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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
Mixed multiquadratic splitting fields 混合多二次分裂场
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-05-14 DOI: 10.1007/s00013-025-02135-9
Fatma Kader Bingöl, Adam Chapman, Ahmed Laghribi
{"title":"Mixed multiquadratic splitting fields","authors":"Fatma Kader Bingöl,&nbsp;Adam Chapman,&nbsp;Ahmed Laghribi","doi":"10.1007/s00013-025-02135-9","DOIUrl":"10.1007/s00013-025-02135-9","url":null,"abstract":"<div><p>We study mixed multiquadratic field extensions as splitting fields for central simple algebras of exponent 2 in characteristic 2. As an application, we provide examples of nonexcellent mixed biquadratic field extensions.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"29 - 37"},"PeriodicalIF":0.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-025-02135-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145165111","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 maximally symmetric subalgebras 关于极大对称子代数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-05-14 DOI: 10.1007/s00013-025-02132-y
Alexander Kleshchev
{"title":"On maximally symmetric subalgebras","authors":"Alexander Kleshchev","doi":"10.1007/s00013-025-02132-y","DOIUrl":"10.1007/s00013-025-02132-y","url":null,"abstract":"<div><p>Let <span>(mathbb {k})</span> be a characteristic zero Dedekind domain, <i>S</i> be a <span>(mathbb {k})</span>-algebra, and <span>(Tsubseteq S)</span> be a full rank subalgebra. Suppose the algebra <i>T</i> is symmetric. It is important to know when <i>T</i> is a <i>maximally symmetric subalgebra</i> of <i>S</i>, i.e., no <span>(mathbb {k})</span>-subalgebra <i>C</i> satisfying <span>(Tsubsetneq Csubseteq S)</span> is symmetric. In this note, we establish a useful sufficient condition for this using a notion of a quasi-unit of an algebra. This condition is used to obtain an old and a new result on maximal symmetricity for generalized Schur algebras corresponding to certain Brauer tree algebras. The old result was used in our work with Evseev on RoCK blocks of symmetric groups. The new result will be used in our forthcoming work on RoCK blocks of double covers of symmetric groups.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 2","pages":"123 - 132"},"PeriodicalIF":0.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-025-02132-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145143652","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
Resolutions over strict complete intersections 在严格的完整交叉点上的分辨率
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-05-14 DOI: 10.1007/s00013-025-02133-x
Tony J. Puthenpurakal
{"title":"Resolutions over strict complete intersections","authors":"Tony J. Puthenpurakal","doi":"10.1007/s00013-025-02133-x","DOIUrl":"10.1007/s00013-025-02133-x","url":null,"abstract":"<div><p>Let <span>((Q, mathfrak {n} ))</span> be a regular local ring and let <span>(f_1, ldots , f_c in mathfrak {n} ^2)</span> be a <i>Q</i>-regular sequence. Set <span>((A, mathfrak {m} ) = (Q/(textbf{f} ), mathfrak {n} /(textbf{f} )))</span>. Further assume that the initial forms <span>(f_1^*, ldots , f_c^*)</span> form a <span>(G(Q) = bigoplus _{n ge 0}mathfrak {n} ^i/mathfrak {n} ^{i+1})</span>-regular sequence. Without loss of any generality, assume <span>(operatorname {ord}_Q(f_1) ge operatorname {ord}_Q(f_2) ge cdots ge operatorname {ord}_Q(f_c))</span>. Let <i>M</i> be a finitely generated <i>A</i>-module and let <span>((mathbb {F} , partial ))</span> be a minimal free resolution of <i>M</i>. Then we prove that <span>(operatorname {ord}(partial _i) le operatorname {ord}_Q(f_1) - 1)</span> for all <span>(i gg 0)</span>. We also construct an MCM <i>A</i>-module <i>M</i> such that <span>(operatorname {ord}(partial _{2i+1}) = operatorname {ord}_Q(f_1) - 1)</span> for all <span>(i ge 0)</span>. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"17 - 28"},"PeriodicalIF":0.5,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145165110","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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