Pub Date : 2023-12-21DOI: 10.1017/s0013091523000743
Luis Paris, Olga Varghese
Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group D with respect to the standard generating set. We give a recursive formula for the spherical growth series of D in terms of the spherical growth series of standard parabolic subgroups. As an application we obtain the rationality of the spherical growth series of a Dyer group. Furthermore, we show that the spherical growth series of D is closely related to the Euler characteristic of D.
循环群的图积和 Coxeter 群是由标记图定义的两个群族。戴尔群族包含这两个群族，为我们提供了统一研究这些群的框架。本文主要研究戴尔群 D 关于标准生成集的球形增长序列。我们根据标准抛物线子群的球形增长数列给出了 D 的球形增长数列的递推公式。作为应用，我们得到了戴尔群球面增长数列的合理性。此外，我们还证明了 D 的球形增长数列与 D 的欧拉特征密切相关。
{"title":"The spherical growth series of Dyer groups","authors":"Luis Paris, Olga Varghese","doi":"10.1017/s0013091523000743","DOIUrl":"https://doi.org/10.1017/s0013091523000743","url":null,"abstract":"<p>Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group <span>D</span> with respect to the standard generating set. We give a recursive formula for the spherical growth series of <span>D</span> in terms of the spherical growth series of standard parabolic subgroups. As an application we obtain the rationality of the spherical growth series of a Dyer group. Furthermore, we show that the spherical growth series of <span>D</span> is closely related to the Euler characteristic of <span>D</span>.</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138825072","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 : 2023-11-24DOI: 10.1017/s0013091523000585
Aleš Drápal, Ian M. Wanless
Let $mathbb F$ be a finite field of odd order and $a,binmathbb Fsetminus{0,1}$ be such that $chi(a) = chi(b)$ and $chi(1-a)=chi(1-b)$, where χ is the extended quadratic character on $mathbb F$. Let $Q_{a,b}$ be the quasigroup over $mathbb F$ defined by $(x,y)mapsto x+a(y-x)$ if $chi(y-x) geqslant 0$, and $(x,y) mapsto x+b(y-x)$ if
Pub Date : 2023-11-22DOI: 10.1017/s0013091523000548
Alan S. Cigoli, Manuel Mancini, Giuseppe Metere
In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.
{"title":"On the representability of actions of Leibniz algebras and Poisson algebras","authors":"Alan S. Cigoli, Manuel Mancini, Giuseppe Metere","doi":"10.1017/s0013091523000548","DOIUrl":"https://doi.org/10.1017/s0013091523000548","url":null,"abstract":"In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542301","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 : 2023-11-16DOI: 10.1017/s0013091523000664
D. T. Nguyen
We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.
我们得到了广义三次除数函数的修正移位卷积的二阶矩的一个新的界，并证明了对应用来说，修正版是充分的。
{"title":"Generalized Divisor Functions in Arithmetic Progressions: II","authors":"D. T. Nguyen","doi":"10.1017/s0013091523000664","DOIUrl":"https://doi.org/10.1017/s0013091523000664","url":null,"abstract":"We obtain a new bound on the second moment of modified shifted convolutions of the generalized threefold divisor function and show that, for applications, the modified version is sufficient.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526200","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 : 2023-11-10DOI: 10.1017/s0013091523000706
Nguyen Xuan Tho
Abstract We investigate the equation $D=x^4-y^4$ in field extensions. As an application, for a prime number p , we find solutions to $p=x^4-y^4$ if $pequiv 11$ (mod 16) and $p^3=x^4-y^4$ if $pequiv 3$ (mod 16) in all cubic extensions of $mathbb{Q}(i)$ .
