Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

英文中文

Desingularization of 2D elliptic free-boundary problem with non-autonomous nonlinearity 具有非自主非线性的二维椭圆自由边界问题的去周期化

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-30 DOI: 10.1017/prm.2024.48

Jie Wan

In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two

[ begin{cases} -Delta u=lambda k(x)f(u) & text{in} D, u= c & displaystyletext{on} partial D, displaystyle - int_{partial D} frac{partial u}{partial nu},{rm d}s=I, end{cases} ]

where

$Dsubseteq mathbb {R}^2$

is a smooth bounded domain,

$nu$

is the outward unit normal to the boundary

$partial D$

$lambda$

and

$I$

are given constants and

$c$

is an unknown constant. Under some assumptions on

$f$

and

$k$

, we prove that there exists a family of solutions concentrating near strict local minimum points of

$Gamma (x)=({1}/{2})h(x,,x)- ({1}/{8pi })l

在本文中，我们考虑了由二维约束等离子体问题引起的半线性椭圆方程的解的存在性和极限行为。frac{partial u}{partialnu},{rm d}s=I, （end{cases}）] where $Ds=I, （end{cases}）.其中 $Dsubseteq mathbb {R}^2$ 是光滑有界域，$nu$ 是边界 $partial D$ 的向外单位法线，$lambda$ 和 $I$ 是给定常数，$c$ 是未知常数。在一些关于 $f$ 和 $k$ 的假设下，我们证明当 $lambda to +infty$ 时，存在一系列解集中在 $Gamma (x)=({1}/{2})h(x,,x)- ({1}/{8pi })ln k(x)$ 的严格局部最小点附近。这里的$h(x,,x)$是$D$中$-Delta$的罗宾函数。规定函数 $f$ 和 $k$ 可以是非常通用的。将 $k$ 视为 $measure$ 并使用涡度方法，即求解涡度最大化问题并分析最大化者的渐近行为，可以证明这一结果。此外，还得到了集中在几个点附近的解的存在性。

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引用次数: 0

Spreading primitive groups of diagonal type do not exist 不存在对角类型的展开原始群

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-26 DOI: 10.1017/prm.2024.53

John Bamberg, Saul D. Freedman, Michael Giudici

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between

$2$

-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

有限置换群的同步层次由介于 2$ - 传递群和原始群之间的群类组成。其中包括平展群类，平展群的定义是包络点的集合和多集合，已知平展群是近简、仿射或对角类型的基元群。在本文中，我们证明事实上不存在对角线类型的展开群。作为证明的一部分，我们证明了除六个零星群之外的所有非阿贝尔有限单纯群都有一个反式作用，其中一个点稳定器的适当正则子群被所有相应的两点稳定器所补充。

引用次数: 0

Topologically free actions and ideals in twisted Banach algebra crossed products 扭曲巴拿赫代数交叉积中的拓扑自由作用和理想

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-26 DOI: 10.1017/prm.2024.37

Krzysztof Bardadyn, Bartosz Kwaśniewski

We generalize the influential

$C^*$

-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full

$L^p$

-operator algebra crossed product for any

$pin [1,,infty ]$

. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted

$L^p$

-operator algebras coincide.

我们将川村-富山（Kawamura-Tomiyama）和阿奇博尔德-斯皮尔伯格（Archbold-Spielberg）对离散群作用交叉积的有影响力的 $C^*$ 代数结果推广到巴拿赫代数和扭曲作用领域。我们证明了拓扑自由性等同于所有来自子群的还原扭曲巴纳赫代数交叉积的交集性质，而在非扭曲情况下，等同于在 [1,,infty ]$ 中任意 $p 的全 $L^p$ 算子代数交叉积的广义交集性质。这为各种巴拿赫代数交叉积提供了有效的简单性标准。我们还用它把一些交叉积的素理想空间确定为作用的准轨道空间。对于可简化的作用，我们证明了完整的和简化的扭曲 $L^p$ -operator 放大系数是重合的。

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引用次数: 0

Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group 具有半小阿尔班尼斯图和残余有限基群的品种的辛格猜想

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-19 DOI: 10.1017/prm.2024.52

Luca F. Di Cerbo, Luigi Lombardi

We prove the Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group.

