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Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants 非交换代数与表示理论》：对称、结构与不变式
Communications in Mathematics Pub Date : 2023-07-31 DOI: 10.46298/cm.11678
S. A. Lopes
{"title":"Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants","authors":"S. A. Lopes","doi":"10.46298/cm.11678","DOIUrl":"https://doi.org/10.46298/cm.11678","url":null,"abstract":"This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra $mathbb{A}_{1}$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139353090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A recursive formula for the product of element orders of finite abelian groups 有限阿贝尔群元素阶积的递推公式
Communications in Mathematics Pub Date : 2023-04-04 DOI: 10.46298/cm.10996
Subhrajyoti Saha
{"title":"A recursive formula for the product of element orders of finite abelian groups","authors":"Subhrajyoti Saha","doi":"10.46298/cm.10996","DOIUrl":"https://doi.org/10.46298/cm.10996","url":null,"abstract":"Let G be a finite group and let ψ(G) denote the sum of element orders of G; later this concept has been used to define R(G) which is the product of the element orders of G. Motivated by the recursive formula for ψ(G), we consider a finite abelian group G and obtain a similar formula for R(G).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44276178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Matrix formulas for multiplicities in the spin module 自旋模中乘法的矩阵公式
Communications in Mathematics Pub Date : 2023-04-04 DOI: 10.46298/cm.11156
Lucas Fresse, S. Mehdi
{"title":"Matrix formulas for multiplicities in the spin module","authors":"Lucas Fresse, S. Mehdi","doi":"10.46298/cm.11156","DOIUrl":"https://doi.org/10.46298/cm.11156","url":null,"abstract":"We obtain inductive and enumerative formulas for the multiplicities of the\u0000weights of the spin module for the Clifford algebra of a Levi subalgebra in a\u0000complex semisimple Lie algebra. Our formulas involve only matrices and\u0000tableaux, and our techniques combine linear algebra, Lie theory, and\u0000combinatorics. Moreover, this suggests a relationship with complex nilpotent\u0000orbits. The case of the special linear Lie algebra $mathfrak{sl}(n,{mathbb\u0000C})$ is emphasized.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49449176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Minkowski's successive minima in convex and discrete geometry 凸和离散几何中的Minkowski连续极小
Communications in Mathematics Pub Date : 2023-03-31 DOI: 10.46298/cm.11155
I. Aliev, M. Henk
{"title":"Minkowski's successive minima in convex and discrete geometry","authors":"I. Aliev, M. Henk","doi":"10.46298/cm.11155","DOIUrl":"https://doi.org/10.46298/cm.11155","url":null,"abstract":"In this short survey we want to present some of the impact of Minkowski's\u0000successive minima within Convex and Discrete Geometry. Originally related to\u0000the volume of an $o$-symmetric convex body, we point out relations of the\u0000successive minima to other functionals, as e.g., the lattice point enumerator\u0000or the intrinsic volumes and we present some old and new conjectures about\u0000them. Additionally, we discuss an application of successive minima to a version\u0000of Siegel's lemma.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49090055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On triviality of uniform Diophantine exponents of lattices 格的一致丢番图指数的平凡性
Communications in Mathematics Pub Date : 2023-03-29 DOI: 10.46298/cm.11137
O. German
{"title":"On triviality of uniform Diophantine exponents of lattices","authors":"O. German","doi":"10.46298/cm.11137","DOIUrl":"https://doi.org/10.46298/cm.11137","url":null,"abstract":"In this paper we prove that uniform Diophantine exponents of lattices attain\u0000only trivial values.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46522794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the number of lattice points in a ball 关于球中格点的数目
Communications in Mathematics Pub Date : 2023-03-27 DOI: 10.46298/cm.11119
