Using a F67{{}_{7}F_{6}} hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long and Ramakrishna.
利用 F 6 7 {{}_{7}F_{6}} 超几何变换公式，我们证明了两个超级共轭。其中一个超共轭证实了郭，刘和施洛瑟最近的猜想，并给出了龙和拉马克里希纳的超共轭的扩展。
{"title":"Supercongruences arising from a 7 F 6 hypergeometric transformation formula","authors":"Chen Wang","doi":"10.1515/forum-2023-0239","DOIUrl":"https://doi.org/10.1515/forum-2023-0239","url":null,"abstract":"Using a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>6</m:mn> <m:none /> <m:mprescripts /> <m:mn>7</m:mn> <m:none /> </m:mmultiscripts> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0239_eq_0198.png\" /> <jats:tex-math>{{}_{7}F_{6}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> hypergeometric transformation formula, we prove two supercongruences. In particular, one of these supercongruences confirms a recent conjecture of Guo, Liu and Schlosser, and gives an extension of a supercongruence of Long and Ramakrishna.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"180 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422925","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}
Serhii Bardyla, Luke Elliott, James D. Mitchell, Yann Péresse
In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0294.png" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or the symmetric inverse monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>I</m:mi> <m:mi>ℕ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0187.png" /> <jats:tex-math>{I_{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0294.png" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>I</m:mi> <m:mi>ℕ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0187.png" /> <jats:tex-math>{I_{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We construct several examples of countable Polish topological semigroups that do not embed into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0294.png" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0230_eq_0294.png" /> <jats:tex-math>{mathb
{"title":"Topological embeddings into transformation monoids","authors":"Serhii Bardyla, Luke Elliott, James D. Mitchell, Yann Péresse","doi":"10.1515/forum-2023-0230","DOIUrl":"https://doi.org/10.1515/forum-2023-0230","url":null,"abstract":"In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0294.png\" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or the symmetric inverse monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>I</m:mi> <m:mi>ℕ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0187.png\" /> <jats:tex-math>{I_{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0294.png\" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>I</m:mi> <m:mi>ℕ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0187.png\" /> <jats:tex-math>{I_{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We construct several examples of countable Polish topological semigroups that do not embed into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0294.png\" /> <jats:tex-math>{mathbb{N}^{mathbb{N}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℕ</m:mi> <m:mi>ℕ</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0230_eq_0294.png\" /> <jats:tex-math>{mathb","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374731","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}
We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.
{"title":"Wells-type exact sequence and crossed extensions of algebras with bracket","authors":"José Manuel Casas, Emzar Khmaladze, Manuel Ladra","doi":"10.1515/forum-2023-0355","DOIUrl":"https://doi.org/10.1515/forum-2023-0355","url":null,"abstract":"We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"60 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374734","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}
Quaternion algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℍ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0389_eq_0331.png" /> <jats:tex-math>{mathbb{H}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℍ</m:mi> <m:mi>M</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0389_eq_0330.png" /> <jats:tex-math>{mathbb{H}^{M}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0389_eq_0275.png" /> <jats:tex-math>{Mgeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Write <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ℂ</m:mi> <m:mi>η</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>ξ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mi>η</m:mi> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0389_eq_0316.png" /> <jats:tex-math>{mathbb{C}_{eta}={xi:xi=xi_{0}+betaeta,,xi_{0},,betainmathbb{R}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>i</m:mi> <m:mo rspace="4.2pt">,</m:mo> <m:mi>j</m:mi> <m:mo rspace="4.2pt">,</m:mo> <m:mi>k</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0389_eq_0298.png" /> <jats:tex-math>{etain{i,,j,,k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We remark that not only <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998