研究了域扩展中的方程$D=x^4-y^4$。作为一个应用，对于素数p，我们找到了$p=x^4-y^4$ if $pequiv 11$ (mod 16)和$p^3=x^4-y^4$ if $pequiv 3$ (mod 16)在$mathbb{Q}(i)$的所有三次扩展中的解。
{"title":"On the Difference of Two Fourth Powers","authors":"Nguyen Xuan Tho","doi":"10.1017/s0013091523000706","DOIUrl":"https://doi.org/10.1017/s0013091523000706","url":null,"abstract":"Abstract We investigate the equation $D=x^4-y^4$ in field extensions. As an application, for a prime number p , we find solutions to $p=x^4-y^4$ if $pequiv 11$ (mod 16) and $p^3=x^4-y^4$ if $pequiv 3$ (mod 16) in all cubic extensions of $mathbb{Q}(i)$ .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092056","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 : 2023-11-09DOI: 10.1017/s0013091523000676
Xinfu Li, Li Xu, Meiling Zhu
Abstract This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities begin{equation*} begin{cases} -Delta u=lambda u+h(epsilon x)|u|^{q-2}u+eta |u|^{p-2}u,quad xin mathbb{R}^N, int_{mathbb{R}^N}|u|^2,textrm{d}x=a^2, end{cases} end{equation*} where $a, epsilon, eta gt 0$ , q is L 2 -subcritical, p is L 2 -supercritical, $lambdain mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h . Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$ .
{"title":"Multiplicity and Stability of Normalized Solutions to Non-autonomous Schrödinger Equation with Mixed Non-linearities","authors":"Xinfu Li, Li Xu, Meiling Zhu","doi":"10.1017/s0013091523000676","DOIUrl":"https://doi.org/10.1017/s0013091523000676","url":null,"abstract":"Abstract This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities begin{equation*} begin{cases} -Delta u=lambda u+h(epsilon x)|u|^{q-2}u+eta |u|^{p-2}u,quad xin mathbb{R}^N, int_{mathbb{R}^N}|u|^2,textrm{d}x=a^2, end{cases} end{equation*} where $a, epsilon, eta gt 0$ , q is L 2 -subcritical, p is L 2 -supercritical, $lambdain mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h . Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$ .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241988","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}
Abstract An element g in a group G is called reversible if g is conjugate to g −1 in G . An element g in G is strongly reversible if g is conjugate to g −1 by an involution in G . The group of affine transformations of $mathbb D^n$ may be identified with the semi-direct product $mathrm{GL}(n, mathbb D) ltimes mathbb D^n $ , where $mathbb D:=mathbb R, mathbb C$ or $ mathbb H $ . This paper classifies reversible and strongly reversible elements in the affine group $mathrm{GL}(n, mathbb D) ltimes mathbb D^n $ .
{"title":"Reversibility of affine transformations","authors":"Krishnendu Gongopadhyay, Tejbir Lohan, Chandan Maity","doi":"10.1017/s001309152300069x","DOIUrl":"https://doi.org/10.1017/s001309152300069x","url":null,"abstract":"Abstract An element g in a group G is called reversible if g is conjugate to g −1 in G . An element g in G is strongly reversible if g is conjugate to g −1 by an involution in G . The group of affine transformations of $mathbb D^n$ may be identified with the semi-direct product $mathrm{GL}(n, mathbb D) ltimes mathbb D^n $ , where $mathbb D:=mathbb R, mathbb C$ or $ mathbb H $ . This paper classifies reversible and strongly reversible elements in the affine group $mathrm{GL}(n, mathbb D) ltimes mathbb D^n $ .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135341269","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 : 2023-11-03DOI: 10.1017/s0013091523000573
Lars Winther Christensen, Luigi Ferraro, Peder Thompson
Abstract Let $mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(mathfrak{p})$ the residue field of the local ring $R_mathfrak{p}$ . We prove that if an R -module M satisfies $operatorname{Ext}_R^{n}(k(mathfrak{p}),M)=0$ for some $ngeqslantdim R$ , then $operatorname{Ext}_R^i(k(mathfrak{p}),M)=0$ holds for all $i geqslant n$ . This improves a result of Christensen, Iyengar and Marley by lowering the bound on n . We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.
{"title":"Rigidity of Ext and Tor via Flat–Cotorsion Theory","authors":"Lars Winther Christensen, Luigi Ferraro, Peder Thompson","doi":"10.1017/s0013091523000573","DOIUrl":"https://doi.org/10.1017/s0013091523000573","url":null,"abstract":"Abstract Let $mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(mathfrak{p})$ the residue field of the local ring $R_mathfrak{p}$ . We prove that if an R -module M satisfies $operatorname{Ext}_R^{n}(k(mathfrak{p}),M)=0$ for some $ngeqslantdim R$ , then $operatorname{Ext}_R^i(k(mathfrak{p}),M)=0$ holds for all $i geqslant n$ . This improves a result of Christensen, Iyengar and Marley by lowering the bound on n . We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775005","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 : 2023-11-03DOI: 10.1017/s001309152300055x
Boram Park, Seonjeong Park
Abstract Given a graph G without loops, the pseudograph associahedron P G is a smooth polytope, so there is a projective smooth toric variety X G corresponding to P G . Taking the real locus of X G , we have the projective smooth real toric variety $X^{mathbb{R}}_G$ . The integral cohomology groups of $X^{mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G ; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.
