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On a Gallai-type problem and illumination of spiky balls and cap bodies 关于尖球和帽体的盖莱型问题及其说明
IF 0.8 3区 数学
Mathematika Pub Date : 2025-03-25 DOI: 10.1112/mtk.70017
Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko
{"title":"On a Gallai-type problem and illumination of spiky balls and cap bodies","authors":"Andrii Arman,&nbsp;Andriy Bondarenko,&nbsp;Andriy Prymak,&nbsp;Danylo Radchenko","doi":"10.1112/mtk.70017","DOIUrl":"10.1112/mtk.70017","url":null,"abstract":"<p>We show that any finite family of pairwise intersecting balls in <span></span><math></math> can be pierced by <span></span><math></math> points improving the previously known estimate of <span></span><math></math>. As a corollary, this implies that any 2-illuminable spiky ball in <span></span><math></math> can be illuminated by <span></span><math></math> directions. For the illumination number of convex spiky balls, that is, cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of <span></span><math></math>, which can be compared with the previous <span></span><math></math> established only for the centrally symmetric cap bodies. We also prove the lower bounds of <span></span><math></math> for the three problems above.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143698904","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 sharp higher order Sobolev embedding 一个高阶索博列夫嵌入
IF 0.8 3区 数学
Mathematika Pub Date : 2025-03-12 DOI: 10.1112/mtk.70012
Raul Hindov, Shahaf Nitzan, Jan-Fredrik Olsen, Eskil Rydhe
{"title":"A sharp higher order Sobolev embedding","authors":"Raul Hindov,&nbsp;Shahaf Nitzan,&nbsp;Jan-Fredrik Olsen,&nbsp;Eskil Rydhe","doi":"10.1112/mtk.70012","DOIUrl":"10.1112/mtk.70012","url":null,"abstract":"<p>We obtain sharp embeddings from the Sobolev space <span></span><math></math> into the space <span></span><math></math> and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595390","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
An exotic calculus of Berezin–Toeplitz operators Berezin-Toeplitz算子的奇异演算
IF 0.8 3区 数学
Mathematika Pub Date : 2025-03-08 DOI: 10.1112/mtk.70016
Izak Oltman
{"title":"An exotic calculus of Berezin–Toeplitz operators","authors":"Izak Oltman","doi":"10.1112/mtk.70016","DOIUrl":"10.1112/mtk.70016","url":null,"abstract":"<p>We develop a calculus of Berezin–Toeplitz operators quantizing exotic classes of smooth functions on compact Kähler manifolds and acting on holomorphic sections of powers of positive line bundles. These functions (classical observables) are exotic in the sense that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sjöstrand's method of complex stationary phase. We obtain a functional calculus result, a trace formula, and a parametrix construction for this larger class of functions. These results are crucially used in proving a probabilistic Weyl-law for randomly perturbed (standard) Berezin–Toeplitz operators in Oltman (arXiv:2207.09599).</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581445","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
Youden's demon is Sylvester's problem 尤登的恶魔是西尔维斯特的问题
IF 0.8 3区 数学
Mathematika Pub Date : 2025-02-18 DOI: 10.1112/mtk.70015
Florian Frick, Andrew Newman, Wesley Pegden
{"title":"Youden's demon is Sylvester's problem","authors":"Florian Frick,&nbsp;Andrew Newman,&nbsp;Wesley Pegden","doi":"10.1112/mtk.70015","DOIUrl":"10.1112/mtk.70015","url":null,"abstract":"<p>If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are <span></span><math></math>. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line (Youden's demon problem) is the same as a natural generalization of Sylvester's four point problem to Gaussian points in <span></span><math></math>. Our main tool is the observation that the Gale dual of independent samples in <span></span><math></math> itself can be taken to be a set of independent points (translated to have barycenter at the origin) when the distribution of the points is Gaussian.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143431696","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
A four-dimensional peabody of constant width 宽度恒定的四维体
IF 0.8 3区 数学
Mathematika Pub Date : 2025-02-13 DOI: 10.1112/mtk.70014
Isaac Arelio, Luis Montejano, Deborah Oliveros
{"title":"A four-dimensional peabody of constant width","authors":"Isaac Arelio,&nbsp;Luis Montejano,&nbsp;Deborah Oliveros","doi":"10.1112/mtk.70014","DOIUrl":"10.1112/mtk.70014","url":null,"abstract":"<p>In this paper, we present a unique four-dimensional body of constant width based on the classical notion of focal conics.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404474","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 new construction of forests with low visibility 低能见度森林的新建筑
IF 0.8 3区 数学
Mathematika Pub Date : 2025-02-07 DOI: 10.1112/mtk.70013
Kirill Kashkan
