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Tiling with monochromatic bipartite graphs of bounded maximum degree 用有界最大度的单色双方形图进行平铺
IF 0.8 3区 数学
Mathematika Pub Date : 2024-09-26 DOI: 10.1112/mtk.12280
António Girão, Oliver Janzer
{"title":"Tiling with monochromatic bipartite graphs of bounded maximum degree","authors":"António Girão,&nbsp;Oliver Janzer","doi":"10.1112/mtk.12280","DOIUrl":"https://doi.org/10.1112/mtk.12280","url":null,"abstract":"<p>We prove that for any <span></span><math></math>, there exists a constant <span></span><math></math> such that the following is true. Let <span></span><math></math> be an infinite sequence of bipartite graphs such that <span></span><math></math> and <span></span><math></math> hold for all <span></span><math></math>. Then, in any <span></span><math></math>-edge-coloured complete graph <span></span><math></math>, there is a collection of at most <span></span><math></math> monochromatic subgraphs, each of which is isomorphic to an element of <span></span><math></math>, whose vertex sets partition <span></span><math></math>. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12280","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142324691","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}
引用次数: 0
Poissonian pair correlation for higher dimensional real sequences 高维实序列的泊松对相关性
IF 0.8 3区 数学
Mathematika Pub Date : 2024-09-26 DOI: 10.1112/mtk.12283
Tanmoy Bera, Mithun Kumar Das, Anirban Mukhopadhyay
{"title":"Poissonian pair correlation for higher dimensional real sequences","authors":"Tanmoy Bera,&nbsp;Mithun Kumar Das,&nbsp;Anirban Mukhopadhyay","doi":"10.1112/mtk.12283","DOIUrl":"https://doi.org/10.1112/mtk.12283","url":null,"abstract":"<p>In this article, we examine the Poissonian pair correlation (PPC) statistic for higher dimensional real sequences. Specifically, we demonstrate that for <span></span><math></math>, almost all <span></span><math></math>, the sequence <span></span><math></math> in <span></span><math></math> has PPC conditionally on the additive energy bound of <span></span><math></math>. This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [Aistleitner, El-Baz, and Munsch, Geom. Funct. Anal. <b>31</b> (2021), 483–512]. More generally, we derive the PPC for <span></span><math></math> for almost all <span></span><math></math>. As a consequence we establish the metric PPC for <span></span><math></math> provided that all of the <span></span><math></math> are greater than two.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12283","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142324692","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}
引用次数: 0
Optimal resistor networks 最佳电阻网络
IF 0.8 3区 数学
Mathematika Pub Date : 2024-09-06 DOI: 10.1112/mtk.12278
J. Robert Johnson, Mark Walters
{"title":"Optimal resistor networks","authors":"J. Robert Johnson,&nbsp;Mark Walters","doi":"10.1112/mtk.12278","DOIUrl":"https://doi.org/10.1112/mtk.12278","url":null,"abstract":"<p>Given a graph on <span></span><math></math> vertices with <span></span><math></math> edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions — graphs like a star, and graphs which are close to regular — with the transition between them occurring when the average degree is 3. However, in this paper, we show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs — namely ‘which rooted graph minimises the average resistance to the root?’. The rooted case is much simpler to analyse that the unrooted, and the one of the main results of this paper is that the two cases are asymptotically equivalent.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12278","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142152263","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}
引用次数: 0
Asymptotic Fermat's last theorem for a family of equations of signature 符号方程组的渐近费马最后定理
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-29 DOI: 10.1112/mtk.12279
Pedro-José Cazorla García
{"title":"Asymptotic Fermat's last theorem for a family of equations of signature","authors":"Pedro-José Cazorla García","doi":"10.1112/mtk.12279","DOIUrl":"https://doi.org/10.1112/mtk.12279","url":null,"abstract":"<p>In this paper, we study the integer solutions of a family of Fermat-type equations of signature <span></span><math></math>, <span></span><math></math>. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant <span></span><math></math> such that if <span></span><math></math>, there are no solutions <span></span><math></math> of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12279","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142100080","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}
引用次数: 0
Colorful vector balancing 彩色矢量平衡
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-27 DOI: 10.1112/mtk.12274
Gergely Ambrus, Rainie Bozzai
{"title":"Colorful vector balancing","authors":"Gergely Ambrus,&nbsp;Rainie Bozzai","doi":"10.1112/mtk.12274","DOIUrl":"https://doi.org/10.1112/mtk.12274","url":null,"abstract":"<p>We extend classical estimates for the vector balancing constant of <span></span><math></math> equipped with the Euclidean and the maximum norms proved in the 1980s by showing that for <span></span><math></math> and <span></span><math></math>, given vector families <span></span><math></math> with <span></span><math></math>, one may select vectors <span></span><math></math> with\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12274","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142089869","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}
引用次数: 0
A note on the squarefree density of polynomials 关于多项式无平方密度的说明
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-26 DOI: 10.1112/mtk.12275
R. C. Vaughan, Yu. G. Zarhin
