A Brief Introduction on Professor Zuxiu Zheng’s Achievements in Mathematical Field

As a student of Professor Zuxiu Zheng, I am greatly honored, in the occasion of his 80th birth- day, to give a brief introduction of what I know about Professor Zheng and his long lasting contribution to mathematical education and research in China over the last few decades.     

BIFURCATION ANALYSIS IN A GENERALIZED FRICTION MODEL WITH TIME-DELAYED FEEDBACK

In this paper, we consider a generalized model of the two friction models, both of which have two different types of control forces with time-delayed feedback proposed by Ashesh Sara et al. By taking the time delay as the bifurcation parameter, we discuss the local stability of the Hopf bifurcations. Under some condition, the generalized model harbors a phenomenon that the equilibrium may undergo finite switches from stability to instability to stability and finally become unstable. By applying the method introduced by Faria and Magalhaes, we compute the normal form on the center manifold to determine the direction and stability of the Hopf bifurcations. Numerical simulations are carried out and more than one periodic solutions may exist according to the bifurcation diagram given by BIFTOOL. Finally a brief conclusion is presented.     

学会观察联想

同学们知道,解答数学问题离不开观察联想.如何根据问题的具体特征,恰当地进行观察联想,常常成为能否较快解决问题的关键.本文拟结合同学们解决几个圆锥曲线问题的思维历程,谈谈如何在解题中有效地实施观察联想.以资同学们参考.     

（3＋1）维破碎孤子方程的变量分离解和局域激发模式

多线性分离变量法已成功地应用于诸多（2＋1）维非线性可积系统.将该方法拓展运用于（3＋1）维破碎孤子方程中,获得了含任意函数的变量分离解.通过适当地设定任意函数的形式,得到了（3＋1）维破碎孤子方程丰富的局域激发模式.     

一类具非线性边界条件的高阶方程的双参数奇摄动问题

研究了一类含双参数的非线性高阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用微分不等式理论证明了原问题解的存在性及所得形式渐近解的一致有效性.     

GSOR方法最优迭代参数的简化推导

对于增广线性系统,Bai等研究了广义SOR方法（Bai Z Z,Parlett B,Wang Z Q.On generaliged successive overrelaxation methods for augmented linear systems.Numerische Mathematik,2005,102（1）：1-38）,并得到其最优迭代参数.给出了另外一种推导最优迭代参数的简化方法,这种方法对于求解其他参数加速定常迭代方法的最优迭代参数非常有意义.     

具有随机资金流的一类效用优化问题研究

现实的金融市场上,当有重大信息出现时,会对股价产生冲击,使得股价产生跳跃,同时投资过程会有随机资金流的介入,考虑股价出现跳跃与随机资金流介入的投资组合优化问题,通过构造倒向-前向随机微分方程并结合随机最优控制理论研究了一般效用函数下的投资组合选择问题,获得最优投资组合策略,然后针对二次效用函数,给出显式表示的最优投资组合策略.     

一道美国大学生数学竞赛题的证明

文献[1,2]中对一道美国大学生数学竞赛题通过设定函数利用插值的方法进行了证明,本文根据问题的特点,给出另外的证明,用同一方法还对另一行列式问题给出一般求解公式.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.     

微分中值定理的应用及推广

介绍应用微分中值定理时,构造所需辅助函数的两种有效方法：观察法和解微分方程法。并通过变量代换法化无限为有限,将罗尔定理的应用推广到无限区间上。