Mathematika最新文献

{"title":"Souplet–Zhang and Hamilton-type gradient estimates for non-linear elliptic equations on smooth metric measure spaces","authors":"Ali Taheri,&nbsp;Vahideh Vahidifar","doi":"10.1112/mtk.12208","DOIUrl":"10.1112/mtk.12208","url":null,"abstract":"<p>In this article, we present new gradient estimates for positive solutions to a class of non-linear elliptic equations  involving the <i>f</i>-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville-type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"751-779"},"PeriodicalIF":0.8,"publicationDate":"2023-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12208","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45790093","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}

{"title":"Distribution of Dirichlet L-functions","authors":"Zikang Dong,&nbsp;Weijia Wang,&nbsp;Hao Zhang","doi":"10.1112/mtk.12205","DOIUrl":"10.1112/mtk.12205","url":null,"abstract":"<p>In this article, we study the distribution of values of Dirichlet <i>L</i>-functions, the distribution of values of the random models for Dirichlet <i>L</i>-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 <mo>&lt;</mo>\u0000 <mo>Re</mo>\u0000 <mi>s</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$frac{1}{2}&lt;operatorname{Re}s&lt;1$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Re</mo>\u0000 <mi>s</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$operatorname{Re}s=1$</annotation>\u0000 </semantics></math> separately.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"719-750"},"PeriodicalIF":0.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41572702","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 chromatic number of \u0000 \u0000 \u0000 R\u0000 n\u0000 \u0000 $mathbb {R}^{n}$\u0000 with multiple forbidden distances","authors":"Eric Naslund","doi":"10.1112/mtk.12197","DOIUrl":"https://doi.org/10.1112/mtk.12197","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>⊂</mo>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$Asubset mathbb {R}_{&gt;0}$</annotation>\u0000 </semantics></math> be a finite set of distances, and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>A</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$G_{A}(mathbb {R}^{n})$</annotation>\u0000 </semantics></math> be the graph with vertex set <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^{n}$</annotation>\u0000 </semantics></math> and edge set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>:</mo>\u0000 <mspace></mspace>\u0000 <mo>∥</mo>\u0000 <mi>x</mi>\u0000 <mo>−</mo>\u0000 <mi>y</mi>\u0000 <msub>\u0000 <mo>∥</mo>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <mi>A</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace (x,y)in mathbb {R}^{n}: Vert x-yVert _{2}in Arbrace$</annotation>\u0000 </semantics></math>, and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>,</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>χ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mi>A</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"692-718"},"PeriodicalIF":0.8,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50126756","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":"Curvatures for unions of WDC sets","authors":"Dušan Pokorný","doi":"10.1112/mtk.12195","DOIUrl":"10.1112/mtk.12195","url":null,"abstract":"<p>We prove the existence of the curvature measures for a class of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>WDC</mi>\u0000 </msub>\u0000 <annotation>${mathcal {U}}_{{rm WDC}}$</annotation>\u0000 </semantics></math> sets, which is a direct generalisation of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mspace></mspace>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>${mathcal {U}}_{rm {P! R}}$</annotation>\u0000 </semantics></math> sets studied by Rataj and Zähle. Moreover, we provide a simple characterisation of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>WDC</mi>\u0000 </msub>\u0000 <annotation>${mathcal {U}}_{{rm WDC}}$</annotation>\u0000 </semantics></math> sets in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math> and prove that in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math>, the class of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>WDC</mi>\u0000 </msub>\u0000 <annotation>${mathcal {U}}_{{rm WDC}}$</annotation>\u0000 </semantics></math> sets contains essentially all classes of sets known to admit curvature measures.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"665-691"},"PeriodicalIF":0.8,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41703859","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":"Sums of triples in Abelian groups","authors":"Ido Feldman,&nbsp;Assaf Rinot","doi":"10.1112/mtk.12200","DOIUrl":"10.1112/mtk.12200","url":null,"abstract":"<p>Motivated by a problem in additive Ramsey theory, we extend Todorčević's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group <i>G</i> of size ℵ₂, there exists a coloring <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mo>:</mo>\u0000 <mi>G</mi>\u0000 <mo>→</mo>\u0000 <mi>Z</mi>\u0000 </mrow>\u0000 <annotation>$c:Grightarrow mathbb {Z}$</annotation>\u0000 </semantics></math> such that for every uncountable <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>⊆</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$Xsubseteq G$</annotation>\u0000 </semantics></math> and every integer <i>k</i>, there are three distinct elements <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>,</mo>\u0000 <mi>z</mi>\u0000 </mrow>\u0000 <annotation>$x,y,z$</annotation>\u0000 </semantics></math> of <i>X</i> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>+</mo>\u0000 <mi>y</mi>\u0000 <mo>+</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation>$c(x+y+z)=k$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"622-664"},"PeriodicalIF":0.8,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42772492","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}

