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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Pub Date : 2023-10-26DOI: 10.1017/s0013091523000603
Ling Li, Yutian Lei
Abstract In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: begin{equation*} left{ begin{aligned} &-Delta u=left(frac{1}{|x|^{n-2}}ast v^pright)v^{p-1},quad u gt 0 text{in} mathbb{R}^{n}, &-Delta v=left(frac{1}{|x|^{n-2}}ast u^qright)u^{q-1},quad v gt 0 text{in} mathbb{R}^{n}, end{aligned} right. end{equation*} where $n geq3$ and $min{p,q} gt 1$ . We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.
{"title":"On Liouville Theorems of a Hartree–Poisson system","authors":"Ling Li, Yutian Lei","doi":"10.1017/s0013091523000603","DOIUrl":"https://doi.org/10.1017/s0013091523000603","url":null,"abstract":"Abstract In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: begin{equation*} left{ begin{aligned} &-Delta u=left(frac{1}{|x|^{n-2}}ast v^pright)v^{p-1},quad u gt 0 text{in} mathbb{R}^{n}, &-Delta v=left(frac{1}{|x|^{n-2}}ast u^qright)u^{q-1},quad v gt 0 text{in} mathbb{R}^{n}, end{aligned} right. end{equation*} where $n geq3$ and $min{p,q} gt 1$ . We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134909890","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 : 2023-10-23DOI: 10.1017/s0013091523000597
Belgacem Rahal, Phuong Le
Abstract In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem begin{equation*} -text{div} (|nabla u|^{p-2} nabla u + w(x)|nabla u|^{q-2} nabla u) = left(frac{1}{|x|^{N-mu}}*f|u|^rright) f(x)|u|^{r-2}u quadtext{in} mathbb{R}^N, end{equation*} where $qge pge2$ , r > q , $0 lt mu lt N$ and $w,f in L^1_{rm loc}(mathbb{R}^N)$ are two non-negative functions such that $w(x) le C_1|x|^a$ and $f(x) ge C_2|x|^b$ for all $|x| gt R_0$ , where $R_0,C_1,C_2 gt 0$ and $a,binmathbb{R}$ . Under some appropriate assumptions on p , q , r , µ , a , b and N , we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of $mathbb{R}^N$ . First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.
摘要本文研究双相问题begin{equation*} -text{div} (|nabla u|^{p-2} nabla u + w(x)|nabla u|^{q-2} nabla u) = left(frac{1}{|x|^{N-mu}}*f|u|^rright) f(x)|u|^{r-2}u quadtext{in} mathbb{R}^N, end{equation*}的可能无界变号弱解，其中$qge pge2$, r ＆gt;Q, $0 lt mu lt N$和$w,f in L^1_{rm loc}(mathbb{R}^N)$是两个非负函数，使得$w(x) le C_1|x|^a$和$f(x) ge C_2|x|^b$适用于所有$|x| gt R_0$，其中$R_0,C_1,C_2 gt 0$和$a,binmathbb{R}$。在p, q, r，µ，a, b和N的适当假设下，我们证明了在$mathbb{R}^N$紧集外稳定或稳定的弱解的各种liouville型定理。首先，利用稳定性性质建立标准积分估计，得到稳定弱解的不存在性结果。然后，利用Pohožaev恒等式，导出了紧集外稳定弱解的liouville型定理。
{"title":"Stable Solutions to Double Phase Problems Involving a Nonlocal Term","authors":"Belgacem Rahal, Phuong Le","doi":"10.1017/s0013091523000597","DOIUrl":"https://doi.org/10.1017/s0013091523000597","url":null,"abstract":"Abstract In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem begin{equation*} -text{div} (|nabla u|^{p-2} nabla u + w(x)|nabla u|^{q-2} nabla u) = left(frac{1}{|x|^{N-mu}}*f|u|^rright) f(x)|u|^{r-2}u quadtext{in} mathbb{R}^N, end{equation*} where $qge pge2$ , r > q , $0 lt mu lt N$ and $w,f in L^1_{rm loc}(mathbb{R}^N)$ are two non-negative functions such that $w(x) le C_1|x|^a$ and $f(x) ge C_2|x|^b$ for all $|x| gt R_0$ , where $R_0,C_1,C_2 gt 0$ and $a,binmathbb{R}$ . Under some appropriate assumptions on p , q , r , µ , a , b and N , we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of $mathbb{R}^N$ . First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135405611","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 : 2023-10-23DOI: 10.1017/s0013091523000561
Tomokuni Takahashi
Abstract We classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of the hyperquadrics containing the surface.
{"title":"Classification of Subpencils for Hyperplane Sections on Certain K3 Surfaces","authors":"Tomokuni Takahashi","doi":"10.1017/s0013091523000561","DOIUrl":"https://doi.org/10.1017/s0013091523000561","url":null,"abstract":"Abstract We classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of the hyperquadrics containing the surface.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368173","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 : 2023-10-23DOI: 10.1017/s0013091523000640
Alastair N. Fletcher, Daniel A. Nicks
Abstract Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.
