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Periodic points of self-maps of products of lens spaces $L(3)times L(3)$ 透镜空间积L(3)乘以L(3)$的自映射的周期点
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.053
J. Jezierski
Let $fcolon Mto M$ be a self-map of a compact manifold and $nin mathbb{N}$.The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps$fcolon Mto M$ the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears.Here we give a criterion which helps to decide whether the necessary condition is also sufficient.We apply this result to show that for self-maps of the product of the lens space $M=L(3)times L(3)$ the necessary condition is also sufficient.
设$f冒号M到M$是紧流形的自映射,$n在mathbb{n}$中。f的光滑同伦类中n个周期点的最小个数可能小于连续同伦类。我们问:对于哪个自映射$f冒号M到M$两个最小值是相同的,对于每个规定的多重性?在环面和紧李群的自映射研究中,出现了一个必要条件。这里我们给出一个判别必要条件是否也是充分条件的判据。我们应用这一结果证明了透镜空间积的自映射$M=L(3)乘以L(3)$的必要条件也是充分的。
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引用次数: 0
Multiple solutions for perturbed quasilinear elliptic problems 扰动拟线性椭圆型问题的多重解
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.069
R. Bartolo, A. M. Candela, A. Salvatore
We investigate the existence of multiple solutionsfor the $(p,q)$-quasilinear elliptic problem[begin{cases}-Delta_p u -Delta_q u = g(x, u) + varepsilon h(x,u)& mbox{in } Omega,u=0 & mbox{on } partialOmega, end{cases}]where $1< p< q< +infty$, $Omega$ is an open bounded domain of${mathbb R}^N$, the nonlinearity $g(x,u)$ behaves at infinity as $|u|^{q-1}$,$varepsilonin{mathbb R}$ and $hin C(overlineOmegatimes{mathbb R},{mathbb R})$.In spite of the possible lack of a variational structure of this problem,from suitable assumptions on $g(x,u)$ andappropriate procedures and estimates,the existence of multiple solutions can be proved for small perturbations.
我们研究了$(p,q)$拟线性椭圆问题的多重解的存在性{cases}-Delta_p u-Delta_q u =g(x,u)+varepsilonh(x,u)&mbox{in}Omega,u=0&mbox{on}partial Omega,end{cases}]其中$1<p<q<+infty$,$Omega$是${mathbb R}^N$的开有界域,非线性$g(x,u)$在无穷大时表现为$|u|^{q-1}$,{mathbb R}$中的$varepsilonin和C中的$h(overlineOmegatimes{math BB R},{mathibb R})$。尽管这个问题可能缺乏变分结构,但通过对$g(x,u)$的适当假设以及适当的程序和估计,可以证明小扰动下存在多个解。
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引用次数: 0
Rank-two solenoidal endomorphisms 秩二螺线管自同态
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.063
K. Ha, Jong Bum Lee
Let $G$ be a torsion-free abelian group of rank two and let$phi$ be an endomorphism of $G$, called a rank-two emph{solenoidal endomorphism}.Then it is represented by a $2times 2$-matrix $M_phi$ with rational entries.The purpose of this article is to prove the following:The group, $mathrm{coker}(phi)$, of the cokernut of $phi$ is finiteif and only if $M_phi$ is nonsingular, and if it is so, thenwe give an explicit formula for the order of $mathrm{coker}(phi)$, $[G:mathrm{im}(phi)]$,in terms of $p$-adic absolute values of the determinant of $M_phi$.Since $G$ is abelian, the Reidemeister number of $phi$ is equal to the order of the cokernut of $mathrm{id}-phi$ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual $widehatphi$ of $phi$.Thereby, we solve completely the problem raised in cite{Miles} of finding the possible sequences of periodic point countsfor emph{all} endomorphisms of the rank-two solenoids.
设$G$为二阶无扭转阿贝尔群,设$phi$为$G$的自同态,称为二阶emph{螺线自同态}。然后用含有有理数项的$2times 2$ -矩阵$M_phi$表示。本文的目的是证明如下:$phi$的椰子的群$mathrm{coker}(phi)$是有限的当且仅当$M_phi$是非奇异的,如果是,那么我们给出一个关于$M_phi$的行列式的$p$进数绝对值的$mathrm{coker}(phi)$, $[G:mathrm{im}(phi)]$的阶的显式公式。因为$G$是阿贝尔的,$phi$的Reidemeister数等于$mathrm{id}-phi$的cokernut的阶数,当它是有限的时候,等于$phi$的Pontryagin对偶$widehatphi$的不动点的个数,从而完全解决了cite{Miles}中提出的寻找二阶螺线管的emph{所有}自同态的周期点计数的可能序列的问题。
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引用次数: 0
The fixed point set of the inverse involution on a Lie group 李群上逆对合的不动点集
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.012
H. Duan, Shali Liu
The inverse involution on a Lie group $G$ is the periodic $2$ transformation$gamma $ that sends each element $gin G$ to its inverse $g^{-1}$. Thevariety of the fixed point set $Fix(gamma )$ is of importance for therelevances with Morse function on the Lie group $G$, and the $G$-representationsof the cyclic group $mathbb{Z}_{2}$. In this paper we develop an approach to calculate the diffeomorphism types of the fixed point sets $Fix(gamma)$ for the simple Lie groups.
