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A note on odd partition numbers 关于奇数分区号的说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-24 DOI: 10.1007/s00013-024-01999-7
Michael Griffin, Ken Ono
{"title":"A note on odd partition numbers","authors":"Michael Griffin,&nbsp;Ken Ono","doi":"10.1007/s00013-024-01999-7","DOIUrl":"10.1007/s00013-024-01999-7","url":null,"abstract":"<div><p>Ramanujan’s partition congruences modulo <span>(ell in {5, 7, 11})</span> assert that </p><div><div><span>$$begin{aligned} p(ell n+delta _{ell })equiv 0pmod {ell }, end{aligned}$$</span></div></div><p>where <span>(0&lt;delta _{ell }&lt;ell )</span> satisfies <span>(24delta _{ell }equiv 1pmod {ell }.)</span> By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes <span>(ell ge 5,)</span> we give a new proof of the conclusion that there are infinitely many <i>m</i> for which <span>(p(ell m+delta _{ell }))</span> is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such <i>m</i> satisfies <span>(m&lt;(ell ^2-1)/24,)</span> representing a significant improvement to the previous bound.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01999-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146584","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
Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces 恒定曲率曲面上磁盘切线束的霍费尔-泽恩德容量
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-16 DOI: 10.1007/s00013-024-02003-y
Johanna Bimmermann
{"title":"Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces","authors":"Johanna Bimmermann","doi":"10.1007/s00013-024-02003-y","DOIUrl":"10.1007/s00013-024-02003-y","url":null,"abstract":"<div><p>We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu’s bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian <i>H</i> will give rise to the non-vanishing Gromov–Witten invariant.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02003-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063958","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 reciprocity law in function fields 函数场中的互易定律
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-16 DOI: 10.1007/s00013-024-02006-9
Yoshinori Hamahata
{"title":"A reciprocity law in function fields","authors":"Yoshinori Hamahata","doi":"10.1007/s00013-024-02006-9","DOIUrl":"10.1007/s00013-024-02006-9","url":null,"abstract":"<div><p>We generalize Gauss’ lemma over function fields, and establish a reciprocity law for power residue symbols. As an application, a reciprocity law for power residue symbols is established in totally imaginary function fields.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140967104","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 vertical Sato-Tate law for GL(4) GL(4) 的垂直萨托-塔特定律
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-15 DOI: 10.1007/s00013-024-01996-w
Tian An Wong
{"title":"A vertical Sato-Tate law for GL(4)","authors":"Tian An Wong","doi":"10.1007/s00013-024-01996-w","DOIUrl":"10.1007/s00013-024-01996-w","url":null,"abstract":"<div><p>We establish a doubly-weighted vertical Sato-Tate law for GL(4) with explicit error terms. The main ingredient is an extension of the orthogonality relation for Maass cusp forms on GL(4) of Goldfeld, Stade, and Woodbury from spherical to general forms, and without their assumption of the Ramanujan conjecture for the error term.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972534","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
Combinatorial upper bounds for the smallest eigenvalue of a graph 图形最小特征值的组合上限
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-14 DOI: 10.1007/s00013-024-01998-8
Aryan Esmailpour, Sara Saeedi Madani, Dariush Kiani
{"title":"Combinatorial upper bounds for the smallest eigenvalue of a graph","authors":"Aryan Esmailpour,&nbsp;Sara Saeedi Madani,&nbsp;Dariush Kiani","doi":"10.1007/s00013-024-01998-8","DOIUrl":"10.1007/s00013-024-01998-8","url":null,"abstract":"<div><p>Let <i>G</i> be a graph, and let <span>(lambda (G))</span> denote the smallest eigenvalue of <i>G</i>. First, we provide an upper bound for <span>(lambda (G))</span> based on induced bipartite subgraphs of <i>G</i>. Consequently, we extract two other upper bounds, one relying on the average degrees of induced bipartite subgraphs and a more explicit one in terms of the chromatic number and the independence number of <i>G</i>. In particular, motivated by our bounds, we introduce two graph invariants that are of interest on their own. Finally, special attention goes to the investigation of the sharpness of our bounds in various classes of graphs as well as the comparison with an existing well-known upper bound.\u0000</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936648","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
Boundary values of analytic semigroups generated by fractional Laplacians 分数拉普拉斯生成的解析半群的边界值