{"title":"A note on the squarefree density of polynomials","authors":"R. C. Vaughan,&nbsp;Yu. G. Zarhin","doi":"10.1112/mtk.12275","DOIUrl":"https://doi.org/10.1112/mtk.12275","url":null,"abstract":"<p>The conjectured squarefree density of an integral polynomial <span></span><math></math> in <span></span><math></math> variables is an Euler product <span></span><math></math> which can be considered as a product of local densities. We show that a necessary and sufficient condition for <span></span><math></math> to be 0 when <span></span><math></math> is a polynomial in <span></span><math></math> variables over the integers, is that either there is a prime <span></span><math></math> such that the values of <span></span><math></math> at all integer points are divisible by <span></span><math></math> or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density <span></span><math></math> satisfies <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12275","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142077770","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}
引用次数: 0
The interplay between bounded ranks of tensors arising from partitions 分区引起的张量有界等级之间的相互作用
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-25 DOI: 10.1112/mtk.12277
Thomas Karam
{"title":"The interplay between bounded ranks of tensors arising from partitions","authors":"Thomas Karam","doi":"10.1112/mtk.12277","DOIUrl":"https://doi.org/10.1112/mtk.12277","url":null,"abstract":"<p>Let <span></span><math></math> be integers. Using a fragmentation technique, we characterise <span></span><math></math>-tuples <span></span><math></math> of non-empty families of partitions of <span></span><math></math> such that it suffices that an order-<span></span><math></math> tensor has bounded <span></span><math></math>-rank for each <span></span><math></math> for it to have bounded <span></span><math></math>-rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142077971","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}
引用次数: 0
On Waring numbers of henselian rings 论恒星环的瓦林数
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-25 DOI: 10.1112/mtk.12276
Tomasz Kowalczyk, Piotr Miska
{"title":"On Waring numbers of henselian rings","authors":"Tomasz Kowalczyk,&nbsp;Piotr Miska","doi":"10.1112/mtk.12276","DOIUrl":"https://doi.org/10.1112/mtk.12276","url":null,"abstract":"<p>Let <span></span><math></math> be a positive integer. Let <span></span><math></math> be a henselian local ring with residue field <span></span><math></math> of <span></span><math></math>th level <span></span><math></math>. We give some upper and lower bounds for the <span></span><math></math>th Waring number <span></span><math></math> in terms of <span></span><math></math> and <span></span><math></math>. In large number of cases, we are able to compute <span></span><math></math>. Similar results for the <span></span><math></math>th Waring number of the total ring of fractions of <span></span><math></math> are obtained. We then provide applications. In particular, we compute <span></span><math></math> and <span></span><math></math> for <span></span><math></math> and any prime <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142077970","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}
引用次数: 0
Inscribable fans I: Inscribed cones and virtual polytopes 可刻画扇形 I：刻画锥形和虚拟多面体
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-14 DOI: 10.1112/mtk.12270
Sebastian Manecke, Raman Sanyal
{"title":"Inscribable fans I: Inscribed cones and virtual polytopes","authors":"Sebastian Manecke,&nbsp;Raman Sanyal","doi":"10.1112/mtk.12270","DOIUrl":"https://doi.org/10.1112/mtk.12270","url":null,"abstract":"<p>We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope <span></span><math></math>. We show that the associated space of polytopes, called the <i>inscribed cone</i> of <span></span><math></math>, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to <span></span><math></math> is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension 2 as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed <i>virtual</i> polytopes. Polytopes with a fixed normal fan <span></span><math></math> form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the <i>type space</i> of <span></span><math></math>. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be nontrivial even if <span></span><math></math> does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called <i>reflection groupoids</i>. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141991604","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}
引用次数: 0
An overdetermined problem related to the Finsler -Laplacian 与芬斯勒-拉普拉奇相关的超定问题
IF 0.8 3区 数学
Mathematika Pub Date : 2024-08-05 DOI: 10.1112/mtk.12267
Antonio Greco, Benyam Mebrate
{"title":"An overdetermined problem related to the Finsler -Laplacian","authors":"Antonio Greco,&nbsp;Benyam Mebrate","doi":"10.1112/mtk.12267","DOIUrl":"https://doi.org/10.1112/mtk.12267","url":null,"abstract":"<p>In this paper, we consider the Finsler <span></span><math></math>-Laplacian torsion equation. The domain of the problem is bounded by a conical surface supporting a Neumann-type condition, and an unknown surface supporting both a Dirichlet and a Neumann condition. The case when the cone coincides with the punctured space is included. We show that the existence of a weak solution implies that the unknown surface lies on the boundary of a Finsler-ball. Incidentally, some properties of the Finsler–Minkowski norms are proved here under mild smoothness assumptions.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12267","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141967317","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}
引用次数: 0
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