Pub Date : 2024-05-02DOI: 10.1017/s0013091524000294
Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang
In his 1984 AMS Memoir, Andrews introduced the family of functions $cphi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $textrm{C}Phi_k(q)$ for $2leq kleq17$ by utilizing the theory of modular forms, where $textrm{C}Phi_k(q)$ denotes the generating function of $cphi_k(n)$. In this paper, we first establish another expression of $textrm{C}Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $cphi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $cphi_{12}(n)$.
{"title":"Some congruences for 12-coloured generalized Frobenius partitions","authors":"Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang","doi":"10.1017/s0013091524000294","DOIUrl":"https://doi.org/10.1017/s0013091524000294","url":null,"abstract":"In his 1984 AMS Memoir, Andrews introduced the family of functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline1.png\"/> <jats:tex-math>$cphi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the number of <jats:italic>k</jats:italic>-coloured generalized Frobenius partitions of <jats:italic>n</jats:italic>. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline2.png\"/> <jats:tex-math>$textrm{C}Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline3.png\"/> <jats:tex-math>$2leq kleq17$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by utilizing the theory of modular forms, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline4.png\"/> <jats:tex-math>$textrm{C}Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the generating function of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline5.png\"/> <jats:tex-math>$cphi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we first establish another expression of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline6.png\"/> <jats:tex-math>$textrm{C}Phi_{12}(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> with integer coefficients, then prove some congruences modulo small powers of 3 for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline7.png\"/> <jats:tex-math>$cphi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline8.png\"/> <jats:tex-math>$cphi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140836424","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}
Pub Date : 2024-05-01DOI: 10.1017/s0013091524000397
{"title":"PEM series 2 volume 67 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0013091524000397","DOIUrl":"https://doi.org/10.1017/s0013091524000397","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141027603","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}
Pub Date : 2024-05-01DOI: 10.1017/s0013091524000385
{"title":"PEM series 2 volume 67 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0013091524000385","DOIUrl":"https://doi.org/10.1017/s0013091524000385","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141032418","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}
Pub Date : 2024-04-30DOI: 10.1017/s0013091524000282
Andreas Hartmann, Xavier Massaneda
We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.
{"title":"Inhomogeneous Poisson processes in the disk and interpolation","authors":"Andreas Hartmann, Xavier Massaneda","doi":"10.1017/s0013091524000282","DOIUrl":"https://doi.org/10.1017/s0013091524000282","url":null,"abstract":"We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140836347","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}
Pub Date : 2024-04-29DOI: 10.1017/s0013091524000300
Amin Mahmoodi
A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].
我们定义并研究了对偶巴拿赫代数的一个合适的弱可亲和性概念，我们称之为弱康恩可亲和性。除其他外，我们还证明了局部紧凑群 G 的度量代数 M(G) 总是弱康恩可亲和性的。它可以作为约翰逊关于 $L^1(G)$ 总是弱可朋性定理的补充[10]。
{"title":"Weak amenability for dual Banach algebras","authors":"Amin Mahmoodi","doi":"10.1017/s0013091524000300","DOIUrl":"https://doi.org/10.1017/s0013091524000300","url":null,"abstract":"<p>A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra <span>M</span>(<span>G</span>) of a locally compact group <span>G</span> is always weakly Connes amenable. It can be a complement to Johnson’s theorem that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094551681-0933:S0013091524000300:S0013091524000300_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$L^1(G)$</span></span></img></span></span> is always weakly amenable [10].</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140811608","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}
Pub Date : 2024-04-25DOI: 10.1017/s0013091524000233
Dmitri Alekseevsky, Gianni Manno, Giovanni Moreno
In Communications in Contemporary Mathematics24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=mathsf{PGL}(n+1)$ (respectively, $G=mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $mathbb{P}^{n+1}$ (respectively, affine $mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms
在Communications in Contemporary Mathematics24 3, (2022)一文中，作者提出了一种方法，用于在对Lie群G的温和假设下，构造施加于$(n+1)$维均质空间$G/H$的超曲面上的G不变偏微分方程（PDEs）。本文将该方法分别应用于 $G=mathsf{PGL}(n+1)$ （分别为 $G=mathsf{Aff}(n+1)$ ）和均相空间 $G/H$ 为 $(n+1)$ 维投影 $mathbb{P}^{n+1}$ （分别为仿射 $mathbb{A}^{n+1}$ ）空间的情况。本文的主要结果是，具有 n 个独立未知变量的投影或仿射不变 PDE 与 n 变量无迹三次方形式空间的不变超曲面一一对应，且与 $mathbb{R}^{d,n-d}$ 的共形变换组 $mathsf{CO}(d,n-d)$ 有关。
{"title":"Projectively and affinely invariant PDEs on hypersurfaces","authors":"Dmitri Alekseevsky, Gianni Manno, Giovanni Moreno","doi":"10.1017/s0013091524000233","DOIUrl":"https://doi.org/10.1017/s0013091524000233","url":null,"abstract":"In <jats:italic>Communications in Contemporary Mathematics</jats:italic>24 3, (2022),the authors have developed a method for constructing <jats:italic>G</jats:italic>-invariant partial differential equations (PDEs) imposed on hypersurfaces of an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline1.png\"/> <jats:tex-math>$(n+1)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional homogeneous space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline2.png\"/> <jats:tex-math>$G/H$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, under mild assumptions on the Lie group <jats:italic>G</jats:italic>. In the present paper, the method is applied to the case when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline3.png\"/> <jats:tex-math>$G=mathsf{PGL}(n+1)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> (respectively, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline4.png\"/> <jats:tex-math>$G=mathsf{Aff}(n+1)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>) and the homogeneous space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline5.png\"/> <jats:tex-math>$G/H$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline6.png\"/> <jats:tex-math>$(n+1)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional projective <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline7.png\"/> <jats:tex-math>$mathbb{P}^{n+1}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> (respectively, affine <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000233_inline8.png\"/> <jats:tex-math>$mathbb{A}^{n+1}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with <jats:italic>n</jats:italic> independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of <jats:italic>trace-free cubic forms</jats:itali","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800728","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}
Pub Date : 2024-04-23DOI: 10.1017/s001309152400018x
Özgür Esentepe, Ryo Takahashi
Given any commutative Noetherian ring R and an element x in R, we consider the full subcategory $mathsf{C}(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between $mathsf{C}(x), mathsf{C}(y)$ and $mathsf{C}(xy)$ for any two ring elements x and y. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.