我们证明了具有半小阿尔班尼斯图和残余有限基群的变种的辛格猜想。

引用次数: 0

The homotopy decomposition of the suspension of a non-simply-connected five-manifold 非简单连接五芒星悬浮的同调分解

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-17 DOI: 10.1017/prm.2024.49

Pengcheng Li, Zhongjian Zhu

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth

$five$

-manifolds. As applications, we compute the reduced

$K$

-groups of

$M$

and show that the suspension map between the third cohomotopy set

$pi ^3(M)$

and the fourth cohomotopy set

$pi ^4(Sigma M)$

is a bijection.

在本文中，我们确定了某些连通可定向封闭光滑 $five$ -manifolds 的还原悬浮空间的同调类型。作为应用，我们计算了 $M$ 的还原 $K$ 群，并证明了第三同调集 $pi ^3(M)$ 和第四同调集 $pi ^4(Sigma M)$ 之间的悬浮映射是双射的。

引用次数: 0

Sharp conditions for the validity of the Bourgain–Brezis–Mironescu formula 布尔干-布雷齐斯-米罗内斯库公式有效性的苛刻条件

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-16 DOI: 10.1017/prm.2024.47

Elisa Davoli, Giovanni Di Fratta, Valerio Pagliari

Following the seminal paper by Bourgain, Brezis, and Mironescu, we focus on the asymptotic behaviour of some nonlocal functionals that, for each $uin L^2(mathbb {R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights ${rho _{varepsilon} }$, $varepsilon in (0,,1)$, we devise sufficient and necessary conditions on ${rho _{varepsilon} }$ for the associated nonlocal functionals to converge as $varepsilon to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

继布尔甘（Bourgain）、布雷齐斯（Brezis）和米罗内斯库（Mironescu）的开创性论文之后，我们将重点放在一些非局部函数的渐近行为上，对于 L^2(mathbb {R}^N)$ 中的每个 $u$，这些函数被定义为 $u$ 的加权平方差商的双积分。给定一个权值系列 ${rho _{varepsilon}$varepsilon 在（0,,1）$ 中，我们设计了关于 ${rho _{varepsilon} 的充分和必要条件。｝相关的非局部函数随着 $varepsilon to 0$ 收敛到迪里希特积分的变体上。最后，我们对我们的结果与现有文献进行了比较。

引用次数: 0

Linear actions of 的线性行动

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-16 DOI: 10.1017/prm.2024.36

Jim Fowler, Courtney Thatcher

For an odd prime $p$, we consider free actions of $(mathbb {Z}_{/{p}})^2$ on $S^{2n-1}times S^{2n-1}$ given by linear actions of $(mathbb {Z}_{/{p}})^2$ on $mathbb {R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the $k$-invariants and the Pontrjagin classes from the rotation numbers.

对于奇素数$p$，我们考虑$S^{2n-1}times S^{2n-1}$上$(mathbb {Z}_{/{p}})^2$ 的自由作用$(mathbb {Z}_{/{p}})^2$ 由$mathbb {R}^{4n}$ 上$(mathbb {Z}_{/{p}})^2$ 的线性作用给出。简单的例子包括透镜空间交叉透镜空间，但 $k$ 不变的计算表明还存在其他商。利用波斯尼科夫塔和外科理论的工具，我们可以通过 $k$ 不变式对商进行同构分类，并通过庞特贾金类对商进行同构分类。我们将介绍这些结果，并演示如何根据旋转数计算 $k$ 变量和庞特贾金类。

引用次数: 0

Quasi-hereditary covers of Temperley–Lieb algebras and relative dominant dimension Temperley-Lieb代数的准遗传盖和相对主维

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-11 DOI: 10.1017/prm.2024.35

Tiago Cruz, Karin Erdmann

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras

$S(2, d)$

and their

$q$

-analogues. Our second main result completely determines the relative dominant dimension of