Jeffrey D. Vaaler
{"title":"On the number of lattice points in a ball","authors":"Jeffrey D. Vaaler","doi":"10.46298/cm.11119","DOIUrl":"https://doi.org/10.46298/cm.11119","url":null,"abstract":"We prove a fairly general inequality that estimates the number of lattice\u0000points in a ball of positive radius in general position in a Euclidean space.\u0000The bound is uniform over lattices induced by a matrix having a bounded\u0000operator norm.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41773419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space 光滑度量空间上非线性椭圆方程的梯度估计
Communications in Mathematics Pub Date : 2023-02-27 DOI: 10.46298/cm.10951
Xiaoshan Wang, Linfen Cao
{"title":"Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space","authors":"Xiaoshan Wang, Linfen Cao","doi":"10.46298/cm.10951","DOIUrl":"https://doi.org/10.46298/cm.10951","url":null,"abstract":"Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a < 0 under the m-dimensions Bakry-Émery Ricci curvature.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41867795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields 一些数域的第一个Hilbert 2-类域的2-类群的循环性
Communications in Mathematics Pub Date : 2023-02-18 DOI: 10.46298/cm.10983
A. Azizi, M. Rezzougui, A. Zekhnini
{"title":"Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields","authors":"A. Azizi, M. Rezzougui, A. Zekhnini","doi":"10.46298/cm.10983","DOIUrl":"https://doi.org/10.46298/cm.10983","url":null,"abstract":"Let $mathds{k}$ be a real quadratic number field. Denote by\u0000$mathrm{Cl}_2(mathds{k})$ its $2$-class group and by $mathds{k}_2^{(1)}$\u0000(resp. $mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field.\u0000The aim of this paper is to study, for a real quadratic number field whose\u0000discriminant is divisible by one prime number congruent to $3$ modulo 4, the\u0000metacyclicity of $G=mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k})$ and the\u0000cyclicity of $mathrm{Gal}(mathds{k}_2^{(2)}/mathds{k}_2^{(1)})$ whenever the\u0000rank of $mathrm{Cl}_2(mathds{k})$ is $2$, and the $4$-rank of\u0000$mathrm{Cl}_2(mathds{k})$ is $1$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46032598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Certain Paracontact Metrics Satisfying the Critical Point Equation 满足临界点方程的若干副接触测度
Communications in Mathematics Pub Date : 2023-02-14 DOI: 10.46298/cm.10549
D. Patra
{"title":"Certain Paracontact Metrics Satisfying the Critical Point Equation","authors":"D. Patra","doi":"10.46298/cm.10549","DOIUrl":"https://doi.org/10.46298/cm.10549","url":null,"abstract":"The aim of this paper is to study theCPE (Critical Point Equation) on some paracontact metric manifolds.First, we prove that if a para-Sasakian metric satisfies the CPE,then it is Einstein with constant scalar curvature -2n(2n+1). Next,we prove that if $(kappa,mu)$-paracontact metric satisfies theCPE, then it is locally isometric to the product of a flat$(n+1)$-dimensional manifold and $n$-dimensional manifold ofnegative constant curvature$-4$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41523437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS MODULO p^2 关于勒让德多项式模p^2的同余
Communications in Mathematics Pub Date : 2023-02-14 DOI: 10.46298/cm.10767
Aeran Kim
{"title":"CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS MODULO p^2","authors":"Aeran Kim","doi":"10.46298/cm.10767","DOIUrl":"https://doi.org/10.46298/cm.10767","url":null,"abstract":"In this article, we extend Z. H. Sun's congruences concerning Legendre polynomials P p−1 2 (x) to P p+1 2 (x) for odd prime p, which enables us to deduce some congruences resembling p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2).\u0000 이 논문에서 우리는 Z. H. Sun의 르장드르 다항식의 합동식 P p−1 2 (x) 에서 P p+1 2 (x) (단, p는 소수) 까지를 이용해서 이 합동식과 비슷한 합동식 p+1 2 ∑ k=0 4pk + 4k 2 − 1 16 k (2k − 1) 2 (2k k)2 (mod p 2) 을 유도한다.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43163509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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