给定任何交换诺特环 R 和 R 中的元素 x，我们考虑其奇异性范畴的全子范畴 $mathsf{C}(x)$ ，这个子范畴由 x 乘以的态量为零的对象组成。我们的主要观察结果是，对于任意两个环元素 x 和 y，我们可以在 $mathsf{C}(x), mathsf{C}(y)$ 和 $mathsf{C}(xy)$ 之间建立一种关系。
{"title":"Annihilators and decompositions of singularity categories","authors":"Özgür Esentepe, Ryo Takahashi","doi":"10.1017/s001309152400018x","DOIUrl":"https://doi.org/10.1017/s001309152400018x","url":null,"abstract":"Given any commutative Noetherian ring <jats:italic>R</jats:italic> and an element <jats:italic>x</jats:italic> in <jats:italic>R</jats:italic>, we consider the full subcategory <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400018X_inline1.png\" /> <jats:tex-math>$mathsf{C}(x)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> of its singularity category consisting of objects for which the morphism that is given by the multiplication by <jats:italic>x</jats:italic> is zero. Our main observation is that we can establish a relation between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400018X_inline2.png\" /> <jats:tex-math>$mathsf{C}(x), mathsf{C}(y)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400018X_inline3.png\" /> <jats:tex-math>$mathsf{C}(xy)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for any two ring elements <jats:italic>x</jats:italic> and <jats:italic>y</jats:italic>. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800731","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}
Pub Date : 2024-04-22DOI: 10.1017/s0013091524000269
Patrik Wahlberg
We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.
{"title":"Propagation of anisotropic Gabor wave front sets","authors":"Patrik Wahlberg","doi":"10.1017/s0013091524000269","DOIUrl":"https://doi.org/10.1017/s0013091524000269","url":null,"abstract":"We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636165","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}
Pub Date : 2024-04-18DOI: 10.1017/s001309152400021x
Indranil Biswas, Manish Kumar, A. J. Parameswaran
Let $f,:,X,longrightarrow ,Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_*{mathcal O}_X)/{mathcal O}_Y)^*$ is virtually globally generated. Moreover, $((f_*{mathcal O}_X)/{mathcal O}_Y)^*$ is ample if and only if f is genuinely ramified.
{"title":"Pushforward of structure sheaf and virtual global generation","authors":"Indranil Biswas, Manish Kumar, A. J. Parameswaran","doi":"10.1017/s001309152400021x","DOIUrl":"https://doi.org/10.1017/s001309152400021x","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417123522473-0871:S001309152400021X:S001309152400021X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f,:,X,longrightarrow ,Y$</span></span></img></span></span> be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417123522473-0871:S001309152400021X:S001309152400021X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$((f_*{mathcal O}_X)/{mathcal O}_Y)^*$</span></span></img></span></span> is virtually globally generated. Moreover, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417123522473-0871:S001309152400021X:S001309152400021X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$((f_*{mathcal O}_X)/{mathcal O}_Y)^*$</span></span></img></span></span> is ample if and only if <span>f</span> is genuinely ramified.</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140613858","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}
Pub Date : 2024-04-15DOI: 10.1017/s0013091524000221
Nicola Fusco, Domenico Angelo La Manna
In this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature.
{"title":"A remark on a conjecture on the symmetric Gaussian problem","authors":"Nicola Fusco, Domenico Angelo La Manna","doi":"10.1017/s0013091524000221","DOIUrl":"https://doi.org/10.1017/s0013091524000221","url":null,"abstract":"In this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593594","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}
$L^1(G)$ is always weakly amenable [10].
$f,:,X,longrightarrow ,Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle 
$((f_*{mathcal O}_X)/{mathcal O}_Y)^*$ is virtually globally generated. Moreover, 
$((f_*{mathcal O}_X)/{mathcal O}_Y)^*$ is ample if and only if f is genuinely ramified.
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