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Matrices for finite group representations that respect Galois automorphisms 尊重伽罗瓦自动形态的有限群代表矩阵
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-03-02 DOI: 10.1007/s00013-023-01963-x
David J. Benson
{"title":"Matrices for finite group representations that respect Galois automorphisms","authors":"David J. Benson","doi":"10.1007/s00013-023-01963-x","DOIUrl":"10.1007/s00013-023-01963-x","url":null,"abstract":"<div><p>We are given a finite group <i>H</i>, an automorphism <span>(tau )</span> of <i>H</i> of order <i>r</i>, a Galois extension <i>L</i>/<i>K</i> of fields of characteristic zero with cyclic Galois group <span>(langle sigma rangle )</span> of order <i>r</i>, and an absolutely irreducible representation <span>(rho :Hrightarrow textsf {GL} (n,L))</span> such that the action of <span>(tau )</span> on the character of <span>(rho )</span> is the same as the action of <span>(sigma )</span>. Then the following are equivalent.</p><p>   <span>(bullet )</span> <span>(rho )</span> is equivalent to a representation <span>(rho ':Hrightarrow textsf {GL} (n,L))</span> such that the action of <span>(sigma )</span> on the entries of the matrices corresponds to the action of <span>(tau )</span> on <i>H</i>, and</p><p>   <span>(bullet )</span> the induced representation <span>(textsf {ind} _{H,Hrtimes langle tau rangle }(rho ))</span> has Schur index one; that is, it is similar to a representation over <i>K</i>.</p><p>    As examples, we discuss a three dimensional irreducible representation of <span>(A_5)</span> over <span>(mathbb {Q}[sqrt{5}])</span> and a four dimensional irreducible representation of the double cover of <span>(A_7)</span> over <span>(mathbb {Q}[sqrt{-7}])</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-023-01963-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140020000","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
Cesàro means in local Dirichlet spaces 局部 Dirichlet 空间中的 Cesàro 均值
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-02-24 DOI: 10.1007/s00013-024-01967-1
J. Mashreghi, M. Nasri, M. Withanachchi
{"title":"Cesàro means in local Dirichlet spaces","authors":"J. Mashreghi,&nbsp;M. Nasri,&nbsp;M. Withanachchi","doi":"10.1007/s00013-024-01967-1","DOIUrl":"10.1007/s00013-024-01967-1","url":null,"abstract":"<div><p>The Cesàro means of Taylor polynomials <span>(sigma _n,)</span> <span>(n ge 0,)</span> are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces <span>({mathcal {D}}_zeta ,)</span> they serve as a proper LPAS. The primary objective of this note is to accurately determine the norm of <span>(sigma _n)</span> when it is considered as an operator on <span>({mathcal {D}}_zeta .)</span> There exist several practical methods to impose a norm on <span>({mathcal {D}}_zeta ,)</span> and each norm results in a distinct operator norm for <span>(sigma _n.)</span> In this context, we explore three different norms on <span>({mathcal {D}}_zeta )</span> and, for each norm, precisely compute the value of <span>(Vert sigma _nVert _{{mathcal {D}}_zeta rightarrow {mathcal {D}}_zeta }.)</span> Furthermore, in all instances, we identify the maximizing functions and demonstrate their uniqueness.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 5","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948830","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
Some new decay estimates for ((2+1))-dimensional degenerate oscillatory integral operators $$(2+1)$$ -dimensional 退化振荡积分算子的一些新衰变估计值
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-02-20 DOI: 10.1007/s00013-024-01966-2
Yuxin Tan, Shaozhen Xu
