Pub Date : 2022-05-01DOI: 10.1017/S0017089523000204
T. Aihara, Qi Wang
Abstract We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ( $=$ the support $tau$ -tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).
{"title":"A symmetry of silting quivers","authors":"T. Aihara, Qi Wang","doi":"10.1017/S0017089523000204","DOIUrl":"https://doi.org/10.1017/S0017089523000204","url":null,"abstract":"Abstract We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ( \u0000$=$\u0000 the support \u0000$tau$\u0000 -tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"687 - 696"},"PeriodicalIF":0.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42107942","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 : 2022-04-27DOI: 10.1017/S0017089522000180
Sam Hughes
Abstract We show via $ell^2$ -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building.
{"title":"A note on the rational homological dimension of lattices in positive characteristic","authors":"Sam Hughes","doi":"10.1017/S0017089522000180","DOIUrl":"https://doi.org/10.1017/S0017089522000180","url":null,"abstract":"Abstract We show via \u0000$ell^2$\u0000 -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"138 - 140"},"PeriodicalIF":0.5,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48568792","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 : 2022-04-26DOI: 10.1017/S0017089523000253
Valeriano Aiello, S. Baader, Livio Ferretti
Abstract We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$ , within a sufficiently large degree range: $4/p^7$ . As an application, we classify knot Jones polynomials modulo two of span up to eight.
{"title":"On the Jones polynomial modulo primes","authors":"Valeriano Aiello, S. Baader, Livio Ferretti","doi":"10.1017/S0017089523000253","DOIUrl":"https://doi.org/10.1017/S0017089523000253","url":null,"abstract":"Abstract We derive an upper bound on the density of Jones polynomials of knots modulo a prime number \u0000$p$\u0000 , within a sufficiently large degree range: \u0000$4/p^7$\u0000 . As an application, we classify knot Jones polynomials modulo two of span up to eight.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"730 - 734"},"PeriodicalIF":0.5,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44999431","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 : 2022-04-12DOI: 10.1017/S0017089522000131
G. Figueiredo, M. Montenegro
Abstract We prove the existence of a solution for a class of activator–inhibitor system of type $- Delta u +u = f(u) -v$ , $-Delta v+ v=u$ in $mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $Ngeq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.
{"title":"Existence of solution for a class of activator–inhibitor systems","authors":"G. Figueiredo, M. Montenegro","doi":"10.1017/S0017089522000131","DOIUrl":"https://doi.org/10.1017/S0017089522000131","url":null,"abstract":"Abstract We prove the existence of a solution for a class of activator–inhibitor system of type \u0000$- Delta u +u = f(u) -v$\u0000 , \u0000$-Delta v+ v=u$\u0000 in \u0000$mathbb{R}^{N}$\u0000 . The function f is a general nonlinearity which can grow polynomially in dimension \u0000$Ngeq 3$\u0000 or exponentiallly if \u0000$N=2$\u0000 . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"98 - 113"},"PeriodicalIF":0.5,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44823830","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 : 2022-04-03DOI: 10.1017/S0017089522000349
A. Klyachko, Mikhail A. Mikheenko
Abstract The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).
{"title":"Yet another Freiheitssatz: Mating finite groups with locally indicable ones","authors":"A. Klyachko, Mikhail A. Mikheenko","doi":"10.1017/S0017089522000349","DOIUrl":"https://doi.org/10.1017/S0017089522000349","url":null,"abstract":"Abstract The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"337 - 344"},"PeriodicalIF":0.5,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688552","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 : 2022-03-04DOI: 10.1017/S0017089522000064
Hongjun Li, Chunhui Qiu
Abstract As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.
{"title":"A note on holomorphic sectional curvature of a hermitian manifold","authors":"Hongjun Li, Chunhui Qiu","doi":"10.1017/S0017089522000064","DOIUrl":"https://doi.org/10.1017/S0017089522000064","url":null,"abstract":"Abstract As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"739 - 745"},"PeriodicalIF":0.5,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43636462","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 : 2022-02-22DOI: 10.1017/S0017089523000010
O. Mata-Gutiérrez, L. Roa-Leguizamón, H. Torres-López
Abstract The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x in X$ to show that there exists an embedding from the Grassmannian variety $mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $mathfrak{M}_{X,H}(n;,c_1,c_2+m)$ which induces an injective morphism from $X times M_{X,H}(n;,c_1,c_2)$ to $Hilb_{, mathfrak{M}_{X,H}(n;,c_1,c_2+m)}$ .
