Forum Mathematicum最新文献

{"title":"Two curious q-supercongruences and their extensions","authors":"Haihong He, Xiaoxia Wang","doi":"10.1515/forum-2023-0164","DOIUrl":"https://doi.org/10.1515/forum-2023-0164","url":null,"abstract":"We prove two single-parameter <jats:italic>q</jats:italic>-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of the <jats:italic>q</jats:italic>-binomial theorem, a Karlsson–Minton-type summation formula due to Gasper, and the method of “creative microscoping” developed by Guo and Zudilin. Incidentally, an assertion of Li, Tang and Wang is also confirmed by establishing its <jats:italic>q</jats:italic>-analogue.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"16 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of fractional heat semigroups associated with Schrödinger operators on Heisenberg groups","authors":"Chuanhong Sun, Pengtao Li, Zengjian Lou","doi":"10.1515/forum-2023-0285","DOIUrl":"https://doi.org/10.1515/forum-2023-0285","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;L&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℍ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0800.png\" /&gt; &lt;jats:tex-math&gt;{L=-{Delta}_{mathbb{H}^{n}}+V}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a Schrödinger operator on Heisenberg groups &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℍ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0890.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{H}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℍ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_1058.png\" /&gt; &lt;jats:tex-math&gt;{{Delta}_{mathbb{H}^{n}}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is the sub-Laplacian, the nonnegative potential &lt;jats:italic&gt;V&lt;/jats:italic&gt; belongs to the reverse Hölder class &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;B&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒬&lt;/m:mi&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0748.png\" /&gt; &lt;jats:tex-math&gt;{B_{mathcal{Q}/2}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Here &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒬&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0895.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{Q}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is the homogeneous dimension of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℍ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0890.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{H}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. In this article, we introduce the fractional heat semigroups &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;{&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;e&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiplicity of solutions for a singular system with sign-changing potential","authors":"Wentao Lin, Yilan Wei","doi":"10.1515/forum-2023-0345","DOIUrl":"https://doi.org/10.1515/forum-2023-0345","url":null,"abstract":"This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0345_eq_0345.png\" /> <jats:tex-math>{mathbb{R}^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center","authors":"Diego García-Lucas, Leo Margolis","doi":"10.1515/forum-2023-0237","DOIUrl":"https://doi.org/10.1515/forum-2023-0237","url":null,"abstract":"We show that the modular isomorphism problem has a positive answer for groups of nilpotency class 2 with cyclic center, i.e., that for such <jats:italic>p</jats:italic>-groups <jats:italic>G</jats:italic> and <jats:italic>H</jats:italic> an isomorphism between the group algebras <jats:italic>FG</jats:italic> and <jats:italic>FH</jats:italic> implies an isomorphism of the groups <jats:italic>G</jats:italic> and <jats:italic>H</jats:italic> for <jats:italic>F</jats:italic> the field of <jats:italic>p</jats:italic> elements. For groups of odd order this implication is also proven for <jats:italic>F</jats:italic> being any field of characteristic <jats:italic>p</jats:italic>. For groups of even order we need either to make an additional assumption on the groups or on the field.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Donoho–Stark and Price uncertainty principles for a class of q-integral transforms with bounded kernels","authors":"Luis P. Castro, Rita C. Guerra","doi":"10.1515/forum-2023-0244","DOIUrl":"https://doi.org/10.1515/forum-2023-0244","url":null,"abstract":"We consider a very global <jats:italic>q</jats:italic>-integral transform, essentially characterized by having a bounded kernel and satisfying a set of natural and useful properties for the realization of applications. The main ambition of this work is to seek conditions that guarantee uncertainty principles of the Donoho–Stark type for that class of <jats:italic>q</jats:italic>-integral transforms. It should be noted that the global character of the <jats:italic>q</jats:italic>-integral transform in question allows one to immediately deduce corresponding Donoho–Stark uncertainty principles for <jats:italic>q</jats:italic>-integral operators that are its particular cases. These particular cases are very well-known operators, namely: a <jats:italic>q</jats:italic>-cosine-Fourier transform, a <jats:italic>q</jats:italic>-sine-Fourier transform, a <jats:italic>q</jats:italic>-Fourier transform, a <jats:italic>q</jats:italic>-Bessel–Fourier transform and a <jats:italic>q</jats:italic>-Dunkl transform. Moreover, generalizations of the local uncertainty principle of Price for the <jats:italic>q</jats:italic>-cosine-Fourier transform, <jats:italic>q</jats:italic>-sine-Fourier transform, <jats:italic>q</jats:italic>-Fourier transform, <jats:italic>q</jats:italic>-Bessel–Fourier transform and <jats:italic>q</jats:italic>-Dunkl transform are also obtained.