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Infinitely many counterexamples of a conjecture of Franušić and Jadrijević Franušić和jadrijeviki猜想的无限反例
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-07-07 DOI: 10.1007/s00013-025-02144-8
Shubham Gupta
{"title":"Infinitely many counterexamples of a conjecture of Franušić and Jadrijević","authors":"Shubham Gupta","doi":"10.1007/s00013-025-02144-8","DOIUrl":"10.1007/s00013-025-02144-8","url":null,"abstract":"<div><p>Let <i>d</i> be a square-free integer such that <span>(d equiv 15 pmod {60})</span> and Pell’s equation <span>(x^2 - dy^2 = -6)</span> is solvable in rational integers <i>x</i> and <i>y</i>. In this paper, we prove that there exist infinitely many Diophantine quadruples in <span>(mathbb {Z}[sqrt{d}])</span> with the property <i>D</i>(<i>n</i>) for certain <i>n</i>’s. As an application of it, we ‘unconditionally’ prove the existence of infinitely many rings <span>(mathbb {Z}[sqrt{d}])</span> for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does ‘not’ hold. This conjecture states a relationship between the existence of a Diophantine quadruple in <span>(mathcal {R})</span> with the property <i>D</i>(<i>n</i>) and the representability of <i>n</i> as a difference of two squares in <span>(mathcal {R})</span>, where <span>(mathcal {R})</span> is a commutative ring with unity.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"173 - 184"},"PeriodicalIF":0.5,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145142331","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 perfect symmetric rank-metric codes 关于完全对称秩-度量码
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-07-07 DOI: 10.1007/s00013-025-02145-7
Usman Mushrraf, Ferdinando Zullo
{"title":"On perfect symmetric rank-metric codes","authors":"Usman Mushrraf,&nbsp;Ferdinando Zullo","doi":"10.1007/s00013-025-02145-7","DOIUrl":"10.1007/s00013-025-02145-7","url":null,"abstract":"<div><p>Let <span>(textrm{Sym}_q(m))</span> be the space of symmetric matrices in <span>({mathbb {F}}_q^{mtimes m})</span>. A subspace of <span>(textrm{Sym}_q(m))</span> equipped with the rank distance is called an <span>({{mathbb {F}}}_{q})</span>-linear symmetric rank-metric code. In this paper, we study the covering properties of <span>({{mathbb {F}}}_{q})</span>-linear symmetric rank-metric codes. First we characterize <span>({{mathbb {F}}}_{q})</span>-linear symmetric rank-metric codes which are perfect, i.e., that satisfy the equality in the sphere-packing like bound. We show that, despite the rank-metric case, there are non-trivial perfect codes. Indeed, we prove that the only perfect non-trivial <span>({{mathbb {F}}}_{q})</span>-linear symmetric rank-metric codes in <span>(textrm{Sym}_q(m))</span> are the symmetric MRD codes with minimum distance 3 and <i>m</i> odd. Also, we characterize families of codes which are quasi-perfect.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 3","pages":"259 - 271"},"PeriodicalIF":0.5,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145011718","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
The Brylinski beta function of a coaxial layer 同轴层的Brylinski beta函数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-07-04 DOI: 10.1007/s00013-025-02138-6
Pooja Rani, M. K. Vemuri
{"title":"The Brylinski beta function of a coaxial layer","authors":"Pooja Rani,&nbsp;M. K. Vemuri","doi":"10.1007/s00013-025-02138-6","DOIUrl":"10.1007/s00013-025-02138-6","url":null,"abstract":"<div><p>In (Differential Geom. Appl. 92: Paper No. 102078, 12 pp., 2024), an analogue of Brylinski’s knot beta function was defined for a compactly supported (Schwartz) distribution <i>T</i> on Euclidean space. Here we consider the Brylinski beta function of the distribution defined by a coaxial layer on a submanifold of Euclidean space. We prove that it has an analytic continuation to the whole complex plane as a meromorphic function with only simple poles, and in the case of a coaxial layer on a space curve, we compute some of the residues in terms of the curvature and torsion.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"213 - 225"},"PeriodicalIF":0.5,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145142415","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 some values which do not belong to the image of Ramanujan’s tau-function 关于一些不属于拉马努金函数象的值
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-06-21 DOI: 10.1007/s00013-025-02139-5
Akihiro Goto
{"title":"On some values which do not belong to the image of Ramanujan’s tau-function","authors":"Akihiro Goto","doi":"10.1007/s00013-025-02139-5","DOIUrl":"10.1007/s00013-025-02139-5","url":null,"abstract":"<div><p>Lehmer conjectured that Ramanujan’s tau-function never vanishes. As a variation of this conjecture, it is proved that </p><div><div><span>$$begin{aligned} tau (n)ne pm ell , pm 2ell , pm 2ell ^2, end{aligned}$$</span></div></div><p>where <span>(ell &lt;100)</span> is an odd prime, by Balakrishnan, Ono, Craig, Tsai, and many people. We prove that </p><div><div><span>$$begin{aligned} tau (n)ne pm ell , pm 2ell , pm 4ell , pm 8ell end{aligned}$$</span></div></div><p>for <span>(ell in L)</span>, where <i>L</i> is an explicit finite subset of odd primes less than 1000.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"157 - 172"},"PeriodicalIF":0.5,"publicationDate":"2025-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145144124","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
Criteria for the compact elements in a locally compact group to form a subgroup 局部紧群中紧元素构成子群的准则
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-06-17 DOI: 10.1007/s00013-025-02137-7
Marwa Gouiaa
