Homotopy of ends and boundaries of CAT(0) groups

For n ≥ 0, we exhibit CAT(0) groups that are n-connected at infinity, and have boundary which is (n − 1)-connected, but this boundary has non-trivial nth-homotopy group. In particular, we construct 1-ended CAT(0) groups that are simply connected at infinity, but have a boundary with non-trivial fundamental group. Our base examples are 1-ended CAT(0) groups that have non-path connected boundaries. In particular, we show all parabolic semidirect products of the free group of rank 2 and have a non-path connected boundary.     

关于循环节长度计算公式的改进

设p为正整数,p=2α.5β.m,α,β为非负整数,(m,10)=1.若m标准分解式为m=p1t1p2t2…pktk,其中pi为不等于2和5的质数,ti∈Z,令τ(m)=[φ(p1t1),φ(p2t2),…,φ(pktk)],则既约真分数q/p转化为循环小数时,其循环节长度h|τ(m)。     

Stationary convection–diffusion equation in an infinite cylinder

We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.     

Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

Riffs on the infinite ping-pong ball conundrum

This article presents a novel re-conceptualisation to a well-known problem – The Ping-Pong Ball Conundrum. We introduce a variant of this super-task by considering it through the lens of ‘measuring infinity’ – a conceptualisation of infinity that extrapolates measuring properties of numbers, rather than cardinal properties. This approach is consistent with a nonstandard analysis approach to infinite numbers, and gives credence to the intuitive (but otherwise normatively incorrect) resolution. We explore the mathematical motivation and consequences of this variant, as well as offer further ‘riffs’ on the infinite ball problem for consideration.     

一类五次微分系统的定性分析

对一类五次平面多项式微分系统进行了定性分析.给出原点的中心与等时中心条件及极限环的存在性.研究了此系统无穷远点的性态,该无穷远点是高次奇点,并运用把大角域分为若干小角域的方法对此高次奇点在不定号情形下轨线的分布情况进行讨论.     

Boundary Regularity for Conformally Compact Einstein Metrics in Even Dimensions

Sufficient conditions are derived for a symmetric hyperbolic system with large variable-coefficient terms to be uniformly well posed. Examples of systems satisfying those conditions are presented.     

基于粒子群算法的隔振体系参数优化研究

首先利用粒子群优化算法,以传递率为适应值函数,分别对单自由度、双自由度隔振体系进行了隔振参数的单目标和多目标优化研究.随之,考虑了状态反馈控制方法,目标函数选为传递函数矩阵的无穷范数;数值试验表明,在无控最优隔振参数的基础上,施加反馈控制,并不能进一步优化隔振体系;而重新在线计算隔振参数及反馈控制器,可以进一步优化原无控最优隔振体系.最后,进行了兼顾控制力传递率的多目标研究,旨在优化隔振体系传递率的同时,尽可能降低控制能源消耗.本文的系统研究,解决了隔振体系参数的最优设计问题;并在隔振基础上施加控制体系,讨论了最优参数设置的重要性;此外,兼顾控制能源消耗的参数优化,对于传统的隔振和控制体系设计具有一定的创新意义.     

关于Smarandache对偶函数

定义Smarandache对偶函数S*(n)为最大的正整数m使得m!|n.定义另一种双阶乘函数S**(n)为最大的正整数2m-1使得(2m-1)!!|n,其中2 n;且当2|n时,为最大的正整数2m使得(2m)!!|n.本文的主要目的是利用初等方法研究一个包含S**(n)的无穷级数的收敛性,并给出一个有趣的恒等式.