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":"1 1","pages":""},"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":"93 12","pages":"0"},"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":" 47","pages":"0"},"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":" 5","pages":"0"},"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":"218 S699","pages":"0"},"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":"32 125","pages":"0"},"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":"237 6","pages":"0"},"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}
Pub Date : 2023-11-01DOI: 10.1017/s001309152300072x
{"title":"PEM series 2 volume 66 issue 4 Cover and Front matter","authors":"","doi":"10.1017/s001309152300072x","DOIUrl":"https://doi.org/10.1017/s001309152300072x","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"14 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139302420","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 : 2023-11-01DOI: 10.1017/s0013091523000731
{"title":"PEM series 2 volume 66 issue 4 Cover and Back matter","authors":"","doi":"10.1017/s0013091523000731","DOIUrl":"https://doi.org/10.1017/s0013091523000731","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"22 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139298789","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 : 2023-11-01DOI: 10.1017/s0013091523000718
Belgacem Rahal, Phuong Le
{"title":"Stable solutions to double phase problems involving a nonlocal term – erratum","authors":"Belgacem Rahal, Phuong Le","doi":"10.1017/s0013091523000718","DOIUrl":"https://doi.org/10.1017/s0013091523000718","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"78 1","pages":"1229 - 1229"},"PeriodicalIF":0.7,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139291868","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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