We prove two single-parameter q-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 q-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 q-analogue.
{"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}
Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_0800.png" /> <jats:tex-math>{L=-{Delta}_{mathbb{H}^{n}}+V}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Schrödinger operator on Heisenberg groups <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>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_0890.png" /> <jats:tex-math>{mathbb{H}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_1058.png" /> <jats:tex-math>{{Delta}_{mathbb{H}^{n}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the sub-Laplacian, the nonnegative potential <jats:italic>V</jats:italic> belongs to the reverse Hölder class <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi mathvariant="script">𝒬</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_0748.png" /> <jats:tex-math>{B_{mathcal{Q}/2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Here <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒬</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_0895.png" /> <jats:tex-math>{mathcal{Q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the homogeneous dimension of <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>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0285_eq_0890.png" /> <jats:tex-math>{mathbb{H}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this article, we introduce the fractional heat semigroups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <
设 L = - Δ ℍ n + V {L=-{Delta}_{mathbb{H}^{n}}+V} 是海森堡群 ℍ n {{mathbb{H}^{n}} 上的薛定谔算子，其中 Δ ℍ n {{Delta}_{mathbb{H}^{n}} 是子拉普拉斯。 其中 Δ ℍ n {{Delta}_{mathbb{H}^{n}} 是子拉普拉卡，非负势 V 属于反向荷尔德类 B 𝒬 / 2 {B_{mathcal{Q}/2}} 。｝ .这里𝒬 {mathcal{Q}} 是ℍ n {mathbb{H}^{n} 的同次元维度。｝ .在本文中，我们引入分数热半群 { e - t L α } t > 0 {{e^{-tL^{alpha}}}_{t>0}} 。 , α > 0 {alpha>0} , 与 L 相关联。 通过热方程的基本解，我们分别估计了分数热核 K α , t L ( ⋅ , ⋅ ) {K_{alpha,t}^{L}(,cdot,,cdot,)} 的梯度和时间分数导数。作为应用，我们通过{ e - t L α } t > 0 {{e^{-tL^{alpha}}}_{t>0}} 来描述空间 BMO L γ ( ℍ n ) {mathrm{BMO}_{L}^{gamma}(mathbb{H}^{n})} 。 .
{"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>L</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0800.png\" /> <jats:tex-math>{L=-{Delta}_{mathbb{H}^{n}}+V}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Schrödinger operator on Heisenberg groups <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>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0890.png\" /> <jats:tex-math>{mathbb{H}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_1058.png\" /> <jats:tex-math>{{Delta}_{mathbb{H}^{n}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the sub-Laplacian, the nonnegative potential <jats:italic>V</jats:italic> belongs to the reverse Hölder class <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi mathvariant=\"script\">𝒬</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0748.png\" /> <jats:tex-math>{B_{mathcal{Q}/2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Here <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒬</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0895.png\" /> <jats:tex-math>{mathcal{Q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the homogeneous dimension of <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>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0285_eq_0890.png\" /> <jats:tex-math>{mathbb{H}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this article, we introduce the fractional heat semigroups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <","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}
This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝd{mathbb{R}^{d}}. 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.
{"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}
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 p-groups G and H an isomorphism between the group algebras FG and FH implies an isomorphism of the groups G and H for F the field of p elements. For groups of odd order this implication is also proven for F being any field of characteristic p. For groups of even order we need either to make an additional assumption on the groups or on the field.
我们证明了模同构问题对于具有循环中心的无幂级数 2 的群有一个肯定的答案，即对于这样的 p 群 G 和 H，群代数 FG 和 FH 之间的同构意味着群 G 和 H 对于 p 元素域 F 的同构。对于奇数阶群，F 是任何特征 p 的域时，这一蕴涵也可得到证明。对于偶数阶群，我们需要对群或域做一个额外的假设。
{"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}
We consider a very global q-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 q-integral transforms. It should be noted that the global character of the q-integral transform in question allows one to immediately deduce corresponding Donoho–Stark uncertainty principles for q-integral operators that are its particular cases. These particular cases are very well-known operators, namely: a q-cosine-Fourier transform, a q-sine-Fourier transform, a q-Fourier transform, a q-Bessel–Fourier transform and a q-Dunkl transform. Moreover, generalizations of the local uncertainty principle of Price for the q-cosine-Fourier transform, q-sine-Fourier transform, q-Fourier transform, q-Bessel–Fourier transform and q-Dunkl transform are also obtained.