{"title":"Sums of distances on graphs and embeddings into Euclidean space","authors":"Stefan Steinerberger","doi":"10.1112/mtk.12198","DOIUrl":"10.1112/mtk.12198","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>,</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$G=(V,E)$</annotation>\u0000 </semantics></math> be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>⋯</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$x_1, dots , x_k$</annotation>\u0000 </semantics></math>, take <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$x_{k+1}$</annotation>\u0000 </semantics></math> to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>≪</mo>\u0000 <mo>|</mo>\u0000 <mi>V</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$m ll |V|$</annotation>\u0000 </semantics></math>. We prove that this suggests that the graph <i>G</i> is, in a suitable sense, “<i>m</i>-dimensional” by exhibiting an explicit 1-Lipschitz embedding <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ϕ</mi>\u0000 <mo>:</mo>\u0000 <mi>V</mi>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>ℓ</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$phi : V rightarrow ell ^1(mathbb {R}^m)$</annotation>\u0000 </semantics></math> with good properties.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"600-621"},"PeriodicalIF":0.8,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44399735","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 \u0000 \u0000 \u0000 K\u0000 \u0000 ℵ\u0000 0\u0000 \u0000 \u0000 $K^{aleph _0}$\u0000 game: Vertex colouring","authors":"Nathan Bowler,&nbsp;Marit Emde,&nbsp;Florian Gut","doi":"10.1112/mtk.12196","DOIUrl":"10.1112/mtk.12196","url":null,"abstract":"<p>We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's aim being to prevent this. We show that if there are only finitely many colours, then Maker can obtain a complete subgraph in which all colours appear infinitely often, but that Breaker can prevent this if there are infinitely many colours. Even when there are infinitely many colours, we show that Maker can obtain a complete subgraph in which infinitely many of the colours each appear infinitely often.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"584-599"},"PeriodicalIF":0.8,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12196","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44538270","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}

{"title":"On the error term in a mixed moment of L-functions","authors":"Rizwanur Khan,&nbsp;Zeyuan Zhang","doi":"10.1112/mtk.12199","DOIUrl":"10.1112/mtk.12199","url":null,"abstract":"<p>There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of Dirichlet <i>L</i>-functions and a closely related mixed moment of <i>L</i>-functions involving automorphic <i>L</i>-functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"573-583"},"PeriodicalIF":0.8,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45908702","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 number of vertices of projective polytopes","authors":"Natalia García-Colín,&nbsp;Luis Pedro Montejano,&nbsp;Jorge Luis Ramírez Alfonsín","doi":"10.1112/mtk.12193","DOIUrl":"10.1112/mtk.12193","url":null,"abstract":"<p>Let <i>X</i> be a set of <i>n</i> points in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^d$</annotation>\u0000 </semantics></math> in general position. What is the maximum number of vertices that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>conv</mi>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathsf {conv}(T(X))$</annotation>\u0000 </semantics></math> can have among all the possible permissible projective transformations <i>T</i>? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether <i>balanced</i> 2-colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance-type problem of finite sets.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"535-561"},"PeriodicalIF":0.8,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12193","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41478429","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}

{"title":"Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line","authors":"Fan Ge","doi":"10.1112/mtk.12194","DOIUrl":"10.1112/mtk.12194","url":null,"abstract":"<p>Assuming the Riemann hypothesis and a hypothesis on small gaps between zeta zeros (see equation (ES 2<i>K</i>) below for a precise definition), we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith [J. Math. Phys. <b>60</b> (2019), no. 8, 083509], which states that for any positive integer <i>K</i> and real number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$a&gt;0$</annotation>\u0000 </semantics></math>,\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"562-572"},"PeriodicalIF":0.8,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46249620","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}