{"title":"Normal Families and Quasiregular Mappings","authors":"Alastair N. Fletcher, Daniel A. Nicks","doi":"10.1017/s0013091523000640","DOIUrl":"https://doi.org/10.1017/s0013091523000640","url":null,"abstract":"Abstract Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135367701","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 : 2023-10-23DOI: 10.1017/s0013091523000652
Shane Chern, Shishuo Fu, Zhicong Lin
Abstract Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li’s conjecture and Burstein’s first conjecture but also two more delicate generating function identities with the $mathsf{ides}$ statistic concerned in the restricted permutation case and the $mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.
最近，Hong和Li对反转序列中的长度- 4模式回避进行了系统的研究，特别是他们推测0021-避免反转序列的数量可以通过OEIS条目A218225来枚举。与此同时，伯斯坦认为，同样的序列也可能包含三组模式受限的排列。本文的目的不仅是对Hong and Li猜想和Burstein第一猜想的证实，而且是对限制置换情况下的$mathsf{ides}$统计量和限制逆序列情况下的$mathsf{asc}$统计量的两个更精细的生成函数恒等式的证实，从而得到一个新的等分布结果。
{"title":"Burstein’s Permutation Conjecture, Hong and Li’s Inversion Sequence Conjecture and Restricted Eulerian Distributions","authors":"Shane Chern, Shishuo Fu, Zhicong Lin","doi":"10.1017/s0013091523000652","DOIUrl":"https://doi.org/10.1017/s0013091523000652","url":null,"abstract":"Abstract Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li’s conjecture and Burstein’s first conjecture but also two more delicate generating function identities with the $mathsf{ides}$ statistic concerned in the restricted permutation case and the $mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135411984","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 : 2023-10-19DOI: 10.1017/s0013091523000639
Hui Li, Jun Wang, Xiao Yao, Zhuan Ye
Abstract Let $f(z)=sumlimits_{j=0}^{infty} a_j z^j$ be a transcendental entire function and let $f_omega(z)=sumlimits_{j=0}^{infty}chi_j(omega) a_j z^j$ be a random entire function, where $chi_j(omega)$ are independent and identically distributed random variables defined on a probability space $(Omega, mathcal{F}, mu)$ . In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(omega)$ and a set $Esubset [e, infty)$ of finite logarithmic measure such that, for $r gt r_0$ and $rnotin E$ , begin{equation*} |log M(r, f)- N(r,0, f_omega)|le (C/A)^{frac1{B}},log^{frac1{B}},log M(r,f) +log,log M(r, f), qquad a.s. end{equation*} where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f . As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions f ω and for any ϵ > 0, there is r 0 such that, for $r gt r_0$ , begin{equation*} T(r, f) le N(r,0, f_omega)+left(tfrac12+epsilonright) log T(r, f). end{equation*}
{"title":"Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions","authors":"Hui Li, Jun Wang, Xiao Yao, Zhuan Ye","doi":"10.1017/s0013091523000639","DOIUrl":"https://doi.org/10.1017/s0013091523000639","url":null,"abstract":"Abstract Let $f(z)=sumlimits_{j=0}^{infty} a_j z^j$ be a transcendental entire function and let $f_omega(z)=sumlimits_{j=0}^{infty}chi_j(omega) a_j z^j$ be a random entire function, where $chi_j(omega)$ are independent and identically distributed random variables defined on a probability space $(Omega, mathcal{F}, mu)$ . In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(omega)$ and a set $Esubset [e, infty)$ of finite logarithmic measure such that, for $r gt r_0$ and $rnotin E$ , begin{equation*} |log M(r, f)- N(r,0, f_omega)|le (C/A)^{frac1{B}},log^{frac1{B}},log M(r,f) +log,log M(r, f), qquad a.s. end{equation*} where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f . As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions f ω and for any ϵ > 0, there is r 0 such that, for $r gt r_0$ , begin{equation*} T(r, f) le N(r,0, f_omega)+left(tfrac12+epsilonright) log T(r, f). end{equation*}","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666548","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 : 2023-10-18DOI: 10.1017/s0013091523000627
Tim Huber, Chang Liu, James McLaughlin, Dongxi Ye, Miaodan Yuan, Sumeng Zhang
Abstract This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $eta(12z)^{2}$ and recent results of Chang on the p th coefficients of $eta(4z)^{6}$ and $eta(6z)^{4}$ . Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.
{"title":"On the Vanishing of the Coefficients of CM Eta Quotients","authors":"Tim Huber, Chang Liu, James McLaughlin, Dongxi Ye, Miaodan Yuan, Sumeng Zhang","doi":"10.1017/s0013091523000627","DOIUrl":"https://doi.org/10.1017/s0013091523000627","url":null,"abstract":"Abstract This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $eta(12z)^{2}$ and recent results of Chang on the p th coefficients of $eta(4z)^{6}$ and $eta(6z)^{4}$ . Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135883459","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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