{"title":"A new construction of forests with low visibility","authors":"Kirill Kashkan","doi":"10.1112/mtk.70013","DOIUrl":"10.1112/mtk.70013","url":null,"abstract":"<p>A set of points with finite density is constructed in <span></span><math></math>, with <span></span><math></math>, by adding points to a Poisson process such that any line segment of length <span></span><math></math> in <span></span><math></math> will contain one of the points of the set within distance <span></span><math></math> of it. The constant implied by the big-<span></span><math></math> notation depends on the dimension only.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362696","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
On convex bodies in , , with directly congruent projections 在凸体上，有直接全等的投影
IF 0.8 3区 数学
Mathematika Pub Date : 2025-02-06 DOI: 10.1112/mtk.70011
Reema A. Sbeih
{"title":"On convex bodies in , , with directly congruent projections","authors":"Reema A. Sbeih","doi":"10.1112/mtk.70011","DOIUrl":"10.1112/mtk.70011","url":null,"abstract":"<p>Let <span></span><math></math> and let <span></span><math></math> and <span></span><math></math> be two convex bodies in <span></span><math></math> such that their orthogonal projections <span></span><math></math> and <span></span><math></math> onto any <span></span><math></math>-dimensional subspace <span></span><math></math> are directly congruent, that is, there exists a rotation <span></span><math></math> and a vector <span></span><math></math> such that <span></span><math></math>. Assume also that the 2-dimensional projections of <span></span><math></math> and <span></span><math></math> are pairwise different and they do not have <span></span><math></math>-symmetries. Then <span></span><math></math> and <span></span><math></math> are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362806","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
Lattices in function fields and applications 函数域中的格及其应用
IF 0.8 3区 数学
Mathematika Pub Date : 2025-01-31 DOI: 10.1112/mtk.70010
Christian Bagshaw, Bryce Kerr
{"title":"Lattices in function fields and applications","authors":"Christian Bagshaw,&nbsp;Bryce Kerr","doi":"10.1112/mtk.70010","DOIUrl":"10.1112/mtk.70010","url":null,"abstract":"<p>In recent decades, the use of ideas from Minkowski's Geometry of Numbers has gained recognition as a helpful tool in bounding the number of solutions to modular congruences with variables from short intervals. In 1941, Mahler introduced an analogue to the Geometry of Numbers in function fields over finite fields. Here, we build on Mahler's ideas and develop results useful for bounding the sizes of intersections of lattices and convex bodies in <span></span><math></math>, which are more precise than what is known over <span></span><math></math>. These results are then applied to various problems regarding bounding the number of solutions to congruences in <span></span><math></math>, such as the number of points on polynomial curves in low-dimensional subspaces of finite fields. Our results improve on a number of previous bounds due to Bagshaw, Cilleruelo, Shparlinski and Zumalacárregui. We also present previous techniques developed by various authors for estimating certain energy/point counts in a unified manner.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143121401","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
On differences of two harmonic numbers 关于两个调和数的差
IF 0.8 3区 数学
Mathematika Pub Date : 2025-01-27 DOI: 10.1112/mtk.70009
Jeck Lim, Stefan Steinerberger
{"title":"On differences of two harmonic numbers","authors":"Jeck Lim,&nbsp;Stefan Steinerberger","doi":"10.1112/mtk.70009","DOIUrl":"10.1112/mtk.70009","url":null,"abstract":"<p>We prove the existence of infinitely many <span></span><math></math> such that the difference of harmonic numbers <span></span><math></math> approximates 1 well\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143119712","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
Coherency properties for monoids of transformations and partitions 变换和划分的模群的相干性
IF 0.8 3区 数学
Mathematika Pub Date : 2025-01-08 DOI: 10.1112/mtk.70005
Matthew Brookes, Victoria Gould, Nik Ruškuc
{"title":"Coherency properties for monoids of transformations and partitions","authors":"Matthew Brookes,&nbsp;Victoria Gould,&nbsp;Nik Ruškuc","doi":"10.1112/mtk.70005","DOIUrl":"10.1112/mtk.70005","url":null,"abstract":"<p>A monoid <span></span><math></math> is <i>right coherent</i> if every finitely generated subact of every finitely presented right <span></span><math></math>-act itself has a finite presentation; it is <i>weakly right coherent</i> if every finitely generated right ideal of <span></span><math></math> has a finite presentation. We show that full and partial transformation monoids, symmetric inverse monoids and partition monoids over an infinite set are all weakly right coherent, but that none of them is right coherent. Left coherency and weak left coherency are defined dually, and the corresponding results hold for these properties. In order to prove the non-coherency results, we give a presentation of an inverse semigroup which does not embed into any left or right coherent monoid.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143113277","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
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