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On the average size of the eigenvalues of the Hecke operators 赫克算子特征值的平均大小
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-23 DOI: 10.1007/s00013-024-02089-4
William Cason, Akash Jim, Charlie Medlock, Erick Ross, Hui Xue
{"title":"On the average size of the eigenvalues of the Hecke operators","authors":"William Cason,&nbsp;Akash Jim,&nbsp;Charlie Medlock,&nbsp;Erick Ross,&nbsp;Hui Xue","doi":"10.1007/s00013-024-02089-4","DOIUrl":"10.1007/s00013-024-02089-4","url":null,"abstract":"<div><p>We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space <span>(S_k(Gamma _0(N)))</span> in both the vertical and the horizontal perspective. The “average size” is measured via the quadratic mean.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"255 - 263"},"PeriodicalIF":0.5,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02089-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184870","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
Products of Catalan numbers which are squares 加泰罗尼亚数的平方乘积
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-23 DOI: 10.1007/s00013-024-02088-5
Lajos Hajdu, Florian Luca, Szabolcs Tengely, Maciej Ulas
{"title":"Products of Catalan numbers which are squares","authors":"Lajos Hajdu,&nbsp;Florian Luca,&nbsp;Szabolcs Tengely,&nbsp;Maciej Ulas","doi":"10.1007/s00013-024-02088-5","DOIUrl":"10.1007/s00013-024-02088-5","url":null,"abstract":"<div><p>Let <span>(C_{n})</span> be the <i>n</i>-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each <span>(kge 3)</span>, there are infinitely many <i>k</i>-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of <span>(xin mathbb {N}_{+})</span> such that <span>(C_{x}C_{x+1})</span> is a power-full number and prove that there are infinitely many such <i>x</i>. Moreover we present some numerical results which motivate further problems.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"265 - 281"},"PeriodicalIF":0.5,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184636","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 further q-analogue of a formula due to Guillera 另一个由吉列拉提出的q-类似公式
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-22 DOI: 10.1007/s00013-024-02094-7
John M. Campbell
{"title":"A further q-analogue of a formula due to Guillera","authors":"John M. Campbell","doi":"10.1007/s00013-024-02094-7","DOIUrl":"10.1007/s00013-024-02094-7","url":null,"abstract":"<div><p>Hou, Krattenthaler, and Sun have introduced two <i>q</i>-analogues of a remarkable series for <span>(pi ^2)</span> due to Guillera, and these <i>q</i>-identities were, respectively, proved with the use of a <i>q</i>-analogue of a Wilf–Zeilberger pair provided by Guillera and with the use of <span>( _{3}phi _{2})</span>-transforms. We prove a <i>q</i>-analogue of Guillera’s formula for <span>(pi ^2)</span> that is inequivalent to previously known <i>q</i>-analogues of the same formula due to Guillera, including the Hou–Krattenthaler–Sun <i>q</i>-identities and a subsequent <i>q</i>-identity due to Wei. In contrast to previously known <i>q</i>-analogues of Guillera’s formula, our new <i>q</i>-analogue involves another free parameter apart from the <i>q</i>-parameter. Our derivation of this new result relies on the <i>q</i>-analogue of Zeilberger’s algorithm.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 5","pages":"511 - 516"},"PeriodicalIF":0.5,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143818162","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 rank of projective modules 关于投影模的秩
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-20 DOI: 10.1007/s00013-024-02081-y
F.E.A. Johnson
{"title":"On the rank of projective modules","authors":"F.E.A. Johnson","doi":"10.1007/s00013-024-02081-y","DOIUrl":"10.1007/s00013-024-02081-y","url":null,"abstract":"<div><p>Let <i>P</i> be a nonzero projective module over an integral group ring. We consider the question of whether the rank of <i>P</i> is necessarily positive.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"233 - 241"},"PeriodicalIF":0.5,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02081-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184864","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
Poincaré inequality for one-forms on four manifolds with bounded Ricci curvature 具有有界里奇曲率的四流形上单形的波恩卡莱不等式
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-20 DOI: 10.1007/s00013-024-02091-w
Shouhei Honda, Andrea Mondino
{"title":"Poincaré inequality for one-forms on four manifolds with bounded Ricci curvature","authors":"Shouhei Honda,&nbsp;Andrea Mondino","doi":"10.1007/s00013-024-02091-w","DOIUrl":"10.1007/s00013-024-02091-w","url":null,"abstract":"<div><p>In this short note, we provide a quantitative global Poincaré inequality for one-forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on the Ricci curvature. This seems to be the first non-trivial result giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov–Hausdorff convergence, via a degeneration result to orbifolds by Anderson.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 4","pages":"449 - 455"},"PeriodicalIF":0.5,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02091-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621815","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
Nonexistence of certain regular maps of 2-power order 某些2-幂阶正则映射的不存在性
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-17 DOI: 10.1007/s00013-024-02093-8
Yao Tian, Xiaogang Li
