Pub Date : 2025-07-14DOI: 10.1016/j.aam.2025.102935
Jiuqiang Liu , Guihai Yu
Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no s pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no d-dimensional Boolean algebras.
{"title":"A relationship for LYM inequalities between Boolean lattices and linear lattices with applications","authors":"Jiuqiang Liu , Guihai Yu","doi":"10.1016/j.aam.2025.102935","DOIUrl":"10.1016/j.aam.2025.102935","url":null,"abstract":"<div><div>Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for studying Sperner property. In this paper, we provide a general relationship for LYM inequalities between Boolean lattices and linear lattices. As applications, we use this relationship to derive generalizations of some well-known theorems on maximum sizes of families containing no copy of certain poset or certain configuration from Boolean lattices to linear lattices, including generalizations of the well-known Kleitman theorem on families containing no <em>s</em> pairwise disjoint members (a non-uniform variant of the famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on families containing no <em>d</em>-dimensional Boolean algebras.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102935"},"PeriodicalIF":1.0,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144613816","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}
Pub Date : 2025-07-09DOI: 10.1016/j.aam.2025.102932
V. Berthé , S. Puzynina
An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.
{"title":"On the rigidity of Arnoux-Rauzy words","authors":"V. Berthé , S. Puzynina","doi":"10.1016/j.aam.2025.102932","DOIUrl":"10.1016/j.aam.2025.102932","url":null,"abstract":"<div><div>An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of the same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words that are generated by iterating a substitution are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words generated by iterating a substitution are rigid. The proof relies on two main ingredients: first, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102932"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579417","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}
Pub Date : 2025-07-09DOI: 10.1016/j.aam.2025.102934
S. Yakubovich
We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.
{"title":"On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values","authors":"S. Yakubovich","doi":"10.1016/j.aam.2025.102934","DOIUrl":"10.1016/j.aam.2025.102934","url":null,"abstract":"<div><div>We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102934"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579758","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}
Pub Date : 2025-07-08DOI: 10.1016/j.aam.2025.102933
Christine Cho, James Oxley
Let M and N be matroids such that N is the image of M under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for x and y positive, if and only if or . We give a number of consequences of this result.
{"title":"Weak maps and the Tutte polynomial","authors":"Christine Cho, James Oxley","doi":"10.1016/j.aam.2025.102933","DOIUrl":"10.1016/j.aam.2025.102933","url":null,"abstract":"<div><div>Let <em>M</em> and <em>N</em> be matroids such that <em>N</em> is the image of <em>M</em> under a rank-preserving weak map. Generalizing results of Lucas, we prove that, for <em>x</em> and <em>y</em> positive, <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>≥</mo><mi>T</mi><mo>(</mo><mi>N</mi><mo>;</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> if and only if <span><math><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mi>x</mi><mi>y</mi></math></span> or <span><math><mi>M</mi><mo>≅</mo><mi>N</mi></math></span>. We give a number of consequences of this result.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102933"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144572076","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}
Pub Date : 2025-07-03DOI: 10.1016/j.aam.2025.102931
Guoce Xin , Chen Zhang
Sylvester's denumerant is a quantity that counts the number of nonnegative integer solutions to the equation , where is a sequence of positive integers with . We present a polynomial time algorithm in N for computing when a is bounded and t is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in Maple under the name Cyc-Denum and demonstrates superior performance when compared to Sills-Zeilberger's Maple package PARTITIONS.
{"title":"A polynomial time algorithm for Sylvester waves when entries are bounded","authors":"Guoce Xin , Chen Zhang","doi":"10.1016/j.aam.2025.102931","DOIUrl":"10.1016/j.aam.2025.102931","url":null,"abstract":"<div><div>Sylvester's denumerant <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> is a quantity that counts the number of nonnegative integer solutions to the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>t</mi></math></span>, where <span><math><mi>a</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>)</mo></math></span> is a sequence of positive integers with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We present a polynomial time algorithm in <em>N</em> for computing <span><math><mi>d</mi><mo>(</mo><mi>t</mi><mo>;</mo><mi>a</mi><mo>)</mo></math></span> when <strong><em>a</em></strong> is bounded and <em>t</em> is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in <span>Maple</span> under the name <span>Cyc-Denum</span> and demonstrates superior performance when <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>500</mn></math></span> compared to Sills-Zeilberger's <span>Maple</span> package <span>PARTITIONS</span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102931"},"PeriodicalIF":1.0,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144535520","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}
Pub Date : 2025-06-30DOI: 10.1016/j.aam.2025.102930
Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann
Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.
{"title":"Algebraic geometry of quantum graphical models","authors":"Eliana Duarte , Dmitrii Pavlov , Maximilian Wiesmann","doi":"10.1016/j.aam.2025.102930","DOIUrl":"10.1016/j.aam.2025.102930","url":null,"abstract":"<div><div>Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"170 ","pages":"Article 102930"},"PeriodicalIF":1.0,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513870","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}
Pub Date : 2025-06-20DOI: 10.1016/j.aam.2025.102928
Arthur Bik , Orlando Marigliano
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 for .
{"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}
Pub Date : 2025-06-20DOI: 10.1016/j.aam.2025.102927
Songlin Guo , Wei Wang , Wei Wang
Suppose G is a controllable graph of order n with adjacency matrix A. Let (e is the all-ones vector) and ('s are eigenvalues of A) be the walk matrix and the discriminant of G, respectively. Wang and Yu (arXiv:1608.01144) [21] showed that if is odd and squarefree, then G is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph G to be DGS without the squarefreeness assumption on . 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.
{"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}
Pub Date : 2025-06-20DOI: 10.1016/j.aam.2025.102929
Giulio Cerbai , Anders Claesson , Bruce E. Sagan
Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with d-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 d-Fishburn permutations.
{"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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号