李群$G$上的逆对合是周期$2$变换$gamma $,它将G$中的每个元素$G 发送到它的逆$G ^{-1}$。不动点集$Fix(gamma)$的变化对于李群$G$上的Morse函数的相关性和循环群$mathbb{Z}_{2}$的$G$-表示具有重要意义。本文给出了一种计算简单李群不动点集$Fix(gamma)$的微分同态类型的方法。
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引用次数: 0
Fixed point indices and fixed words at infinity of selfmaps of graphs II 图的自映射的不动点指标和无穷远处的不动字
4区 数学 Q2 MATHEMATICS Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.007
Qiang Zhang, Xuezhi Zhao
The index $mathrm{ind}(mathbf{F})$ of a fixed point class $mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper cite{ZZ}, the authors introduced a new invariant $mathrm{ichr}(mathbf{F})$ called the improved characteristic, and proved that $mathrm{ind}(mathbf{F})leq mathrm{ichr}(mathbf{F})$ for all fixed point classes of $pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.
不动点类的指标$mathrm{ind}(mathbf{F})$$mathbf{F}$是Nielsen不动点理论中的经典不变量。在最近的论文cite{ZZ}中,作者引入了一个新的不变量$mathrm{ichr}(mathbf{F})$,称为改进特征,并证明了$pi_1$ -连通有限图的内射自映射的所有不动点类的$mathrm{ind}(mathbf{F})leq mathrm{ichr}(mathbf{F})$。在这篇笔记中,我们证明了上面提到的两个同伦不变量是完全相同的。由于改进的特征完全由基群的自同态决定,我们给出了计算图自映射不动点类指标的群论方法。因此,我们给出了一个新的不动点判据,它推广了Gaboriau, Jaeger, Levitt和Lustig的判据。
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引用次数: 0
Realization of a graph as the Reeb graph of a height function on an embedded surface 在嵌入曲面上将图实现为高度函数的Reeb图
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-01-25 DOI: 10.12775/tmna.2021.058
Irina Gelbukh
We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $mathbb{R}^3$such that the Reeb graph of the associated height function has the structure of $G$.In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011.We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functionsand in the class of round Morse-Bott functions.In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott;we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.
我们证明了对于给定的没有循环边和孤立顶点的有限图$G$,在$mathbb{R}^3$中存在闭可定向曲面的嵌入,使得相关高度函数的Reeb图具有$G$的结构。特别地,这对Masumoto和Saeki在2011年提出的相应问题给出了肯定的答案。我们还给出了给定曲面允许给定图实现的标准,并研究了Morse函数类和圆Morse Bott函数类中的问题。在实现同胚的情况下,高度函数可以选择Morse Bott;根据图结构,我们从下面估计它的临界圆的数量和它的孤立临界点的数量。
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引用次数: 0
Existence of solutions for the Brezis-Nirenberg problem 布雷齐斯-尼伦堡问题解的存在性
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2023-01-25 DOI: 10.12775/tmna.2022.029
Francisco Odair de Paiva, O. Miyagaki, Adilson E. Presoto
We are concerned with of existence of solutions to the semilinear elliptic problem$$ begin{cases} - Delta u=lambda_{k}u+u^3 &text{in } Omega, u= 0 &text{on }partial Omega, end{cases}$$%in a bounded domain $Omega subset mathbb{R}^{4}$. Here $lambda_k$is an eigenvalue of the $-Delta$ in $H_0^1(Omega)$. We prove that this problem has a nontrivial solution.