{"title":"On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces","authors":"O. Mata-Gutiérrez, L. Roa-Leguizamón, H. Torres-López","doi":"10.1017/S0017089523000010","DOIUrl":"https://doi.org/10.1017/S0017089523000010","url":null,"abstract":"Abstract The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point \u0000$x in X$\u0000 to show that there exists an embedding from the Grassmannian variety \u0000$mathbb{G}(E_x,m)$\u0000 into the moduli space of torsion-free sheaves \u0000$mathfrak{M}_{X,H}(n;,c_1,c_2+m)$\u0000 which induces an injective morphism from \u0000$X times M_{X,H}(n;,c_1,c_2)$\u0000 to \u0000$Hilb_{, mathfrak{M}_{X,H}(n;,c_1,c_2+m)}$\u0000 .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"414 - 429"},"PeriodicalIF":0.5,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44945142","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 : 2022-02-21DOI: 10.1017/S0017089522000027
L. T. Thuy, N. Toan
Abstract In this study, we consider the nonclassical diffusion equations with time-dependent memory kernels begin{equation*} u_{t} - Delta u_t - Delta u - int_0^infty k^{prime}_{t}(s) Delta u(t-s) ds + f( u) = g end{equation*} on a bounded domain $Omega subset mathbb{R}^N,, Ngeq 3$ . Firstly, we study the existence and uniqueness of weak solutions and then, we investigate the existence of the time-dependent global attractors $mathcal{A}={A_t}_{tinmathbb{R}}$ in $H_0^1(Omega)times L^2_{mu_t}(mathbb{R}^+,H_0^1(Omega))$ . Finally, we prove that the asymptotic dynamics of our problem, when $k_t$ approaches a multiple $mdelta_0$ of the Dirac mass at zero as $tto infty$ , is close to the one of its formal limit begin{equation*}u_{t} - Delta u_{t} - (1+m)Delta u + f( u) = g. end{equation*} The main novelty of our results is that no restriction on the upper growth of the nonlinearity is imposed and the memory kernel $k_t(!cdot!)$ depends on time, which allows for instance to describe the dynamics of aging materials.
{"title":"The nonclassical diffusion equations with time-dependent memory kernels and a new class of nonlinearities","authors":"L. T. Thuy, N. Toan","doi":"10.1017/S0017089522000027","DOIUrl":"https://doi.org/10.1017/S0017089522000027","url":null,"abstract":"Abstract In this study, we consider the nonclassical diffusion equations with time-dependent memory kernels \u0000begin{equation*} u_{t} - Delta u_t - Delta u - int_0^infty k^{prime}_{t}(s) Delta u(t-s) ds + f( u) = g end{equation*}\u0000 on a bounded domain \u0000$Omega subset mathbb{R}^N,, Ngeq 3$\u0000 . Firstly, we study the existence and uniqueness of weak solutions and then, we investigate the existence of the time-dependent global attractors \u0000$mathcal{A}={A_t}_{tinmathbb{R}}$\u0000 in \u0000$H_0^1(Omega)times L^2_{mu_t}(mathbb{R}^+,H_0^1(Omega))$\u0000 . Finally, we prove that the asymptotic dynamics of our problem, when \u0000$k_t$\u0000 approaches a multiple \u0000$mdelta_0$\u0000 of the Dirac mass at zero as \u0000$tto infty$\u0000 , is close to the one of its formal limit \u0000begin{equation*}u_{t} - Delta u_{t} - (1+m)Delta u + f( u) = g. end{equation*}\u0000 The main novelty of our results is that no restriction on the upper growth of the nonlinearity is imposed and the memory kernel \u0000$k_t(!cdot!)$\u0000 depends on time, which allows for instance to describe the dynamics of aging materials.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"716 - 733"},"PeriodicalIF":0.5,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49075907","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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