{"title":"Criteria for the compact elements in a locally compact group to form a subgroup","authors":"Marwa Gouiaa","doi":"10.1007/s00013-025-02137-7","DOIUrl":"10.1007/s00013-025-02137-7","url":null,"abstract":"<div><p>An element in a topological group is called compact or periodic if it is contained in a compact subgroup. In a general locally compact group, the compact elements will not be closed under multiplication. We show that the set of all compact elements forms a subgroup if a more general periodicity property is satisfied.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"185 - 191"},"PeriodicalIF":0.5,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145141946","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
Application of Meyer’s theorem on quasicrystals to exponential polynomials and Dirichlet series 准晶体Meyer定理在指数多项式和狄利克雷级数中的应用
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-06-17 DOI: 10.1007/s00013-025-02140-y
Sergii Yu. Favorov
{"title":"Application of Meyer’s theorem on quasicrystals to exponential polynomials and Dirichlet series","authors":"Sergii Yu. Favorov","doi":"10.1007/s00013-025-02140-y","DOIUrl":"10.1007/s00013-025-02140-y","url":null,"abstract":"<div><p>A simple necessary and sufficient condition is given for exponential polynomials and absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer’s theorem on quasicrystals.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"193 - 200"},"PeriodicalIF":0.5,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145141947","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
Variations on Pascal’s theorem 帕斯卡定理的变体
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-06-09 DOI: 10.1007/s00013-025-02122-0
Ciro Ciliberto, Rick Miranda
{"title":"Variations on Pascal’s theorem","authors":"Ciro Ciliberto,&nbsp;Rick Miranda","doi":"10.1007/s00013-025-02122-0","DOIUrl":"10.1007/s00013-025-02122-0","url":null,"abstract":"<div><p>In this paper, we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the “Mystic Hexagon”. We give explicit equations describing the conditions for <span>(d+4)</span> points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in <span>({mathbb {P}}^3)</span>. Finally we re-prove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown for quadrics in <span>({mathbb {P}}^4)</span> containing five general lines.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"39 - 51"},"PeriodicalIF":0.5,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145163869","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
Comparing cohomology via exact split pairs in diagram algebras 图代数中精确分裂对上同调的比较
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-06-03 DOI: 10.1007/s00013-025-02127-9
Sulakhana Chowdhury, Geetha Thangavelu
{"title":"Comparing cohomology via exact split pairs in diagram algebras","authors":"Sulakhana Chowdhury,&nbsp;Geetha Thangavelu","doi":"10.1007/s00013-025-02127-9","DOIUrl":"10.1007/s00013-025-02127-9","url":null,"abstract":"<div><p>In this article, we compare the cohomology between the categories of modules of the diagram algebras and the categories of modules of its input algebras. Our main result establishes a sufficient condition for exact split pairs between these two categories, analogous to a work by Diracca and Koenig (J Pure Appl Algebra 212:471–485, 2008). To be precise, we prove the existence of the exact split pairs in <i>A</i>-Brauer algebras, cyclotomic Brauer algebras, and walled Brauer algebras with their respective input algebras.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"79 - 92"},"PeriodicalIF":0.5,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145161562","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 multisecants of the Kummer variety of a Jacobian 关于雅可比矩阵Kummer变换的多割线的注记
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-05-15 DOI: 10.1007/s00013-025-02134-w
Robert Auffarth, Sebastian Rahausen
{"title":"A note on multisecants of the Kummer variety of a Jacobian","authors":"Robert Auffarth,&nbsp;Sebastian Rahausen","doi":"10.1007/s00013-025-02134-w","DOIUrl":"10.1007/s00013-025-02134-w","url":null,"abstract":"<div><p>We show that if <i>C</i> is a smooth projective curve and <span>(mathfrak {d})</span> is a <span>(mathfrak {g}^{n}_{2n})</span> on <i>C</i>, then we obtain a rational map <span>(textrm{Sym}^{n}(C)dashrightarrow mathfrak {d})</span> whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of <i>JC</i>. This generalizes previous work done by the first author with Codogni and Salvati Manni.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 3","pages":"273 - 281"},"PeriodicalIF":0.5,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145011450","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
Genus of division algebras over fields with infinite transcendence degree 无限超越度域上的除法代数的属
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-05-14 DOI: 10.1007/s00013-025-02131-z
Sergey V. Tikhonov
{"title":"Genus of division algebras over fields with infinite transcendence degree","authors":"Sergey V. Tikhonov","doi":"10.1007/s00013-025-02131-z","DOIUrl":"10.1007/s00013-025-02131-z","url":null,"abstract":"<div><p>We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let <i>K</i> be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if <span>({mathcal D})</span> is a central division <i>K</i>-algebra, then <span>(textbf{gen}({mathcal D}))</span> consists of Brauer classes <span>([{mathcal D}'])</span> such that <span>([{mathcal D}])</span> and <span>([{mathcal D}'])</span> generate the same subgroup of <span>(text {Br} (K))</span>. In particular, the genus of any division <i>K</i>-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if <span>(text {char}(K) ne 2)</span>, we prove that the genus of a simple algebraic group of type <span>(textrm{G}_2)</span> over such a field <i>K</i> is trivial.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"115 - 121"},"PeriodicalIF":0.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145143336","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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