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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
Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence 沃尔什系统的勒贝格常数和范德科普特序列的差异
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-12-24 DOI: 10.1007/s00013-024-02087-6
Josef Dick, Friedrich Pillichshammer
{"title":"Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence","authors":"Josef Dick,&nbsp;Friedrich Pillichshammer","doi":"10.1007/s00013-024-02087-6","DOIUrl":"10.1007/s00013-024-02087-6","url":null,"abstract":"<div><p>In this short note, we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important role in approximation theory, and on the other hand, there is the star discrepancy of the van der Corput sequence that plays a prominent role in uniform distribution theory. Over the decades, these two quantities have been examined in great detail independently of each other and important results have been proven. Work in these areas has been carried out independently, but as we show here, they actually coincide. Interestingly, many theorems have been discovered in both areas independently, but some results have only been known in one area but not in the other.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 4","pages":"407 - 414"},"PeriodicalIF":0.5,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02087-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622037","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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