我们关注的是一个半线性椭圆问题$$bbegin{cases}-Delta u=lambda解的存在性_{k}u+在有界域$Omegasubetmathbb{R}^{4}$中,u^3&&text{in}Omega,u=0&&text{on}partialOmega、end{cases}$$%。这里$lambda_k$是$H_0^1(Omega)$中$-Delta$的特征值。我们证明了这个问题有一个非平凡的解决方案。
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引用次数: 0
Fourth-order elliptic problems involving concave-superlinear nonlinearities 涉及凹-超线性非线性的四阶椭圆问题
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2022-12-11 DOI: 10.12775/tmna.2022.011
T. Cavalcante, Edcarlos D. Silva
The existence of solutions for a huge class of superlinear elliptic problems involving fourth-order elliptic problems defined on bounded domainsunder Navier boundary conditions is established. To this end we do not apply the well-knownAmbrosetti-Rabinowitz condition. Instead, we assume that the nonlinear termis nonquadratic at infinity. Furthermore, the nonlinear term is a concave-superlinear function which can be indefinite in sign. In order to apply variational methods we employ some delicate arguments recovering some kind of compactness.
在Navier边界条件下,建立了一大类超线性椭圆问题的解的存在性,该问题涉及定义在有界域上的四阶椭圆问题。为此,我们不应用众所周知的Ambrosetti-Rabinowitz条件。相反,我们假设非线性终端在无穷大处是非二次的。此外,非线性项是一个符号不定的凹超线性函数。为了应用变分方法,我们使用了一些精细的自变量来恢复某种紧致性。
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引用次数: 3
On radial solutions for some elliptic equations involving operators with unbounded coefficients in exterior domains 一类外域无界系数算子椭圆型方程的径向解
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2022-12-11 DOI: 10.12775/tmna.2022.026
Anderson L. A. de Araujo, Luiz F. O. Faria, S. Alarcón, Leonelo Iturriaga
We study existence and multiplicity of radial solutions for some quasilinear elliptic problems involving the operator $L_N=Delta - xcdot nabla$ on$mathbb{R}^Nsetminus B_1$, where $Delta$ is the Laplacian, $xcdot nabla$ is an unbounded drift term, $Ngeq 3$ and $B_1$ is the unit ball centered at the origin.We consider: (i) Eigenvalue problems, and (ii) Problems involving a nonlinearity of concave and convex type. On the first class of problems we get a compact embedding result, whereas on the second, we address the well-known questionof Ambrosetti, Brezis and Cerami from 1993 concerning the existence of two positive solutions for some problems involving the supercritical Sobolev exponent in symmetric domains for the Laplacian. Specifically, we providelinebreak a new approach of answering the ABC-question for elliptic problems with unbounded coefficients in exterior domains and we find asymptotic properties of the radial solutions. Furthermore, we study the limit case, namely when nonlinearity involvesa sublinear term and a linear term. As far as we know, this is the first work that deals with such a case, even for the Laplacian. In our approach, we use both topological and variational arguments.
我们研究了一类拟线性椭圆问题径向解的存在性和多重性,这些问题涉及算子$L_N=Delta-xcdotnabla$在$mathbb{R}^Nsetminus B_1$上,其中$Delta$是拉普拉斯算子,$xcdotnapla$是无界漂移项,$Ngeq3$和$B_1$是以原点为中心的单位球。我们考虑:(i)特征值问题,和(ii)涉及凹凸型非线性的问题。在第一类问题上,我们得到了一个紧凑的嵌入结果,而在第二类问题中,我们解决了1993年Ambrosetti、Brezis和Cerami的著名问题,即拉普拉斯算子在对称域中涉及超临界Sobolev指数的一些问题的两个正解的存在性。具体地说,我们提供了一种新的方法来回答外域中系数无界的椭圆问题的ABC问题,并发现了径向解的渐近性质。此外,我们还研究了非线性包含次线性项和线性项的极限情况。据我们所知,这是第一个处理这种情况的工作,即使对于拉普拉斯算子也是如此。在我们的方法中,我们同时使用拓扑和变分自变量。
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引用次数: 0
An accelerated variant of the projection based parallel hybrid algorithm for split null point problems 求解分裂零点问题的基于投影的并行混合算法的一种加速变体
IF 0.7 4区 数学 Q2 MATHEMATICS Pub Date : 2022-12-10 DOI: 10.12775/tmna.2022.015
Yasir Arfat, P. Kumam, M. A. A. Khan, P. S. Ngiamsunthorn
In this paper, we consider an accelerated shrinking projection based parallel hybrid algorithm to study the split null point problem (SNPP) associated with the maximal monotone operators in Hilbert spaces. The analysis of the proposed algorithm provides strong convergence results under suitable set of control conditions as well as viability with the help of a numerical experiment. The results presented in this paper improve various existing results in the current literature.
本文考虑一种基于加速收缩投影的并行混合算法来研究Hilbert空间中与最大单调算子相关的分裂零点问题。对所提出算法的分析在适当的控制条件下提供了强大的收敛性结果,并在数值实验的帮助下提供了可行性。本文给出的结果改进了现有文献中的各种现有结果。
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引用次数: 0
期刊
Topological Methods in Nonlinear Analysis
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