泛函微分方程的弱指数渐近稳定性

本文提出一种新的稳定性概念,即弱指数渐近稳定,并给出两个关于弱指数渐近稳定的判别定理和一个较广泛的指数渐近稳定判别结果.从而使得许多具有一致渐近稳定性的解的趋零速度,得到了一种估计.文中还深入地揭露了一致渐近稳定性和弱指数渐近稳定性之间的内在联系以及弱指数渐近稳定性和指数渐近稳定性的关系.     

强混合样本下最近邻密度估计的渐近正态性

研究了α-混合样本下最近邻密度估计的渐近性质,证明了估计的渐近正态性并且给出了其渐近方差的显式表达式,由此构造了α-混合样本下概率密度的渐近置信区间.     

中立型随机泛函微分方程的稳定性

该文研究了一般中立型随机微分方程解的渐近性质,利用Lyapunov函数和上鞅收敛定理,得到 了该方程解的一些渐近稳定性、多项式渐近稳定性及指数稳定性等渐近性质,其结果涵盖并 推广了已有文献的结论。     

一类小周期椭圆方程的三重尺度渐近分析

利用三重尺度方法对一类小周期椭圆方程进行了三重尺度渐近展开分析,构造了对应的三重尺度形式渐近展开式,得到了均匀化常数和均匀化方程.在形式渐近展开的基础上,构造了对应边值问题解的三重尺度渐近近似解,并分析了对应三重尺度形式渐近误差估计.     

线性模型误差方差的经验似然估计

本文应用经验似然方法得到了线性模型误差方差的一类新的估计,证明了估计的渐近分布为正态分布且渐近方差不超过常用的误差方差估计的渐近方差,同时给出了渐近方差的显式表达.     

Marachkov-Barbashin-Krasovskii型渐近稳定性定理

本文研究非自治系统的渐近稳定性.且得到了不要求Liapunov函数正定,也不要求其沿系统的解的导数负定的渐近稳定性定理.一致渐近稳定性定理及全局渐近稳定性定理.     

无穷维随机微分方程的渐近概周期解

该文引入了渐近θ-概周期随机过程的概念,并在算子半群理论框架下研究了一类带有渐近概周期系数的无穷维随机微分方程,利用随机分析理论建立了此类随机微分方程渐近θ-概周期解的存在性.此外该文还引入了依路径分布渐近概周期过程的概念,并证明了上述渐近θ-概周期解还是依路径分布渐近概周期的.值得注意的是,在早期的研究结果中,建立的均是更弱的一维分布渐近概周期解的存在性.     

Hilbert空间中一类新的渐近几乎非扩张曲线的渐近行为与遍历性

引入了Ｈｉｌｂｅｒｔ空间中一类新的渐近几乎非扩张曲线，并讨论了这类新的渐近几乎非扩张曲线的渐近行为与遍历性，进一步，作为应用，得到了Ｈｉｌｂｅｒｔ空间中非Ｌｉｐｓｃｈｉｔｚ映象的非线性算子族的渐近行为与遍历性方面的结果。     

微分方程指数形式渐近解

本文对常微分方程的指数形式渐近解作了进一步讨论,首先给出了二阶方程一致有效指数形式渐近解的正交条件;接着讨论了匹配渐近法中的指数形式渐近解;最后举例说明。     

一类非线性奇摄动方程的匹配解法和解的精度估计

利用匹配渐近展开法,研究了一类非线性奇异摄动方程.在适当的条件下,得出了该类问题解的渐近展开式.并将结果应用于例子,对渐近解与精确解和用两变量方法求得的解进行比较,可知所得到的渐近解达到了较高精度.     

关于4阶极小渐近基的一个结果

设h≥2,若h阶渐近基A的任一真子集均不是h阶渐近基,则称集合A是自然数集N的h阶极小渐近基.为进一步刻画渐近基与极小渐近基之间的关系,本文综合运用自然数的b进制表示理论及分类讨论的方法,证明了存在一个集合是4阶渐近基且其任何子集均不是4阶极小渐近基.     

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

Asymptotic equivalence of probability distributions with applications to some problems of asymptotic independence

Summary The problem of asymptotic approximation is formulated in a generalized form, and the basic concepts of asymptotic equivalence of probability distributions, which are fundamental to the study of asymptotic approximation problems, are established precisely. For the purpose of practical applications, some useful criterions for the asymptotic equivalence are given.     

Power and exponential asymptotic forms of correlation functions

Using the Ornstein.Zernike equation, we obtain two asymptotic equations, one describing the exponential asymptotic behavior and the other describing the power asymptotic behavior of the total correlation function h(r). We show that the exponential asymptotic form is applicable only on a bounded distance interval l < r < L. The power asymptotic form is always applicable for r > L and reproduces the form of the interaction potential. In this case, as the density of a rarified gas decreases, L → l, the exponential asymptotic form vanishes, and only the power asymptotic form remains. Conversely, as the critical point is approached, L → ℞, and the applicability domain of the exponential asymptotic form increases without bound. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 454–464, September, 2008.     

渐近正态随机变量函数的极限分布

基于渐近正态随机变量,导出随机变量函数极限分布的两个一般性理论结果.作为应用,证明了渐近正态随机变量一系列具体函数的极限分布,其中包括泊松随机变量平方根的渐近正态性,以及随机变量部分和在正则化常数是随机变量情况下的渐近正态性.     

Estimating function with asymptotic bias and its estimator

We introduce the estimating function with asymptotic bias and investigate the asymptotic behavior of the estimator based on it by using their relationship. The estimator based on the estimating function with asymptotic bias has the asymptotic normality with asymptotic bias. We show that this theory has several interesting applications in practical statistics.     

基于一类时滞不等式的时滞微分方程的渐近稳定性

利用Lyapunov的方法讨论了时滞微分方程x.(t)=f(t,x(t),x(t-τ(t)))的全局指数渐近稳定性和全局渐近稳定性.     

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.