Proceedings of the Edinburgh Mathematical Society最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}