高阶泛函微分方程解的振动定理

证明了一类高阶非线性泛函微分方程的若干振动准则，推广了文献中的一些振动结果．     

Generalized Solutions of Linear Parabolic Stochastic Partial Differential Equations

Existence and uniqueness theorems for parabolic stochastic partial differential equations with space—time white noise are proved. The method is a combination of the characterization theorem for Hida distributions with the Feynman—Kac and Girsanov formulae. Accepted 3 December 1996     

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.     

On the Multiplicative Complexity of the Inversion and Division of Hamiltonian Quaternions

The multiplicative complexity of a finite set of rational functions is the number of essential multiplications and divisions that are necessary and sufficient to compute these rational functions. We prove that the multiplicative complexity of inversion in the division algebra \H of Hamiltonian quaternions over the reals, that is, the multiplicative complexity of the coordinates of the inverse of a generic element from \H , is exactly eight. Furthermore, we show that the multiplicative complexity of the left and right division of Hamiltonian quaternions is at least eleven. July 17, 2001. Final version received: October 8, 2001.     

Abstract Volterra Integro-Differential Equations: Approximation and Convergence of Resolvent Operator Families

In this article, we analyze approximation and convergence of resolvent operator families associated with various types of abstract Volterra equations and abstract multi-term fractional differential equations in locally convex spaces.     

一阶线性中立型微分方程解的稳定性

本文讨论方程其中c、p_i∈c([t_0,∞),R),0≤c(t)≤1.τ≥0,i=1,2,…,n.通过对方程(1)的非振动解及振动解的渐近性研究,我们得到了方程(1)的平凡解渐近稳定的新的充分条件.     

Banach空间中多类1-集-压缩映象的逼近定理和不动点定理

本文证明了 Ｋｙ Ｆａｎ定理［１］对定义在 Ｂａｎａｃｈ空间中含有内点的有界闭凸子集上的１－集－压缩映象正确．１－集－压缩映象类包含凝聚映象、非扩张映象、半压缩映象、局部几乎非扩张映象和其它映象．作为定理的应用,得到了一些非自映象在许多众知边界条件下的不动点定理,改进和推广了许多作者的最近结果．     

Quick Approximation to Matrices and Applications

m ×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, R?dl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. From our matrix approximation, the above graph algorithms and the Regularity Lemma and several other results follow in a simple way. We generalize our approximations to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems. Received: July 25, 1997     

A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. Received March 6, 1998     

Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.     

Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs

Received: August 22, 1996     

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.     

Generalized Semicontinuity and Existence Theorems for Cone Saddle Points

This paper is concerned with existence theorems for generalized saddle points (cone saddle points) of vector-valued functions. A concept of lower semicontinuity for vector-valued functions is introduced and its properties are investigated. Previous results of the author are extended by using the lower semicontinuity. Accepted 26 April 1996     

General Linear Methods for Stiff Differential Equations

A new type of general linear method is constructed which combines A-stability or L-stability with ease of implementation. The method is structured in such a manner that its stability region is identical with that of a Runge-Kutta method, using a restriction known as inherent RK stability.This revised version was published online in October 2005 with corrections to the Cover Date.     

二维周期系数线性微分方程的特征指数

给出利用系数直接判定二维周期系数线性微分方程特征指数符号的若干结论．     

Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations

This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.     

线性泛函微分方程解的渐近表示

本文利用一类特殊方程的一致稳定性与收敛性定理研究Ｒ^ｎ中线性泛函微分方程解的渐近表示，在较弱的条件下得到了与文［１］同样的渐近积分公式．文中还讨论了一些特殊解的存在性、零解的吸引性等等     

一类二阶非线性泛函微分方程的振动性定理

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换、积分平均技巧和分类讨论的方法建立了两个新的振动准则,改进和推广了已知的一些结果.     

正倒向随机微分方程解的比较定理

讨论了正倒向随机微分方程解的比较问题.阐述了正倒向随机微分方程在随机最优控制、现代金融理论中的广泛而深刻的应用, 对于一类正倒向随机微分方程, 利用Ito公式、停时等随机分析方法,通过构造辅助正倒向随机微分方程,得到了正倒向随机微分方程解的比较定理.     

一类二阶非线性微分方程的区间振动准则

The main desire of this paper is to present several new interval oscillation criteria related to a generalized Riccati technique for certain second-order nonlinear differential equations.Our results extend some known equations.Finally,several examples illustrate the effectiveness of our results.