen-Shoo <span><span>[11]</span></span>. Using yet another identity, we prove that these refinements are positive integer polynomials in <em>q</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102911"},"PeriodicalIF":1.0,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116089","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":"Equivalence classes of lower and upper descent weak Bruhat intervals","authors":"Seung-Il Choi , Sun-Young Nam , Young-Tak Oh","doi":"10.1016/j.aam.2025.102910","DOIUrl":"10.1016/j.aam.2025.102910","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the set of nonempty left weak Bruhat intervals in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We investigate the equivalence relation <figure><img></figure> on <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <figure><img></figure> if and only if there exists a descent-preserving poset isomorphism between <em>I</em> and <em>J</em>. For each equivalence class <em>C</em> of <figure><img></figure>, a partial order ⪯ is defined by <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><mi>ρ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub><mo>⪯</mo><msub><mrow><mo>[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> if and only if <span><math><mi>σ</mi><msub><mrow><mo>⪯</mo></mrow><mrow><mi>R</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. Kim–Lee–Oh (2024) showed that the poset <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mo>⪯</mo><mo>)</mo></math></span> is isomorphic to a right weak Bruhat interval.</div><div>In this paper, we focus on lower and upper descent weak Bruhat intervals, specifically those of the form <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>,</mo><mi>σ</mi><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span> or <span><math><msub><mrow><mo>[</mo><mi>σ</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>L</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element in the parabolic subgroup <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, generated by <span><math><mo>{</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>}</mo></math></span> for a subset <span><math><mi>S</mi><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>, and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span> is the longest element among the minimal-length representatives of left <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo><mo>∖</mo><mi>S</mi></mrow></msub></math></span>-cosets in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We begin by providing a poset-theoretic characterization of the equivalence relation <figure><img></figure>. Using","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"169 ","pages":"Article 102910"},"PeriodicalIF":1.0,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144088875","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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