{"title":"Nonexistence of certain regular maps of 2-power order","authors":"Yao Tian,&nbsp;Xiaogang Li","doi":"10.1007/s00013-024-02093-8","DOIUrl":"10.1007/s00013-024-02093-8","url":null,"abstract":"<div><p>In a recent paper, Hou et al. conjectured that there exist no regular maps of order <span>(2^n)</span> and of type <span>({2^k,2^s})</span>, where <i>n</i>, <i>k</i>, and <i>s</i> are positive integers satisfying <span>(2le s&lt;k&lt;n-1)</span> and <span>(s+k&gt;n)</span>. In this paper, we give an affirmative answer to this conjecture.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 4","pages":"389 - 394"},"PeriodicalIF":0.5,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622101","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
Correction to: A reciprocity law in function fields 修正：函数场中的互易律
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-17 DOI: 10.1007/s00013-024-02092-9
Yoshinori Hamahata
{"title":"Correction to: A reciprocity law in function fields","authors":"Yoshinori Hamahata","doi":"10.1007/s00013-024-02092-9","DOIUrl":"10.1007/s00013-024-02092-9","url":null,"abstract":"","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"355 - 356"},"PeriodicalIF":0.5,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02092-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184800","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 the Krull dimension of rings of integer-valued rational functions 整数有理函数环的Krull维数
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-03 DOI: 10.1007/s00013-024-02086-7
M. M. Chems-Eddin, B. Feryouch, H. Mouanis, A. Tamoussit
{"title":"On the Krull dimension of rings of integer-valued rational functions","authors":"M. M. Chems-Eddin,&nbsp;B. Feryouch,&nbsp;H. Mouanis,&nbsp;A. Tamoussit","doi":"10.1007/s00013-024-02086-7","DOIUrl":"10.1007/s00013-024-02086-7","url":null,"abstract":"<div><p>Let <i>D</i> be an integral domain with quotient field <i>K</i> and <i>E</i> a subset of <i>K</i>. The <i>ring of integer-valued rational functions on</i> <i>E</i> is defined as </p><div><div><span>$$begin{aligned} mathrm {Int^R}(E,D):=lbrace varphi in K(X);; varphi (E)subseteq Drbrace . end{aligned}$$</span></div></div><p>The main goal of this paper is to investigate the Krull dimension of the ring <span>(mathrm {Int^R}(E,D).)</span> Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"243 - 254"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184674","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 seminorm characterization of infinite Banach direct sums 无限巴拿赫直接和的半规范表征
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-03 DOI: 10.1007/s00013-024-02080-z
Hojjatollah Samea
{"title":"A seminorm characterization of infinite Banach direct sums","authors":"Hojjatollah Samea","doi":"10.1007/s00013-024-02080-z","DOIUrl":"10.1007/s00013-024-02080-z","url":null,"abstract":"<div><p>In this paper, the notion of a <span>(Delta )</span>-direct sum of a family of Banach spaces indexed by a set <i>I</i>, where <span>(Delta )</span> is a union-closed subnet of <span>(textsf{Fin}(I))</span> (the family of all finite subsets of <i>I</i>), is introduced. A seminorm characterization of <span>(Delta )</span>-direct sums and some results are presented. Necessary and sufficient conditions are found that a direct sum of a family of Banach spaces is a <span>(Delta )</span>-direct sum. Elements of a direct sum of Banach spaces that are <span>(Delta )</span>-sectionally convergent are introduced and studied. Examples of <span>(Delta )</span>-direct sums and applications of <span>(Delta )</span>-direct sums to Fourier analysis on compact groups are given.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"283 - 299"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184675","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
Choquet integrals, Hausdorff content and sparse operators Choquet积分，Hausdorff内容和稀疏算子
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2025-01-03 DOI: 10.1007/s00013-024-02083-w
Naoya Hatano, Ryota Kawasumi, Hiroki Saito, Hitoshi Tanaka
{"title":"Choquet integrals, Hausdorff content and sparse operators","authors":"Naoya Hatano,&nbsp;Ryota Kawasumi,&nbsp;Hiroki Saito,&nbsp;Hitoshi Tanaka","doi":"10.1007/s00013-024-02083-w","DOIUrl":"10.1007/s00013-024-02083-w","url":null,"abstract":"<div><p>Let <span>(H^d)</span>, <span>(0&lt;d&lt;n)</span>, be the dyadic Hausdorff content of the <i>n</i>-dimensional Euclidean space <span>({{mathbb {R}}}^n)</span>. It is shown that <span>(H^d)</span> counts a Cantor set of the unit cube <span>([0, 1)^n)</span> as <span>(approx 1)</span>, which implies the unboundedness of the sparse operator <span>({{mathcal {A}}}_{{{mathcal {S}}}})</span> on the Choquet space <span>({mathcal L}^p(H^d))</span>, <span>(p&gt;0)</span>. In this paper, the sparse operator <span>({mathcal A}_{{{mathcal {S}}}})</span> is proved to map <span>({{mathcal {L}}}^p(H^d))</span>, <span>(1le p&lt;infty )</span>, into an associate space of the Orlicz-Morrey space <span>({{{mathcal {M}}}^{p'}_{Phi _0}(H^d)}')</span>, <span>(Phi _0(t)=tlog (e+t))</span>. Further, another characterization of those associate spaces is given by means of the tiling <span>({{mathcal {T}}})</span> of <span>({{mathbb {R}}}^n)</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 3","pages":"311 - 324"},"PeriodicalIF":0.5,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143184676","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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