A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.     

A global rigidity theorem for weakly Landsberg manifolds

In this paper we study a global rigidity property for weakly Landsberg manifolds and prove that a closed weakly Landsberg manifold with the negative flag curvature must be Riemannian.     

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：     

流形元法固定边界约束处理研究

在数值流形方法中,对于材料的固定边界,一般采用罚函数的方法进行处理,即在固定边界上设置刚性弹簧约束其位移来实现固定约束条件的近似满足。罚函数法在理论上不是严格的固定约束处理方法,罚弹簧的布置与弹簧刚度的大小对模拟的效果都会产生影响。基于流形单元上位移函数的组成提出了流形方法固定边界约束处理的新方法,在组成流形单元的物理覆盖上,通过取消相应的覆盖函数在流形单元位移函数中的组成来实现双向固定的约束条件,通过使用只包含单方向位移的覆盖函数使x向固定约束条件和y向固定约束条件得到实现,推导了相应固定约束条件下的流形单元刚度矩阵的数值计算格式。该方法严格满足固定约束的物理意义,简化了固定边界的处理,并经算例证明是有效和准确的,有利于数值流形方法的程序实现和工程应用。     

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.