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Sums of infinite series involving the Riemann zeta function 涉及黎曼zeta函数的无穷级数之和
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-07-16 DOI: 10.1007/s00013-024-02008-7
Raymond Mortini, Rudolf Rupp
{"title":"Sums of infinite series involving the Riemann zeta function","authors":"Raymond Mortini,&nbsp;Rudolf Rupp","doi":"10.1007/s00013-024-02008-7","DOIUrl":"10.1007/s00013-024-02008-7","url":null,"abstract":"<div><p>We determine the values of several infinite series involving the Riemann zeta function. In particular, degree one rationally weighted summands involving the Riemann function evaluated at even numbers give a finite sum involving only the Riemann function evaluated at odd numbers.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 2","pages":"163 - 172"},"PeriodicalIF":0.5,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141642199","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 short note on the similarity of operator-valued multishifts 关于算子值多移位相似性的简短说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-07-11 DOI: 10.1007/s00013-024-02026-5
Soumitra Ghara, Surjit Kumar, Shailesh Trivedi
{"title":"A short note on the similarity of operator-valued multishifts","authors":"Soumitra Ghara,&nbsp;Surjit Kumar,&nbsp;Shailesh Trivedi","doi":"10.1007/s00013-024-02026-5","DOIUrl":"10.1007/s00013-024-02026-5","url":null,"abstract":"<div><p>A complete characterization of the similarity between two operator-valued multishifts with invertible operator weights is obtained purely in terms of operator weights. This generalizes several existing results of the unitary equivalence of two (multi)shifts. Further, we utilize the aforementioned similarity criteria to determine the similarity between two tuples of operators of multiplication by the coordinate functions on certain reproducing kernel Hilbert spaces determined by diagonal kernels.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 3","pages":"263 - 274"},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584937","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
Vector-valued holomorphic functions and abstract Fubini-type theorems 矢量值全态函数和抽象富比尼型定理
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-07-10 DOI: 10.1007/s00013-024-02019-4
Bernhard H. Haak, Markus Haase
{"title":"Vector-valued holomorphic functions and abstract Fubini-type theorems","authors":"Bernhard H. Haak,&nbsp;Markus Haase","doi":"10.1007/s00013-024-02019-4","DOIUrl":"10.1007/s00013-024-02019-4","url":null,"abstract":"<div><p>Let <span>(f = f(z,t))</span> be a function holomorphic in <span>(z in O subseteq {mathbb {C}}^d)</span> for fixed <span>(tin Omega )</span> and measurable in <i>t</i> for fixed <i>z</i> and such that <span>(z mapsto f(z,cdot ))</span> is bounded with values in <span>(E:= textrm{L}_{p}(Omega ))</span>, <span>(1le p le infty )</span>. It is proved (among other things) that </p><div><div><span>$$begin{aligned} langle tmapsto varphi ( f(cdot ,t)),mu rangle = varphi (z mapsto langle f(z, cdot ),mu rangle ) end{aligned}$$</span></div></div><p>whenever <span>(mu in E')</span> and <span>(varphi )</span> is a bp-continuous linear functional on <span>(textrm{H}^infty (O))</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 3","pages":"275 - 290"},"PeriodicalIF":0.5,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584936","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 quest for convergence: exploring series in non-linear environments 追求趋同：探索非线性环境中的数列
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-07-08 DOI: 10.1007/s00013-024-02022-9
Geivison Ribeiro
{"title":"A quest for convergence: exploring series in non-linear environments","authors":"Geivison Ribeiro","doi":"10.1007/s00013-024-02022-9","DOIUrl":"10.1007/s00013-024-02022-9","url":null,"abstract":"<div><p>This note presents an extension of a result within the concept of <span>(left[ mathcal {S}right] )</span>-lineability, originally due to Bernal-González, Conejero, Murillo-Arcila, and Seoane-Sepúlveda. Additionally, we provide a characterization of lineability in the context of complements of unions of closed subspaces in <i>F</i>-spaces in terms of <span>(left[ ell _{infty }right] )</span>-lineability. We also present a negative result in both normed spaces and <i>p</i>-Banach spaces. These findings contribute to the understanding of linearity within exotic settings in vector spaces.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 4","pages":"405 - 412"},"PeriodicalIF":0.5,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141576280","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