{"title":"Some new decay estimates for ((2+1))-dimensional degenerate oscillatory integral operators","authors":"Yuxin Tan,&nbsp;Shaozhen Xu","doi":"10.1007/s00013-024-01966-2","DOIUrl":"10.1007/s00013-024-01966-2","url":null,"abstract":"<div><p>In this paper, we consider the <span>((2+1))</span>-dimensional oscillatory integral operators with cubic homogeneous polynomial phases, which are degenerate in the sense of (Forum Math. 18:427–444, 2006). We improve the previously known <span>(L^2rightarrow L^2)</span> decay rate to 3/8 and also establish a sharp <span>(L^2rightarrow L^6)</span> decay estimate based on the fractional integration method.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923906","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 finiteness of radii of resolving subcategories 论解析子范畴半径的有限性
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-02-17 DOI: 10.1007/s00013-024-01965-3
Yuki Mifune
{"title":"On the finiteness of radii of resolving subcategories","authors":"Yuki Mifune","doi":"10.1007/s00013-024-01965-3","DOIUrl":"10.1007/s00013-024-01965-3","url":null,"abstract":"<div><p>Let <i>R</i> be a commutative Noetherian ring. Denote by <span>({text {mod}}R)</span> the category of finitely generated <i>R</i>-modules. In this paper, we investigate the finiteness of the radii of resolving subcategories of <span>({text {mod}}R)</span> with respect to a fixed semidualizing module. As an application, we give a partial positive answer to a conjecture of Dao and Takahashi: we prove that for a Cohen–Macaulay local ring <i>R</i>, a resolving subcategory of <span>({text {mod}}R)</span> has infinite radius whenever it contains a canonical module and a non-MCM module of finite injective dimension.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139751044","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 characterization of translation and modulation invariant Hilbert space of tempered distributions 钢化分布的平移和调制不变希尔伯特空间的表征
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-02-17 DOI: 10.1007/s00013-023-01964-w
Shubham R. Bais, Pinlodi Mohan, D. Venku Naidu
{"title":"A characterization of translation and modulation invariant Hilbert space of tempered distributions","authors":"Shubham R. Bais,&nbsp;Pinlodi Mohan,&nbsp;D. Venku Naidu","doi":"10.1007/s00013-023-01964-w","DOIUrl":"10.1007/s00013-023-01964-w","url":null,"abstract":"<div><p>Let <span>(mathcal {S}(mathbb {R}^n))</span> be the Schwartz space and <span>(mathcal {S'}(mathbb {R}^n))</span> be the space of tempered distributions on <span>(mathbb {R}^n)</span>. In this article, we prove that if <span>(mathcal {H} subseteq mathcal {S'}(mathbb {R}^n))</span> is a non-zero Hilbert space of tempered distributions which is translation and modulation invariant such that </p><div><div><span>$$begin{aligned} |(f,g)| le C Vert fVert _{mathcal {H}} end{aligned}$$</span></div></div><p>for some <span>(C&gt;0)</span> and for all <span>(fin mathcal {H})</span>, then <span>(mathcal {H}=L^2(mathbb {R}^n))</span>, where <span>(g(x) = e^{-x^2})</span> for all <span>(xin mathbb {R}^n)</span> and <span>((cdot , cdot ))</span> denotes the standard duality pairing between <span>(mathcal {S'}(mathbb {R}^n))</span> and <span>(mathcal {S}(mathbb {R}^n))</span> with respect to which <span>((mathcal {S}(mathbb {R}^n))^*=mathcal {S'}(mathbb {R}^n))</span>.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139751154","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
Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields 虚二次数域的截面梅尔滕斯和米尔斯基公式
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-02-05 DOI: 10.1007/s00013-023-01952-0
Jouni Parkkonen, Frédéric Paulin
{"title":"Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields","authors":"Jouni Parkkonen,&nbsp;Frédéric Paulin","doi":"10.1007/s00013-023-01952-0","DOIUrl":"10.1007/s00013-023-01952-0","url":null,"abstract":"<div><p>We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-023-01952-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139690335","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
Vectorial analogues of Cauchy’s surface area formula 柯西表面积公式的矢量类似公式
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-01-29 DOI: 10.1007/s00013-023-01962-y