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-13 DOI: 10.1007/s00013-024-02004-x
Chung-Sik Sin
{"title":"Boundary values of analytic semigroups generated by fractional Laplacians","authors":"Chung-Sik Sin","doi":"10.1007/s00013-024-02004-x","DOIUrl":"10.1007/s00013-024-02004-x","url":null,"abstract":"<div><p>In the present paper, using the theory of boundary values of analytic semigroups, we find necessary and sufficient conditions to guarantee that the operator <span>(i(-Delta )^{{alpha }/{2}})</span> generates a strongly continuous semigroup in <span>(L^p(mathbb {R}^n))</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936943","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 Ambarzumian’s theorem for quantum graphs 关于量子图的 Ambarzumian 定理的说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-05-04 DOI: 10.1007/s00013-024-01997-9
Patrizio Bifulco, Joachim Kerner
{"title":"A note on Ambarzumian’s theorem for quantum graphs","authors":"Patrizio Bifulco,&nbsp;Joachim Kerner","doi":"10.1007/s00013-024-01997-9","DOIUrl":"10.1007/s00013-024-01997-9","url":null,"abstract":"<div><p>Based on the main result presented in Bifulco and Kerner (Proc Am Math Soc 152:295–306, 2024), we derive Ambarzumian–type theorems for Schrödinger operators defined on quantum graphs.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01997-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886845","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
Non-disjoint strong external difference families can have any number of sets 非相交强外差族可以有任意数量的集合
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-30 DOI: 10.1007/s00013-024-01982-2
Sophie Huczynska, Siaw-Lynn Ng
{"title":"Non-disjoint strong external difference families can have any number of sets","authors":"Sophie Huczynska,&nbsp;Siaw-Lynn Ng","doi":"10.1007/s00013-024-01982-2","DOIUrl":"10.1007/s00013-024-01982-2","url":null,"abstract":"<div><p>Strong external difference families (SEDFs) are much-studied combinatorial objects motivated by an information security application. A well-known conjecture states that only one abelian SEDF with more than 2 sets exists. We show that if the disjointness condition is replaced by non-disjointness, then abelian SEDFs can be constructed with more than 2 sets (indeed any number of sets). We demonstrate that the non-disjoint analogue has striking differences to, and connections with, the classical SEDF and arises naturally via another coding application.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01982-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140834145","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
Generic norm growth of powers of homogeneous unimodular Fourier multipliers 同质单模态傅里叶乘数幂的通用规范增长
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-22 DOI: 10.1007/s00013-024-01994-y
Aleksandar Bulj
{"title":"Generic norm growth of powers of homogeneous unimodular Fourier multipliers","authors":"Aleksandar Bulj","doi":"10.1007/s00013-024-01994-y","DOIUrl":"10.1007/s00013-024-01994-y","url":null,"abstract":"<div><p>For an integer <span>(dge 2)</span>, <span>(tin mathbb {R})</span>, and a 0-homogeneous function <span>(Phi in C^{infty }(mathbb {R}^{d}{setminus }{0},mathbb {R}))</span>, we consider the family of Fourier multiplier operators <span>(T_{Phi }^t)</span> associated with symbols <span>(xi mapsto exp (itPhi (xi )))</span> and prove that for a generic phase function <span>(Phi )</span>, one has the estimate <span>(Vert T_{Phi }^tVert _{L^prightarrow L^p} gtrsim _{d,p, Phi }langle trangle ^{d|frac{1}{p}-frac{1}{2}|})</span>. That is the maximal possible order of growth in <span>(trightarrow pm infty )</span>, according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions <span>(Phi )</span> that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz’ya actually exhibit the same general phenomenon.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799364","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
Rational numbers with small denominators in short intervals 短区间小分母有理数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-04-17 DOI: 10.1007/s00013-024-01993-z
Igor E. Shparlinski
{"title":"Rational numbers with small denominators in short intervals","authors":"Igor E. Shparlinski","doi":"10.1007/s00013-024-01993-z","DOIUrl":"10.1007/s00013-024-01993-z","url":null,"abstract":"<div><p>We use bounds on bilinear forms with Kloosterman fractions and improve the error term in the asymptotic formula of Balazard and Martin (Bull Sci Math 187:Art. 103305, 2023) on the average value of the smallest denominators of rational numbers in short intervals.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-01993-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140615152","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
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