Sharper bounds for the numerical radius of ({n}times {n}) operator matrices $${n}/times {n}$ 算子矩阵数值半径的更精确边界
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-07-01 DOI: 10.1007/s00013-024-02017-6
Pintu Bhunia
{"title":"Sharper bounds for the numerical radius of ({n}times {n}) operator matrices","authors":"Pintu Bhunia","doi":"10.1007/s00013-024-02017-6","DOIUrl":"10.1007/s00013-024-02017-6","url":null,"abstract":"<div><p>Let <span>(A=begin{bmatrix} A_{ij} end{bmatrix})</span> be an <span>(ntimes n)</span> operator matrix, where each <span>(A_{ij})</span> is a bounded linear operator on a complex Hilbert space. Among other numerical radius bounds, we show that <span>(w(A)le w({hat{A}}),)</span> where <span>({hat{A}}=begin{bmatrix} {hat{a}}_{ij} end{bmatrix})</span> is an <span>(ntimes n)</span> complex matrix, with </p><div><div><span>$$begin{aligned} {hat{a}}_{ij}= {left{ begin{array}{ll} w(A_{ii}) &amp;{}text {when }i=j, left| | A_{ij}|+ | A_{ji}^*| right| ^{1/2} left| | A_{ji}|+ | A_{ij}^*| right| ^{1/2} &amp;{}text {when }i&lt;j, 0 &amp;{}hbox {when} i&gt;j . end{array}right. } end{aligned}$$</span></div></div><p>This is a considerable improvement of the existing bound <span>(w(A)le w({tilde{A}}),)</span> where <span>({tilde{A}}=begin{bmatrix} {tilde{a}}_{ij} end{bmatrix})</span> is an <span>(ntimes n)</span> complex matrix, with </p><div><div><span>$$begin{aligned} {tilde{a}}_{ij}= {left{ begin{array}{ll} w(A_{ii}) &amp;{}hbox {when} i=j, Vert A_{ij}Vert &amp;{}hbox {when} ine j. end{array}right. } end{aligned}$$</span></div></div><p>Further, applying the bounds, we develop the numerical radius bounds for the product of two operators and the commutator of operators. Also, we develop an upper bound for the spectral radius of the sum of the product of <i>n</i> pairs of operators, which improves the existing bound.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 2","pages":"173 - 183"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510206","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 transfer in finite group theory 关于有限群论中的转移的说明
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-26 DOI: 10.1007/s00013-024-02000-1
Morton E. Harris
{"title":"A note on transfer in finite group theory","authors":"Morton E. Harris","doi":"10.1007/s00013-024-02000-1","DOIUrl":"10.1007/s00013-024-02000-1","url":null,"abstract":"<div><p>Finite groups are ubiquitous in mathematics and often arise as symmetry groups of objects. Consequently, finite group structure is of great interest. The transfer is a classical homomorphism of any finite group <i>G</i> into certain commutative sections of <i>G</i>. It has several basic applications in and has inspired new developments in finite group structure. In this article, we present a new characterization of the image of the transfer. Then we obtain new consequences and immediate proofs of old transfer consequences in finite group structure.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 2","pages":"113 - 116"},"PeriodicalIF":0.5,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510207","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
Dovgoshey–Hariri–Vuorinen’s metric and Gromov hyperbolicity 多夫戈舍伊-哈里里-武里宁度量和格罗莫夫双曲性
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-20 DOI: 10.1007/s00013-024-02021-w
Qingshan Zhou, Zhoucheng Zheng, Saminathan Ponnusamy, Tiantian Guan
