The attractor of the scalar reaction diffusion equation is a smooth graph

For the scalar reaction diffusion equation with Dirichlet boundary conditions, it is proved that its maximal compact attractor is the graph of a C¹ function from a subset with nonempty interior of a subspace of the state space the dimension of which is equal to the maximal Morse index of the equilibria of the equation.     

Closed relativistic strings in geometrically nontrivial spaces

We find exact solutions of the equations of motion for a closed relativistic string that carries a point mass and moves in the space given by the direct product of Minkowski space and a compact manifold (multidimensional torus). We investigate physical characteristics of the system states described by these solutions.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 72–82, January, 2005.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

复Grassmann流形上陈氏示性式的积分公式──谨以此文纪念我们的老师吴光

在本文中给出了复Ｇｒａｓｓｍａｎｎ流形上陈氏示性式Ｃ＿ｋ（Ω）的积分公式如下：     

Great circle links and virtually fibered knots

We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such manifolds are finitely covered by great circle link complements.     

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra_s, the stationary solution is a pitchfork bifurcation point of the system.     

Exact Finite Dimensional Feedback Control via Inertial Manifold Theory with Application to the Chafee–Infante Equation

A class of nonlinear parabolic partial differential equations is considered, and an exact finite dimensional feedback control law is designed in order to force the systems to behave in a prescribed way. The feedback law is obtained via inertial manifold theory by reducing the system to finite dimensions. The control achieved is exact, as opposed to approximate, as obtained in a previous work. The result is applied to the Chafee–Infante equation, a one-dimensional scalar reaction-diffusion equation, with distributed control.     

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.     

基于离散网格的流形曲面构造综述

基于离散网格的流形曲面构造技术不仅能够生成具有高阶光滑性的曲面,并且该曲面可以是任意拓扑结构的.此外,在构造流形曲面时,无需进行额外的拼接操作,克服了传统曲面造型技术在进行面片之间的拼接时,计算量增大以及曲面光滑性难以保证的难题.本文介绍了流形曲面构造的流程以及构造过程中的难点,然后将目前已有的流形曲面构造技术分为三大类:传统意义上的流形构造方法;基于规范区域的流形构造方法;基于样条曲面推广的流形构造方法.并对每一类都进行详细地分类介绍.最后,对其作一个总结以及对未来的展望.     

正交约束优化问题的一阶算法

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用.由于正交约束的可行域是Stiefel流形,一直以来流形上的优化方法是求解这一问题的主要方法.近年来,随着实际应用问题所要求的变量规模的扩大,传统的流形优化方法在计算上的劣势显现出来,而一些迭代简单、收敛快的新算法逐渐被提出.通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法.通过分析这些方法的主要特性,以及应用问题的要求,对这类问题算法设计的研究进行了展望.