{"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}
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.
{"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}
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.
{"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}
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 a3b{a^{3}b}-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.
全等四边形对球面的边到边倾斜在一系列三篇论文中得到了完整的分类。最后一篇论文对具有某种无理角的 a 3 b {a^{3}b} - 四边形进行了分类：有一系列可进行 2 层地球映射平铺的 1 参数四边形族，以及它们在额外条件下的基本翻转修正，还有 5 个零星四边形，每个都可进行特殊平铺。最后是完整分类的摘要。
{"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}
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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒮</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-2023-0220_eq_0359.png" /> <jats:tex-math>{operatorname{Filt}(mathcal{S})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0220_eq_0332.png" /> <jats:tex-math>{mathcal{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> consists of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℵ</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-2023-0220_eq_0211.png" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules for some fixed <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo><</m:mo> <m:mi>ω</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0220_eq_0475.png" /> <jats:tex-math>{n<omega}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">ℵ</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-2023-0220_eq_0211.png" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒮</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-2023-0220_eq_0359.png" /> <jats:tex-math>{operatorname{Filt}(m
{"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"script\">𝒮</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-2023-0220_eq_0359.png\" /> <jats:tex-math>{operatorname{Filt}(mathcal{S})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒮</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0332.png\" /> <jats:tex-math>{mathcal{S}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> consists of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">ℵ</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-2023-0220_eq_0211.png\" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules for some fixed <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo><</m:mo> <m:mi>ω</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0220_eq_0475.png\" /> <jats:tex-math>{n<omega}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">ℵ</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-2023-0220_eq_0211.png\" /> <jats:tex-math>{aleph_{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Filt</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"script\">𝒮</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-2023-0220_eq_0359.png\" /> <jats:tex-math>{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}
We introduce and study a notion of module nuclear dimension for a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0204.png" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:italic>A</jats:italic> which is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0204.png" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-module over another <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0204.png" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0639.png" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with compatible actions. We show that the module nuclear dimension of <jats:italic>A</jats:italic> is zero if <jats:italic>A</jats:italic> is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0639.png" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-NF. The converse is shown to hold when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0283_eq_0639.png" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</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-2023-0283_eq_0344.png" /> <jats:tex-math>{C(X)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra with simple fibers, with <jats:italic>X</jats:italic> compact and totally d
我们引入并研究了 C * mathrm{C}^{*} -代数 A 的模核维度概念。 -代数 A 的模块核维度概念。 上的另一个 C * mathrm{C}^{*} -模块 -代数𝔄 {mathfrak{A}} 的兼容作用。我们证明，如果 A 是 {mathfrak{A}} ，那么 A 的模块核维度为零。 -NF。当𝔄 {mathfrak{A}} 是具有简单纤维的 C ( X ) {C(X)}-代数，且 X 紧凑且完全断开时，反之成立。我们还引入了一个模块分解秩的概念，并证明当 𝔄 {mathfrak{A}} 是单价且简单时，如果 A 的模块分解秩是有限的，那么 A 是 𝔄 {mathfrak{A}} -QD。 -QD。我们研究𝔄 {mathcal{T}_{mathfrak{A}}(A)} 的𝔄 {mathfrak{A}} 的集合 𝒯 ( A ) {mathcal{T}_{mathfrak{A}}(A)} 。 -A 上的有值模量迹，并将 A 的 Cuntz 半群与集合 𝒯 ( A ) {mathcal{T}_{mathfrak{A}}(A)} 上的下半连续仿射函数联系起来。同时，我们还证明了一个模块崔-埃夫罗斯提升定理。我们给出了一类例子的模块核维度估计值。
{"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:italic>A</jats:italic> which is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-module over another <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0204.png\" /> <jats:tex-math>mathrm{C}^{*}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with compatible actions. We show that the module nuclear dimension of <jats:italic>A</jats:italic> is zero if <jats:italic>A</jats:italic> is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-NF. The converse is shown to hold when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔄</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0283_eq_0639.png\" /> <jats:tex-math>{mathfrak{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</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-2023-0283_eq_0344.png\" /> <jats:tex-math>{C(X)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra with simple fibers, with <jats:italic>X</jats:italic> 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}
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