{"title":"Dovgoshey–Hariri–Vuorinen’s metric and Gromov hyperbolicity","authors":"Qingshan Zhou,&nbsp;Zhoucheng Zheng,&nbsp;Saminathan Ponnusamy,&nbsp;Tiantian Guan","doi":"10.1007/s00013-024-02021-w","DOIUrl":"10.1007/s00013-024-02021-w","url":null,"abstract":"<div><p>In 2016, Dovgoshey et al. introduced a metric <span>(zeta )</span> on proper subdomains <i>G</i> of Euclidean spaces <span>(mathbb {R}^k)</span> and studied its connection with several hyperbolic type metrics. In this paper, we consider the Gromov hyperbolicity of <span>((G,zeta ))</span> and show that there is a natural quasisymmetric homeomorphism between the Euclidean boundary of <i>G</i> and the Gromov boundary of <span>((G,zeta ))</span> equipped with a visual metric.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 3","pages":"319 - 327"},"PeriodicalIF":0.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510211","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 protrusive ordering of 5 points not witnessed by any finite multiset 任何有限多集都无法见证的 5 点突出排序
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-20 DOI: 10.1007/s00013-024-02020-x
Adrian Beker
{"title":"A protrusive ordering of 5 points not witnessed by any finite multiset","authors":"Adrian Beker","doi":"10.1007/s00013-024-02020-x","DOIUrl":"10.1007/s00013-024-02020-x","url":null,"abstract":"<div><p>Given a finite set of points <span>(C subseteq {mathbb {R}}^d)</span>, we say that an ordering of <i>C</i> is <i>protrusive</i> if every point lies outside the convex hull of the points preceding it. We give an example of a set <i>C</i> of 5 points in the Euclidean plane possessing a protrusive ordering that cannot be obtained by ranking the points of <i>C</i> according to the sum of their distances to a finite multiset of points. This answers a question of Alon, Defant, Kravitz, and Zhu.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 4","pages":"399 - 403"},"PeriodicalIF":0.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510210","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
Counterexamples to maximal regularity for operators in divergence form 发散形式算子最大正则性的反例
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-20 DOI: 10.1007/s00013-024-02014-9
Sebastian Bechtel, Connor Mooney, Mark Veraar
{"title":"Counterexamples to maximal regularity for operators in divergence form","authors":"Sebastian Bechtel,&nbsp;Connor Mooney,&nbsp;Mark Veraar","doi":"10.1007/s00013-024-02014-9","DOIUrl":"10.1007/s00013-024-02014-9","url":null,"abstract":"<div><p>In this paper, we present counterexamples to maximal <span>(L^p)</span>-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal <span>(L^2)</span>-regularity on <span>(H^{-1})</span> under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal <span>(L^p)</span>-regularity on <span>(H^{-1}(mathbb {R}^d))</span> or <span>(L^2)</span>-regularity on <span>(L^2(mathbb {R}^d))</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 2","pages":"199 - 209"},"PeriodicalIF":0.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-024-02014-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510208","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 interpretation of Fourier coefficients of Zagier type lifts 关于扎吉尔式升降机傅里叶系数的解释
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-06-13 DOI: 10.1007/s00013-024-02005-w
Vaibhav Kalia
{"title":"On interpretation of Fourier coefficients of Zagier type lifts","authors":"Vaibhav Kalia","doi":"10.1007/s00013-024-02005-w","DOIUrl":"10.1007/s00013-024-02005-w","url":null,"abstract":"<div><p>Jeon, Kang, and Kim defined the Zagier lifts between harmonic weak Maass forms of negative integral weights and half integral weights. These lifts were defined by establishing that traces related to cycle integrals of harmonic weak Maass forms of integral weights appear as Fourier coefficients of harmonic weak Maass forms of half integral weights. For fundamental discriminants <i>d</i> and <span>(delta ,)</span> they studied <span>(delta )</span>-th Fourier coefficients of the <i>d</i>-th Zagier lift with respect to the condition that <span>(ddelta )</span> is not a perfect square. For <span>(ddelta )</span> being a perfect square, the interpretation of coefficients in terms of traces is not possible due to the divergence of cycle integrals. In this paper, we provide an alternate definition of traces called <i>modified trace</i> in the condition that <span>(ddelta )</span> is a perfect square and interpret such coefficients in terms of the <i>modified trace</i>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"123 2","pages":"147 - 162"},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141350095","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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