四元代数ℍ {mathbb{H}} 是一种非交换关联代数。近年来，四元数傅里叶分析法因其在信号分析和图像处理中的应用而受到越来越多的关注。本文讨论的是四元欧几里得空间ℍ M {mathbb{H}^{M}} 中的共轭相位检索问题，其中 M ≥ 2 {Mgeq 2} 。写 ℂ η = { ξ : ξ = ξ 0 + β η , ξ 0 , β ∈ ℝ }. {mathbb{C}_{eta}={xi:xi=xi_{0}+betaeta,,xi_{0},,betainmathbb{R}}} for η ∈ { i , j , k } {etain{i,,j,,k}} .我们注意到不仅 η M {mathbb{C}_{eta}^{M}}. -向量无法在ℍ M {mathbb{H}^{M}} 中进行传统的共轭相位检索，而且ℍ M {mathbb{H}^{M}} 也无法进行传统的共轭相位检索。 而且 ℂ i M ∪ ℂ j M {mathbb{C}_{i}^{M}cupmathbb{C}_{j}^{M}} -复数向量也不能在 ℂ M {mathbb{H}^{M}} 中进行传统的共轭相位检索。 -复数向量无法在ℍ M {mathbb{H}^{M}} 中进行相位检索。 .我们致力于 ℂ i M ∪ ℂ j M {mathbb{C}_{i}^{M}cupmathbb{C}_{j}^{M}} 的共轭相位检索。 -ℍ M {mathbb{H}^{M}} 中的复数向量。 这里的 "共轭 "并非传统意义上的共轭。我们介绍了共轭、最大交换子集和共轭相位检索等概念。利用相位提升技术，我们提出了一些允许共轭相位检索的复杂向量的充分条件。我们还提供了一些例子来说明我们理论的普遍性。
{"title":"What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces","authors":"Yun-Zhang Li, Ming Yang","doi":"10.1515/forum-2023-0389","DOIUrl":"https://doi.org/10.1515/forum-2023-0389","url":null,"abstract":"Quaternion algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℍ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0389_eq_0331.png\" /> <jats:tex-math>{mathbb{H}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℍ</m:mi> <m:mi>M</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0389_eq_0330.png\" /> <jats:tex-math>{mathbb{H}^{M}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>M</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0389_eq_0275.png\" /> <jats:tex-math>{Mgeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Write <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>ℂ</m:mi> <m:mi>η</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mi>ξ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mi>η</m:mi> </m:mrow> </m:mrow> <m:mo rspace=\"4.2pt\">,</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:mrow> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0389_eq_0316.png\" /> <jats:tex-math>{mathbb{C}_{eta}={xi:xi=xi_{0}+betaeta,,xi_{0},,betainmathbb{R}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>η</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mi>i</m:mi> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>j</m:mi> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>k</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0389_eq_0298.png\" /> <jats:tex-math>{etain{i,,j,,k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We remark that not only <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374732","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}
In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.
在这篇文章中，我们在（局部紧凑）半超群的大背景下介绍并探讨了拓扑可亲性的概念。我们从某些概率度量的收敛性、与概率度量卷积的总变化和某些函数的平移，以及相关度量代数的 F 代数性质等方面，获得了拓扑可亲性的几种静态、遍历和巴拿赫代数特性。我们进一步研究了卷积的限制与子半超群上度量的限制卷积之间的相互作用。最后，我们从父半超群的相应度量代数上获得的某些不变性质出发，讨论并描述了子半超群的拓扑可亲性。这反过来为我们提供了对 J. Wong 在 1980 年提出的一个开放问题的肯定答案。
{"title":"Topological amenability of semihypergroups","authors":"Choiti Bandyopadhyay","doi":"10.1515/forum-2022-0326","DOIUrl":"https://doi.org/10.1515/forum-2022-0326","url":null,"abstract":"In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"49 5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374999","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}
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 𝐒𝐆{mathbf{SG}}-class.
{"title":"Multilinear Fourier integral operators on modulation spaces","authors":"Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal","doi":"10.1515/forum-2023-0158","DOIUrl":"https://doi.org/10.1515/forum-2023-0158","url":null,"abstract":"In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐒𝐆</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0158_eq_0327.png\" /> <jats:tex-math>{mathbf{SG}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-class.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"104 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104681","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}
We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.
{"title":"Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications","authors":"João Marcos do Ó, Guozhen Lu, Raoní Ponciano","doi":"10.1515/forum-2023-0292","DOIUrl":"https://doi.org/10.1515/forum-2023-0292","url":null,"abstract":"We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078678","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}
Dirceu Bagio, Daniel Gonçalves, Paula Savana Estácio Moreira, Johan Öinert
Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R⋊αG{Rrtimes_{alpha}G}, which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R⋊αG{Rrtimes_{alpha}G}. We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R⋊αG{Rrtimes_{alpha}G}. We show that every ideal of R⋊αG{Rrtimes_{alpha}G} is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.
