微分周形式与稀疏微分结式

代数周(Chow)形式和代数结式是代数几何的基本概念,同时还是消去理论的强大工具.一个自然的想法是在微分代数几何中发展相应的周形式和结式理论.但是由于微分结构的复杂性,在本文的研究工作之前,微分结式只有部分结果,而微分周形式与稀疏微分结式理论一直没有得到发展.本文的主要结果包括:第一,发展一般(generic)情形的微分相交理论,作为应用,证明一般情形的微分维数猜想.第二,初步建立微分周形式理论.对不可约微分代数簇定义微分周形式并证明其基本性质,特别地,给出微分周形式的Poisson分解公式,引入微分代数簇的主微分次数这一不变量并证明一类微分代数闭链的周簇和周坐标的存在性.作为应用,首次严格定义微分结式,证明其基本性质.第三,初步建立稀疏微分结式理论.引入Laurent微分本性系统的概念,定义稀疏微分结式,证明其基本性质,特别地,引入微分环面簇的概念,给出稀疏微分结式阶数和次数界的估计,并基于此给出计算稀疏微分结式的单指数时间算法.     

微分特征列法在拟微分算子和Lax表示中的应用

微分特征列法用于拟微分算子和非线性发展方程Lax表示的计算.首先,利用微分特征列法和微分带余除法计算拟微分算子的逆和方根,由于不必求解常微分方程组,并将解代入,因此,使得计算得以简化.其次,利用微分特征列法,约化从广义Lax方程和Zakharov-Shabat推出的非线性偏微分方程,并得到相应的非线性发展方程.在Mathematica计算机代数系统上,编写了相关程序,从而可以利用计算机辅助完成一些非线性发展方程Lax表示的计算.     

The Loewner driving function of trajectory arcs of quadratic differentials

We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of a certain type of quadratic differential. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper half-plane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

带随机跳跃的线性二次非零和微分对策问题

对于一类以布朗运动和泊松过程为噪声源的正倒向随机微分方程,在单调性假设下,给出了解的存在性和唯一性的结果．然后将这些结果应用于带随机跳跃的线性二次非零和微分对策问题之中,由上述正倒向随机微分方程的解得到了开环Nash均衡点的显式形式．     

无限滞后测度泛函微分方程的平均化

本文研究了无限滞后测度泛函微分方程的平均化.利用广义常微分方程的平均化方法,在无限滞后测度泛函微分方程可以转化为广义常微分方程的基础上,获得了这类方程的周期和非周期平均化定理,推广了一些相关的结果.     

变系数线性齐次常微分方程组的λ-矩阵求解法

进一步讨论了微分变换矩阵的性质 ,指出了变系数线性齐次微分方程组 ,通过因变量变换化为常系数线性齐次方程组的充要条件     

有界超格上的微分

在有界超格上引入微分,研究了有界超格上微分的一些性质.定义并研究了微分超格的微分超理想和微分超同余,并证明了如果$(L,d)$是一个有界强单微分超格并且$R$是$(L,d)$的一个强微分同余,则$(L/R,g)$仍是一个强单微分超格,其中$g$是由$d$诱导的商超格上的单强微分.     

Localized triangular differential quadrature

A localized triangular differential quadrature method is introduced in this article. Not only is the existing limitation on the approximation order in the triangular differential quadrature eliminated but also the convergent rate is enhanced in the new method. As an example to validate the new method, elastic torsion of prismatic shaft with regular polygonal cross section is studied and excellent agreement with available theoretical and analytic solutions is reached. It is believed that the present work further widens the applicability of the triangular differential quadrature technique. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 682–692, 2003     

Charpit System of Partial Differential Equations with Algebraic Constraints

In this paper the Charpit system of partial differential equations with algebraic constraints is considered. So, first the compatibility conditions of a system of algebraic equations and also of the Charpit system of partial differential equations are separately considered. For the combined system of equations of both types sufficient conditions for the existence of a solution are found. They lead to an algorithm for reducing the combined system to a Charpit system of partial differential equations of dimension less than the initial system and without algebraic constraints. Moreover, it is proved that this system identically satisfies the compatibility conditions if so does the initial system.     

Tight estimates for general averaging applied to almost-periodic differential equations

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

一类常微分方程解的存在唯一性及其应用

提出并证明了一类常微分方程解的存在唯一性成立的一个充要条件,并给出了多项式形式增长函数的一列上界.最终将此结果应用到证明一类倒向随机微分方程的唯一解问题.     

A Type of General Forward-Backward Stochastic Differential Equations and Applications

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with It?o’s stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations. The existence and uniqueness results of the general FBSDEs are obtained. In the framework of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.     

FORCED OSCILLATION AND APPLICATION OF FUNCTIONAL DIFFERENTIAL INEQUALITIES

In this paper we study the oscillations for a class of functional differential inequalities. By using these properties some forced oscillations to the boundary value problems of functional partial differential equations are established.     

A new bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation

The complexity of computing the Galois group of a linear differential equation is of general interest. In a recent work, Feng gave the first degree bound on Hrushovski’s algorithm for computing the Galois group of a linear differential equation. This bound is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group (see Definition 2.7) and is sextuply exponential in the order of the differential equation. In this paper, we use Szántó’s algorithm of triangular representation for algebraic sets to analyze the complexity of computing the Galois group of a linear differential equation and we give a new bound which is triple exponential in the order of the given differential equation.     

Differential algebraic equations with after-effect

In this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.     

线性齐次常微分方程(组)的λ-矩阵求解法

本文在文 [2 ]的基础上 ,应用λ-矩阵及微分算子性质给出了一种变系数齐次常微分线性方程 (组 )的λ-矩阵求解法 ,对文 [2 ]作出了更一般的推广 .     

加权微分建模的初步研究

加权建模是必要的,微分建模是重要的,把二者结合起来,进行加权微分建模既必要也重要.给出了常用模型的微分建模结果,讨论了加权建模中的计算和权重选择问题,探讨了加权微分建模的思路和方法,并结合典型数据验证了该方法的有效性和稳定性.象加权建模一样,加权微分建模的精度、实用价值等,是和权重确定得合理与否紧密相联;应先进行模拟,以与近期实际值或典型样本相差最小的参数所对应的模型为准.     

关于拟线性混合型边界问题的概率表示

关于某些抛物型和椭圆型偏微分方程的混合边界问题的解被表示为一类联系于Ito正向反射边界随机微分方程的反向随机微分方程的解.