{"title":"Cohomology of a Real Toric Variety and Shellability of Posets Arising from a Graph","authors":"Boram Park, Seonjeong Park","doi":"10.1017/s001309152300055x","DOIUrl":"https://doi.org/10.1017/s001309152300055x","url":null,"abstract":"Abstract Given a graph G without loops, the pseudograph associahedron P G is a smooth polytope, so there is a projective smooth toric variety X G corresponding to P G . Taking the real locus of X G , we have the projective smooth real toric variety $X^{mathbb{R}}_G$ . The integral cohomology groups of $X^{mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G ; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135820852","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 : 2023-11-03DOI: 10.1017/s0013091523000688
Vasudevarao Allu, Himadri Halder
Abstract Let $H^{infty}(Omega,X)$ be the space of bounded analytic functions $f(z)=sum_{n=0}^{infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $mathbb{D}:={zin mathbb{C}:|z| lt 1}$ into a complex Banach space X with $leftlVert frightrVert_{H^{infty}(Omega,X)} leq 1$ . Let $phi={phi_{n}(r)}_{n=0}^{infty}$ with $phi_{0}(r)leq 1$ such that $sum_{n=0}^{infty} phi_{n}(r)$ converges locally uniformly with respect to $r in [0,1)$ . For $1leq p,q lt infty$ , we denote begin{equation*} R_{p,q,phi}(f,Omega,X)= sup left{r geq 0: leftlVert x_{0}rightrVert^p phi_{0}(r) + left(sum_{n=1}^{infty} leftlVert x_{n}rightrVertphi_{n}(r)right)^q leq phi_{0}(r)right} end{equation*} and define the Bohr radius associated with ϕ by begin{equation*}R_{p,q,phi}(Omega,X)=inf left{R_{p,q,phi}(f,Omega,X): leftlVert frightrVert_{H^{infty}(Omega,X)} leq 1right}.end{equation*} In this article, we extensively study the Bohr radius $R_{p,q,phi}(Omega,X)$ , when X is an arbitrary Banach space, and $X=mathcal{B}(mathcal{H})$ is the algebra of all bounded linear operators on a complex Hilbert space $mathcal{H}$ . Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.
{"title":"Bohr Radius for Banach Spaces on Simply Connected Domains","authors":"Vasudevarao Allu, Himadri Halder","doi":"10.1017/s0013091523000688","DOIUrl":"https://doi.org/10.1017/s0013091523000688","url":null,"abstract":"Abstract Let $H^{infty}(Omega,X)$ be the space of bounded analytic functions $f(z)=sum_{n=0}^{infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $mathbb{D}:={zin mathbb{C}:|z| lt 1}$ into a complex Banach space X with $leftlVert frightrVert_{H^{infty}(Omega,X)} leq 1$ . Let $phi={phi_{n}(r)}_{n=0}^{infty}$ with $phi_{0}(r)leq 1$ such that $sum_{n=0}^{infty} phi_{n}(r)$ converges locally uniformly with respect to $r in [0,1)$ . For $1leq p,q lt infty$ , we denote begin{equation*} R_{p,q,phi}(f,Omega,X)= sup left{r geq 0: leftlVert x_{0}rightrVert^p phi_{0}(r) + left(sum_{n=1}^{infty} leftlVert x_{n}rightrVertphi_{n}(r)right)^q leq phi_{0}(r)right} end{equation*} and define the Bohr radius associated with ϕ by begin{equation*}R_{p,q,phi}(Omega,X)=inf left{R_{p,q,phi}(f,Omega,X): leftlVert frightrVert_{H^{infty}(Omega,X)} leq 1right}.end{equation*} In this article, we extensively study the Bohr radius $R_{p,q,phi}(Omega,X)$ , when X is an arbitrary Banach space, and $X=mathcal{B}(mathcal{H})$ is the algebra of all bounded linear operators on a complex Hilbert space $mathcal{H}$ . Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775120","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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