Pub Date : 2022-08-17DOI: 10.4153/S0008439523000097
F. Ambrosio, G. Carnovale, F. Esposito
Abstract Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
{"title":"A parametrization of sheets of conjugacy classes in bad characteristic","authors":"F. Ambrosio, G. Carnovale, F. Esposito","doi":"10.4153/S0008439523000097","DOIUrl":"https://doi.org/10.4153/S0008439523000097","url":null,"abstract":"Abstract Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs \u0000$(M,{mathcal O})$\u0000 where M is the identity component of the centralizer of a semisimple element in G and \u0000${mathcal O}$\u0000 is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"976 - 983"},"PeriodicalIF":0.6,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46194631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-14DOI: 10.4153/s0008439523000267
E. Lecouturier, Jun Wang
Sharifi has constructed a map from the first homology of the modular curve $X_1(M)$ to the $K$-group $K_2(mathbf{Z}[zeta_M, frac{1}{M}])$, where $zeta_M$ is a primitive $M$th root of unity. We study how these maps relate when $M$ varies. Our method relies on the techniques developed by Sharifi and Venkatesh.
{"title":"Level compatibility in Sharifi’s conjecture","authors":"E. Lecouturier, Jun Wang","doi":"10.4153/s0008439523000267","DOIUrl":"https://doi.org/10.4153/s0008439523000267","url":null,"abstract":"Sharifi has constructed a map from the first homology of the modular curve $X_1(M)$ to the $K$-group $K_2(mathbf{Z}[zeta_M, frac{1}{M}])$, where $zeta_M$ is a primitive $M$th root of unity. We study how these maps relate when $M$ varies. Our method relies on the techniques developed by Sharifi and Venkatesh.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48184100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-12DOI: 10.4153/S0008439523000103
Bata Krishna Das, Poornendu Kumar, H. Sau
Abstract A subset ${mathcal D}$ of a domain $Omega subset {mathbb C}^d$ is determining for an analytic function $f:Omega to overline {{mathbb D}}$ if whenever an analytic function $g:Omega rightarrow overline {{mathbb D}}$ coincides with f on ${mathcal D}$ , equals to f on whole $Omega $ . This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any $Ngeq 1$ , a set consisting of $N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.
摘要域$Omegasubet｛mathbb C｝^D$的子集${mathcal D｝$为分析函数$f:Omegatooverline｛math bb D｝｝$确定，如果分析函数$g:Omegarightarrowoverline{mathbbD｝}$与${math cal D｝$上的f重合，则整个$Omega上的f等于f。这个注释找到了对称化bidisk的子集被确定的几个充分条件。对于任何$Ngeq1$，构造了一个由$N^2-N+1$多个点组成的集合，该集合确定了具有度约束的任何有理内函数。我们还研究了对称二次空间与一些特殊代数变种的交集何时可以确定有理内函数。
{"title":"Determining sets for holomorphic functions on the symmetrized bidisk","authors":"Bata Krishna Das, Poornendu Kumar, H. Sau","doi":"10.4153/S0008439523000103","DOIUrl":"https://doi.org/10.4153/S0008439523000103","url":null,"abstract":"Abstract A subset \u0000${mathcal D}$\u0000 of a domain \u0000$Omega subset {mathbb C}^d$\u0000 is determining for an analytic function \u0000$f:Omega to overline {{mathbb D}}$\u0000 if whenever an analytic function \u0000$g:Omega rightarrow overline {{mathbb D}}$\u0000 coincides with f on \u0000${mathcal D}$\u0000 , equals to f on whole \u0000$Omega $\u0000 . This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any \u0000$Ngeq 1$\u0000 , a set consisting of \u0000$N^2-N+1$\u0000 many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"984 - 996"},"PeriodicalIF":0.6,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48628260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-29DOI: 10.4153/S0008439522000509
Genival da Silva, James D. Lewis
Abstract Let $X/{mathbb C}$ be a smooth projective variety. We consider two integral invariants, one of which is the level of the Hodge cohomology algebra $H^*(X,{mathbb C})$ and the other involving the complexity of the higher Chow groups ${mathrm {CH}}^*(X,m;{mathbb Q})$ for $mgeq 0$ . We conjecture that these two invariants are the same and accordingly provide some strong evidence in support of this.
{"title":"The complexity of higher Chow groups","authors":"Genival da Silva, James D. Lewis","doi":"10.4153/S0008439522000509","DOIUrl":"https://doi.org/10.4153/S0008439522000509","url":null,"abstract":"Abstract Let \u0000$X/{mathbb C}$\u0000 be a smooth projective variety. We consider two integral invariants, one of which is the level of the Hodge cohomology algebra \u0000$H^*(X,{mathbb C})$\u0000 and the other involving the complexity of the higher Chow groups \u0000${mathrm {CH}}^*(X,m;{mathbb Q})$\u0000 for \u0000$mgeq 0$\u0000 . We conjecture that these two invariants are the same and accordingly provide some strong evidence in support of this.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"903 - 911"},"PeriodicalIF":0.6,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48046316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-24DOI: 10.4153/s0008439523000425
O. Bordellès
In this note, we provide an explicit upper bound for $h_K mathcal{R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.