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"51 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139079798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A note on the post quantum-Sheffer polynomial sequences","authors":"Subuhi Khan, Mehnaz Haneef","doi":"10.1515/forum-2023-0004","DOIUrl":"https://doi.org/10.1515/forum-2023-0004","url":null,"abstract":"In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"17 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds","authors":"Changhua Wei","doi":"10.1515/forum-2023-0139","DOIUrl":"https://doi.org/10.1515/forum-2023-0139","url":null,"abstract":"We are interested in the one-dimensional nonlinear wave equations with multiple wave speeds by the energy method. By choosing different multipliers corresponding to the different wave speeds, we show that the one-dimensional nonlinear wave equations also have globally smooth solutions provided that the nonlinearities satisfy certain structural conditions when the initial data are small. Furthermore, we can prove that the global solutions will converge to the solutions of the linearized system based on the decay properties of the nonlinearities.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139080012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles","authors":"Yixi Liao, Pinren Qian, Erxiao Wang, Yingyun Xu","doi":"10.1515/forum-2023-0209","DOIUrl":"https://doi.org/10.1515/forum-2023-0209","url":null,"abstract":"Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>a</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mi>b</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0209_eq_1951.png\" /> <jats:tex-math>{a^{3}b}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139080264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enochs’ conjecture for cotorsion pairs and more","authors":"Silvana Bazzoni, Jan Šaroch","doi":"10.1515/forum-2023-0220","DOIUrl":"https://doi.org/10.1515/forum-2023-0220","url":null,"abstract":"Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;Filt&lt;/m:mi&gt; &lt;m:mo&gt;⁡&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒮&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0359.png\" /&gt; &lt;jats:tex-math&gt;{operatorname{Filt}(mathcal{S})}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒮&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0332.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{S}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; consists of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;ℵ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0211.png\" /&gt; &lt;jats:tex-math&gt;{aleph_{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-presented modules for some fixed &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;ω&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0475.png\" /&gt; &lt;jats:tex-math&gt;{n&lt;omega}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;ℵ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0211.png\" /&gt; &lt;jats:tex-math&gt;{aleph_{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;Filt&lt;/m:mi&gt; &lt;m:mo&gt;⁡&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒮&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0359.png\" /&gt; &lt;jats:tex-math&gt;{operatorname{Filt}(m","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite approximation properties of C*-modules III","authors":"Massoud Amini","doi":"10.1515/forum-2023-0283","DOIUrl":"https://doi.org/10.1515/forum-2023-0283","url":null,"abstract":"We introduce and study a notion of module nuclear dimension for a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi mathvariant=\"normal\"&gt;C&lt;/m:mi&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /&gt; &lt;jats:tex-math&gt;mathrm{C}^{*}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-algebra &lt;jats:italic&gt;A&lt;/jats:italic&gt; which is a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi mathvariant=\"normal\"&gt;C&lt;/m:mi&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /&gt; &lt;jats:tex-math&gt;mathrm{C}^{*}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-module over another &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi mathvariant=\"normal\"&gt;C&lt;/m:mi&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /&gt; &lt;jats:tex-math&gt;mathrm{C}^{*}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-algebra &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;𝔄&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /&gt; &lt;jats:tex-math&gt;{mathfrak{A}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; with compatible actions. We show that the module nuclear dimension of &lt;jats:italic&gt;A&lt;/jats:italic&gt; is zero if &lt;jats:italic&gt;A&lt;/jats:italic&gt; is &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;𝔄&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /&gt; &lt;jats:tex-math&gt;{mathfrak{A}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-NF. The converse is shown to hold when &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;𝔄&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /&gt; &lt;jats:tex-math&gt;{mathfrak{A}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;C&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;X&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0344.png\" /&gt; &lt;jats:tex-math&gt;{C(X)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-algebra with simple fibers, with &lt;jats:italic&gt;X&lt;/jats:italic&gt; compact and totally d","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}