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A new proof of Rédei’s theorem on the number of directions 雷代方向数定理的新证明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-08 DOI: 10.1007/s00013-024-01979-x
Gábor Somlai
{"title":"A new proof of Rédei’s theorem on the number of directions","authors":"Gábor Somlai","doi":"10.1007/s00013-024-01979-x","DOIUrl":"10.1007/s00013-024-01979-x","url":null,"abstract":"<div><p>Rédei and Megyesi proved that the number of directions determined by a <i>p</i>-element subset of <span>({mathbb F}_p^2)</span> is either 1 or at least <span>(frac{p+3}{2})</span>. The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 6","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01979-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces 关于有限度量空间上的利普齐兹单位球的极值点和表示定理
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-04 DOI: 10.1007/s00013-024-01978-y
Kristian Bredies, Jonathan Chirinos Rodriguez, Emanuele Naldi
{"title":"On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces","authors":"Kristian Bredies,&nbsp;Jonathan Chirinos Rodriguez,&nbsp;Emanuele Naldi","doi":"10.1007/s00013-024-01978-y","DOIUrl":"10.1007/s00013-024-01978-y","url":null,"abstract":"<div><p>In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 6","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01978-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Berezin transform on the generalized Hartogs triangles 关于广义哈特三角形的贝雷津变换的说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-03 DOI: 10.1007/s00013-024-01971-5
Qingyang Zou
{"title":"A note on the Berezin transform on the generalized Hartogs triangles","authors":"Qingyang Zou","doi":"10.1007/s00013-024-01971-5","DOIUrl":"10.1007/s00013-024-01971-5","url":null,"abstract":"<div><p>In this note, we study the regularity of the Berezin transform on the generalized Hartogs triangles. By introducing a rotation invariant weight function, we show the unboundedness of the Berezin transform of weighted Hilbert spaces defined on the generalized Hartogs triangles.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 the modular isomorphism problem for groups with center of index at most (p^3) 关于指数中心最多为 $$p^3$$ 的群的模态同构问题
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-30 DOI: 10.1007/s00013-024-01977-z
Sofia Brenner, Diego García-Lucas
{"title":"On the modular isomorphism problem for groups with center of index at most (p^3)","authors":"Sofia Brenner,&nbsp;Diego García-Lucas","doi":"10.1007/s00013-024-01977-z","DOIUrl":"10.1007/s00013-024-01977-z","url":null,"abstract":"<div><p>Let <i>p</i> be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite <i>p</i>-groups whose center has index <span>(p^3)</span>, which is a strong contrast to the analogous situation for <span>(p = 2)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01977-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140364657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Morphisms between Grassmannians, II 格拉斯曼之间的变形，II
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-26 DOI: 10.1007/s00013-024-01986-y
Gianluca Occhetta, Eugenia Tondelli
{"title":"Morphisms between Grassmannians, II","authors":"Gianluca Occhetta,&nbsp;Eugenia Tondelli","doi":"10.1007/s00013-024-01986-y","DOIUrl":"10.1007/s00013-024-01986-y","url":null,"abstract":"<div><p>Denote by <span>({mathbb {G}}(k,n))</span> the Grassmannian of linear subspaces of dimension <i>k</i> in <span>({mathbb {P}}^n)</span>. We show that if <span>(varphi :{mathbb {G}}(l,n) rightarrow {mathbb {G}}(k,n))</span> is a nonconstant morphism and <span>(l not =0,n-1)</span>, then <span>(l=k)</span> or <span>(l=n-k-1)</span> and <span>(varphi )</span> is an isomorphism.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01986-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Genus and crosscap of solvable conjugacy class graphs of finite groups 有限群可解共轭类图的属和交盖
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-24 DOI: 10.1007/s00013-024-01974-2
Parthajit Bhowal, Peter J. Cameron, Rajat Kanti Nath, Benjamin Sambale