给定类群 G 在环 R 上的部分作用 α，我们研究相关的偏斜类群环 R ⋊ α G {Rrtimes_{alpha}G} ，它带有自然的 G 评等。 ，它带有一个自然的 G 等级。我们证明，R 的 G 不变理想与 G 分级环 R ⋊ α G {Rrtimes_{alpha}G} 的分级理想之间存在一一对应关系。我们提供了 R ⋊ α G {Rrtimes_{alpha}G} 原始性的充分条件，以及简单性的必要条件和充分条件。我们证明，当且仅当 α 具有残差交集性质时，R ⋊ α G {Rrtimes_{alpha}G} 的每个理想都是有等级的。此外，如果α是由拓扑局部作用θ诱导的，那么我们证明θ的最小性等同于 R 的 G-简单性，θ的拓扑传递性等同于 R 的 G-简单性，而θ在底层拓扑空间的每个封闭不变子集上的拓扑自由性等同于α具有剩余交集性质。作为应用，我们从相关部分作用的拓扑性质和相关超图代数的代数性质来描述超图的条件（K）。
{"title":"The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras","authors":"Dirceu Bagio, Daniel Gonçalves, Paula Savana Estácio Moreira, Johan Öinert","doi":"10.1515/forum-2023-0117","DOIUrl":"https://doi.org/10.1515/forum-2023-0117","url":null,"abstract":"Given a partial action α of a groupoid <jats:italic>G</jats:italic> on a ring <jats:italic>R</jats:italic>, we study the associated partial skew groupoid ring <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:msub> <m:mo>⋊</m:mo> <m:mi>α</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0117_eq_0451.png\" /> <jats:tex-math>{Rrtimes_{alpha}G}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which carries a natural <jats:italic>G</jats:italic>-grading. We show that there is a one-to-one correspondence between the <jats:italic>G</jats:italic>-invariant ideals of <jats:italic>R</jats:italic> and the graded ideals of the <jats:italic>G</jats:italic>-graded ring <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:msub> <m:mo>⋊</m:mo> <m:mi>α</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0117_eq_0451.png\" /> <jats:tex-math>{Rrtimes_{alpha}G}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:msub> <m:mo>⋊</m:mo> <m:mi>α</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0117_eq_0451.png\" /> <jats:tex-math>{Rrtimes_{alpha}G}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that every ideal of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:msub> <m:mo>⋊</m:mo> <m:mi>α</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0117_eq_0451.png\" /> <jats:tex-math>{Rrtimes_{alpha}G}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to <jats:italic>G</jats:italic>-simplicity of <jats:italic>R</jats:italic>, topological transitivity of θ is equivalent to <jats:italic>G</jats:italic>-primeness of <jats:italic>R</jats:italic>, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078565","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}
Karlheinz Gröchenig, Christine Pfeuffer, Joachim Toft
We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p<1{p<1}. As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.
我们将卷积主导矩阵的稳定性和谱不变性扩展到准巴纳赫代数 p < 1 {p<1} 的情况。作为应用，我们构建了具有非光滑符号的伪微分算子的谱不变性准巴纳赫代数，从而推广了西约斯特兰德的结果。
{"title":"Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators","authors":"Karlheinz Gröchenig, Christine Pfeuffer, Joachim Toft","doi":"10.1515/forum-2023-0212","DOIUrl":"https://doi.org/10.1515/forum-2023-0212","url":null,"abstract":"We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0212_eq_0605.png\" /> <jats:tex-math>{p<1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078615","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}
We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms.
我们定义了弱全形准模态的 L 序列，并推导出这些 L 序列的函数方程。我们还证明了弱全形准模态的逆定理。
{"title":"L-series of weakly holomorphic quasimodular forms and a converse theorem","authors":"Mrityunjoy Charan","doi":"10.1515/forum-2023-0194","DOIUrl":"https://doi.org/10.1515/forum-2023-0194","url":null,"abstract":"We define <jats:italic>L</jats:italic>-series of weakly holomorphic quasimodular forms and we derive functional equations of those <jats:italic>L</jats:italic>-series. We also prove a converse theorem for weakly holomorphic quasimodular forms.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139078620","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}
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