Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry

Multicomponent evolution equations associated with linear connections on complex manifolds are considered. It is proved that under some general assumptions an equation from this class is integrable by inverse scattering method if the corresponding linear connection is the Levi-Civita connection of an indefinite Kählerian metric of constant holomorphic sectional curvature. This result is based on a certain characterization of the above-mentioned Levi-Civita connections. It is shown that the obtained integrable equations are generalized ferromagnetics, and recurrent formulas for their local conservation laws are given.     

Focker-Planck equation on a manifold. Effective diffusion and spectrum

The main objectives of the present paper are: (i) to construct a proper generalization of Ornstein-Uhlenbeck process for the case of a smooth Riemannian manifold (also under the action of an external potential field); (ii) to prove a Central Limit Theorem for a certain class of additive functionals, making use of the estimations how fast the process converges to equilibrium; (iii) to investigate spectral properties of the corresponding generator; and (iv) to discuss some geometrical aspects of averaging for this process.     

计算代数方程组流形解的特征值方法

本文研究如何运用特征值方法计算代数方程组的流形解,结果表明利用特征值方法能计算多项式方程组解集的不可约分支的一般点,从而特征值方法是求解代数方程组的整体解法。     

Equidistribution for meromorphic transforms and the dd~c-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the ddc-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kahler manifolds: K-mixing, exponential decay of correlations and central limit theorem.     

一类三维生态动力系统的Hopf分支

考虑一类具偏食习惯的捕食者与被捕食者模型．利用中心流形定理和 Hopf分支理论讨论并证明了该系统在一定条件下产生Hopf分支，得到中心流形、小振幅空间周期解的渐近表达式，同时给出了周期解稳定性判据．     

级联系统的惯性流形

本文主要研究级联系统的解的长时间性态，利用非自治无穷维动力系统发展起来的斜积流和延伸相平面，能量估计等方法。在满足一定的谱间断条件下。运用技巧难度大的三锥相嵌的方法，证明了其惯性流形的存在性。同时揭示出级联系统与非自治系统的长时间性态本质上的关联性。     

Mean Curvature Flow with Dirichlet Boundary Conditions in Riemannian Manifolds with Symmetries

Consider a hypermanifold M ₀ of a Riemannian manifold N whose Riccicurvature is bounded from below. If M ₀ is transversal to a conformalvector field on N, then conditions are given, such that the meancurvature evolution of M ₀ with Dirichlet boundary conditions has asolution for all times.     

强Khler-Finsler流形上(p,q)形式的中值Laplace算子(英文)

本文给出了强Khler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等。与Khler流形上利用逆变基本张量及其在Finsler流形上的变形作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强Khler-Finsler流形上的逆变密切Khler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强Khler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广。     

线性流形上矩阵方程B^TXB=D的反对称解

该文讨论了两类线性流形上矩阵方程B^TXB=D的反对称解和反对称最佳逼近解存在的条件,给出了通解的一般表达式,同时解决了解对给定矩阵的唯一最佳逼近问题.     

The equations determining intermediate integrals for Monge-Ampère PDE

In this note we will find the differential equations determining the intermediate integrals for Monge-Ampère equations in an arbitrary number of variables.