{"title":"Genus and crosscap of solvable conjugacy class graphs of finite groups","authors":"Parthajit Bhowal,&nbsp;Peter J. Cameron,&nbsp;Rajat Kanti Nath,&nbsp;Benjamin Sambale","doi":"10.1007/s00013-024-01974-2","DOIUrl":"10.1007/s00013-024-01974-2","url":null,"abstract":"<div><p>The solvable conjugacy class graph of a finite group <i>G</i>, denoted by <span>(Gamma _{sc}(G))</span>, is a simple undirected graph whose vertices are the non-trivial conjugacy classes of <i>G</i> and two distinct conjugacy classes <i>C</i>, <i>D</i> are adjacent if there exist <span>(x in C)</span> and <span>(y in D)</span> such that <span>(langle x, yrangle )</span> is solvable. In this paper, we discuss certain properties of the genus and crosscap of <span>(Gamma _{sc}(G))</span> for the groups <span>(D_{2n})</span>, <span>(Q_{4n})</span>, <span>(S_n)</span>, <span>(A_n)</span>, and <span>({{,mathrm{mathop {textrm{PSL}}},}}(2,2^d))</span>. In particular, we determine all positive integers <i>n</i> such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of <span>(Gamma _{sc}(G))</span> in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of <span>(Gamma _{sc}(G))</span> and the commuting probability of certain finite non-solvable group.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Reproducing kernel Hilbert spaces cannot contain all continuous functions on a compact metric space 重现核希尔伯特空间不能包含紧凑公元空间上的所有连续函数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-23 DOI: 10.1007/s00013-024-01976-0
Ingo Steinwart
{"title":"Reproducing kernel Hilbert spaces cannot contain all continuous functions on a compact metric space","authors":"Ingo Steinwart","doi":"10.1007/s00013-024-01976-0","DOIUrl":"10.1007/s00013-024-01976-0","url":null,"abstract":"<div><p>Given an uncountable, compact metric space <i>X</i>, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on <i>X</i>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01976-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Newly appointed editor 新任命的编辑
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-14 DOI: 10.1007/s00013-024-01973-3
{"title":"Newly appointed editor","authors":"","doi":"10.1007/s00013-024-01973-3","DOIUrl":"10.1007/s00013-024-01973-3","url":null,"abstract":"","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Half-factorial real quadratic orders 半因子实二次阶
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-12 DOI: 10.1007/s00013-024-01969-z
Paul Pollack
{"title":"Half-factorial real quadratic orders","authors":"Paul Pollack","doi":"10.1007/s00013-024-01969-z","DOIUrl":"10.1007/s00013-024-01969-z","url":null,"abstract":"<div><p>Recall that <i>D</i> is a <span>half-factorial domain</span> (HFD) when <i>D</i> is atomic and every equation <span>(pi _1cdots pi _k = rho _1 cdots rho _ell )</span>, with all <span>(pi _i)</span> and <span>(rho _j)</span> irreducible in <i>D</i>, implies <span>(k=ell )</span>. We explain how techniques introduced to attack Artin’s primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we prove that (a) there are infinitely many real quadratic orders that are half-factorial domains, and (b) under the generalized Riemann hypothesis, <span>({mathbb {Q}}(sqrt{2}))</span> contains infinitely many HFD orders.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 note on blocks of finite groups with TI Sylow p-subgroups 关于具有 TI Sylow p 子群的有限群块的说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-07 DOI: 10.1007/s00013-024-01968-0
Deniz Yılmaz
{"title":"A note on blocks of finite groups with TI Sylow p-subgroups","authors":"Deniz Yılmaz","doi":"10.1007/s00013-024-01968-0","DOIUrl":"10.1007/s00013-024-01968-0","url":null,"abstract":"<div><p>Let <span>(mathbb {F})</span> be an algebraically closed field of characteristic zero. We prove that functorial equivalence over <span>(mathbb {F})</span> and perfect isometry between blocks of finite groups do not imply each other.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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