Daniel Hug, Rolf Schneider
{"title":"Vectorial analogues of Cauchy’s surface area formula","authors":"Daniel Hug,&nbsp;Rolf Schneider","doi":"10.1007/s00013-023-01962-y","DOIUrl":"10.1007/s00013-023-01962-y","url":null,"abstract":"<div><p>Cauchy’s surface area formula says that for a convex body <i>K</i> in <i>n</i>-dimensional Euclidean space, the mean value of the <span>((n-1))</span>-dimensional volumes of the orthogonal projections of <i>K</i> to hyperplanes is a constant multiple of the surface area of <i>K</i>. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation on convex bodies.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 3","pages":"343 - 352"},"PeriodicalIF":0.5,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-023-01962-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139590110","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
Unbounded periodic constant mean curvature graphs on calibrable Cheeger Serrin domains 可校准切格-塞林域上的无界周期恒均值曲率图
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-01-25 DOI: 10.1007/s00013-023-01960-0
Ignace Aristide Minlend
{"title":"Unbounded periodic constant mean curvature graphs on calibrable Cheeger Serrin domains","authors":"Ignace Aristide Minlend","doi":"10.1007/s00013-023-01960-0","DOIUrl":"10.1007/s00013-023-01960-0","url":null,"abstract":"<div><p>We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains are calibrable and Cheeger in a suitable sense, and they solve the 1-Laplacian equation.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 3","pages":"319 - 329"},"PeriodicalIF":0.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556375","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 a theorem of Knörr 关于克诺尔的一个定理
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-01-25 DOI: 10.1007/s00013-023-01961-z
Burkhard Külshammer
{"title":"On a theorem of Knörr","authors":"Burkhard Külshammer","doi":"10.1007/s00013-023-01961-z","DOIUrl":"10.1007/s00013-023-01961-z","url":null,"abstract":"<div><p>Knörr has constructed an ideal, in the center of the <i>p</i>-modular group algebra of a finite group <i>G</i>, whose dimension is the number of <i>p</i>-blocks of defect zero in <i>G</i>/<i>Q</i>; here <i>p</i> is a prime and <i>Q</i> is a normal <i>p</i>-subgroup of <i>G</i>. We generalize his construction to symmetric algebras.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-023-01961-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556372","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 sine polynomials of Fejér and Lukács 论费耶尔和卢卡奇的正弦多项式
IF 0.5 4区 数学
Archiv der Mathematik Pub Date : 2024-01-24 DOI: 10.1007/s00013-023-01950-2
Horst Alzer, Man Kam Kwong
{"title":"On the sine polynomials of Fejér and Lukács","authors":"Horst Alzer,&nbsp;Man Kam Kwong","doi":"10.1007/s00013-023-01950-2","DOIUrl":"10.1007/s00013-023-01950-2","url":null,"abstract":"<div><p>The sine polynomials of Fejér and Lukács are defined by </p><div><div><span>$$begin{aligned} F_n(x)=sum _{k=1}^nfrac{sin (kx)}{k} quad text{ and } quad L_n(x)=sum _{k=1}^n (n-k+1)sin (kx), end{aligned}$$</span></div></div><p>respectively. We prove that for all <span>(nge 2)</span> and <span>(xin (0,pi ))</span>, we have </p><div><div><span>$$begin{aligned} F_n(x)le lambda , L_n(x) quad text{ and } quad mu le frac{1}{F_n(x)}-frac{1}{L_n(x)} end{aligned}$$</span></div></div><p>with the best possible constants </p><div><div><span>$$begin{aligned} lambda = frac{8-3sqrt{2}}{12(2-sqrt{2})} quad text{ and } quad mu =frac{2}{9}sqrt{3}. end{aligned}$$</span></div></div><p>An application of the first inequality leads to a class of absolutely monotonic functions involving the arctan function.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"122 3","pages":"307 - 317"},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139556518","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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