$S(2, d)$

with respect to

$Q=V^{otimes d}$

, the

$d$

-th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of

$q$

-Schur algebras

$S(2,d)$

give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of th

表示理论和李理论中的许多联系和对偶性都可以用鲁基耶意义上的准遗传盖来解释。第一作者的最新研究表明，相对主维（和共主维）是对有限维代数的不同准遗传封面进行分类和区分的自然工具。在本文中，我们证明了具有简单保留对偶性的准遗传代数相对于特征倾斜模的直和的相对主维总是偶数或无限，而且这个同调不变式控制着具有简单保留对偶性的准遗传封面的质量。为了解决滕伯里-李卜代数问题，我们将这一结果应用于舒尔代数$S(2, d)$及其$q$类似物。我们的第二个主要结果完全确定了$S(2, d)$ 相对于$Q=V^{otimes d}$，即自然二维模块的$d$张量幂的相对主维。作为副产品，我们推导出 $q$ -Schur 对象 $S(2,d)$ 的 Ringel 对偶产生了 Temperley-Lieb 对象的准遗传盖。此外，我们精确地得到了当 Temperley-Lieb 代数与 $q$ -Schur 代数 $S(2,d)$的 Ringel 对偶的莫里塔等价时，以及当这两个代数不等价时，它们离莫里塔等价有多远。这些结果与积分设定是相容的，我们用它们来推导出，整数上劳伦多项式环上的 $q$ -Schur 代数的 Ringel 对偶与某个投影模是积分 Temperley-Lieb 代数的最佳准继承盖。

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引用次数: 0

Global centres in a class of quintic polynomial differential systems 一类五次多项式微分系统的全局中心

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-11 DOI: 10.1017/prm.2024.43

Leonardo P. C. da Cruz, Jaume Llibre

A centre of a differential system in the plane

$ {mathbb {R}}^2$

is an equilibrium point

$p$

having a neighbourhood

$U$

such that

$Usetminus {p}$

is filled with periodic orbits. A centre

$p$

is global when

$ {mathbb {R}}^2setminus {p}$

is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems

begin{align*} dot{x}= y,quad dot{y}={-}x+a_{05},y^5+a_{14},x,y^4+a_{23},x^2,y^3+a_{32},x^3,y^2+a_{41},x^4,y+a_{50},x^5, end{align*}

in the plane

$ {mathbb {R}}^2$

平面 $ {mathbb {R}}^2$ 中微分系统的中心是一个平衡点 $p$，它有一个邻域 $U$，使得 $Usetminus {p}$充满了周期性的轨道。当 $ {mathbb {R}}^2setminus {p}$ 充满周期性轨道时，中心 $p$ 是全局的。一般来说，对于给定类别的微分系统，区分中心和焦点是一个难题，而区分中心内部的全局中心也很困难。本文的目标是对以下一类五次多项式微分系统的中心和全局中心进行分类 begin{align*}dot{x}= y,quad dot{y}={-}x+a_{05},y^5+a_{14},x,y^4+a_{23},x^2,y^3+a_{32},x^3,y^2+a_{41},x^4,y+a_{50},x^5, end{align*} 在平面 $ {mathbb {R}}^2$ 中。

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引用次数: 0

Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential 带谐波势的分数非线性椭圆方程基态的唯一性

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-04-11 DOI: 10.1017/prm.2024.44

Tianxiang Gou

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,

[ (-Delta)^s u+ left(omega+|x|^2right) u=|u|^{p-2}u quad mbox{in} mathbb{R}^n, ]

where

$n geq 1$

$0< s<1$

$omega >-lambda _{1,s}$

$2< p< {2n}/{(n-2s)^+}$

$lambda _{1,s}>0$

is the lowest eigenvalue of

$(-Delta )^s + |x|^2$

. The fractional Laplacian

$(-Delta )^s$

is characterized as

$mathcal {F}((-Delta )^{s}u)(xi )=|xi |^{2s} mathcal {F}(u)(xi )$

for

$xi in mathbb {R}^n$

在本文中，我们证明了以下带谐波势的分数非线性椭圆方程基态的唯一性：[ (-Delta)^s u+ left(omega+|x|^2right) u=|u|^{p-2}u quad mbox{in} mathbb{R}^n, ] 其中 $n geq 1$ , $0<；s<1$ , $omega >-lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $lambda _{1,s}>0$ 是 $(-Delta )^s + |x|^2$ 的最小特征值。分数拉普拉斯函数 $(-Delta )^s$ 的特征为 $mathcal {F}((-Delta )^{s}u)(xi )=|xi |^{2s} 。对于 $xi in mathbb {R}^n$ 来说，这里的 $mathcal {F}$ 表示傅立叶变换。这解决了[M. Stanislavova and A. G. Stefanov.

引用次数: 0

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Proceedings of the Royal Society of Edinburgh Section A-Mathematics

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