{"title":"From the Ideal Theorem to the class number","authors":"O. Bordellès","doi":"10.4153/s0008439523000425","DOIUrl":"https://doi.org/10.4153/s0008439523000425","url":null,"abstract":"In this note, we provide an explicit upper bound for $h_K mathcal{R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44082621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-21DOI: 10.4153/S0008439522000492
D. Aadi, Y. Omari
Abstract We characterize zero sets for which every subset remains a zero set too in the Fock space $mathcal {F}^p$ , $1leq p
摘要我们刻画了在Fock空间$mathcal｛F｝^p$，$1leq p
{"title":"Zero and uniqueness sets for Fock spaces","authors":"D. Aadi, Y. Omari","doi":"10.4153/S0008439522000492","DOIUrl":"https://doi.org/10.4153/S0008439522000492","url":null,"abstract":"Abstract We characterize zero sets for which every subset remains a zero set too in the Fock space \u0000$mathcal {F}^p$\u0000 , \u0000$1leq p<infty $\u0000 . We are also interested in the study of a stability problem for some examples of uniqueness set with zero excess in Fock spaces.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"532 - 543"},"PeriodicalIF":0.6,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49358167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-17DOI: 10.4153/S0008439522000406
J. Oliva-Maza
Abstract Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like $L^2(mathbb R^+)$ or $H^2(mathbb C^+)$ . These kernels entail an algebraic $L^1$ -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the $L^2(mathbb R^+)$ case turn out to be Hardy kernels as well. In the $H^2(mathbb C^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.
{"title":"On Hardy kernels as reproducing kernels","authors":"J. Oliva-Maza","doi":"10.4153/S0008439522000406","DOIUrl":"https://doi.org/10.4153/S0008439522000406","url":null,"abstract":"Abstract Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like \u0000$L^2(mathbb R^+)$\u0000 or \u0000$H^2(mathbb C^+)$\u0000 . These kernels entail an algebraic \u0000$L^1$\u0000 -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the \u0000$L^2(mathbb R^+)$\u0000 case turn out to be Hardy kernels as well. In the \u0000$H^2(mathbb C^+)$\u0000 scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"428 - 442"},"PeriodicalIF":0.6,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45337449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-17DOI: 10.4153/S0008439522000418
Lijian An
Abstract The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.
{"title":"Groups whose Chermak–Delgado lattice is a subgroup lattice of an abelian group","authors":"Lijian An","doi":"10.4153/S0008439522000418","DOIUrl":"https://doi.org/10.4153/S0008439522000418","url":null,"abstract":"Abstract The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"443 - 449"},"PeriodicalIF":0.6,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49098697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-15DOI: 10.4153/S0008439523000152
E. Parini, Giorgio Saracco
Abstract Given an open, bounded set $Omega $ in $mathbb {R}^N$ , we consider the minimization of the anisotropic Cheeger constant $h_K(Omega )$ with respect to the anisotropy K, under a volume constraint on the associated unit ball. In the planar case, under the assumption that K is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if $Omega $ is a ball, we show that the optimal anisotropy K is not a ball and that, among all regular polygons, the square provides the minimal value.
{"title":"Optimization of the anisotropic Cheeger constant with respect to the anisotropy","authors":"E. Parini, Giorgio Saracco","doi":"10.4153/S0008439523000152","DOIUrl":"https://doi.org/10.4153/S0008439523000152","url":null,"abstract":"Abstract Given an open, bounded set \u0000$Omega $\u0000 in \u0000$mathbb {R}^N$\u0000 , we consider the minimization of the anisotropic Cheeger constant \u0000$h_K(Omega )$\u0000 with respect to the anisotropy K, under a volume constraint on the associated unit ball. In the planar case, under the assumption that K is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if \u0000$Omega $\u0000 is a ball, we show that the optimal anisotropy K is not a ball and that, among all regular polygons, the square provides the minimal value.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":"66 1","pages":"1030 - 1043"},"PeriodicalIF":0.6,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46083684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-14DOI: 10.4153/s0008439523000243
Maria S. Esipova, K. Yeats
The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$ by proving a relation modulo $p$ between certain coefficients of powers of products of particularly nice polynomials. The relation at the level of the $c_2$ invariant provides evidence for a conjecture of Schnetz.
{"title":"A result on the c2 invariant for powers of primes","authors":"Maria S. Esipova, K. Yeats","doi":"10.4153/s0008439523000243","DOIUrl":"https://doi.org/10.4153/s0008439523000243","url":null,"abstract":"The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$ by proving a relation modulo $p$ between certain coefficients of powers of products of particularly nice polynomials. The relation at the level of the $c_2$ invariant provides evidence for a conjecture of Schnetz.","PeriodicalId":55280,"journal":{"name":"Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47433116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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