Markov积分算子半群的限制及关于增加积分算子半群的生成

证明了转移函数是l∞的一个子空C^1上的正的压缩C0半群，其极小生成元恰好是Markov积分算子半群的生成元在C^1中的部分；Markov积分算子半群的生成元稠定的充分必要条件是q-矩阵Q一致有界；同时转移函数是Feller—Reuter—Riley的充要条件是Markov积分算子半群的生成元在Cn中的部分产生一个强连续半群．最后，在序Banach空间给出了增加的压缩积分算子半群的生成定理．     

一类推广的q-Bernstein-Schurer-Kantorovich算子的逼近性质

基于q-整数的概念构造一类修正的Stancu型q-Bernstein-Schurer-Kantorovic算子,文中研究该算子的一些逼近性质,验证该算子的收敛性,并且利用光滑模和Lipschitz型极大函数的估计其收敛速度,同时,利用Korovkin型统计逼近定理研究Stancu型Bernstein-Schurer-Kantorovic算子的统计逼近性质.     

Markov积分半群的扰动和逼近

为了研究Markov积分半群的扰动和逼近,根据转移函数与Markov积分半群之间一一对应关系,以及转移函数的扰动和逼近,通过积分的方法,获得了Markov积分半群的广义Phillips扰动定理和Trotter-Kato逼近定理.     

多元推广的Bernstein算子的逼近性质

构造了一类一致收敛于被逼近函数的多元序列,以此序列为基础,运用多元函数的全连续模及部分连续模来刻画这种多元推广的Bernstein算子的逼近性质,不仅得出了理论逼近结果,而且给出了数值逼近的例子.     

Bernstein-Fan插值算子的逼近阶估计及其算法程序

本文研究了具有三角形波基函数的Bernstein-Fan值算子的收敛定理和逼近阶估计，并给出了它的算法程序。     

标准转移函数的多项式一致收敛性

设P=(p ij(t);i,j∈E,t 0)是一个遍历转移函数,π=(π i,,i∈E)是其平稳分布, 何时存在常数v>0和C>0,使得成立? 本文称满足(0.1)的P为多项式一致收敛的.对一类标准转移函数得到了它是多项式一致收敛的充要条件.作为其应用,可以得到, 对生灭q-矩阵, 其最小Q-过程是多项式一致收敛的当且仅当R=∞且S<∞. 同时, 还得到,当R<∞,S<∞时, 其唯一的可配称诚实Q-过程是多项式一致收敛的.     

标准转移函数的多项式一致收敛性

M.F.Chen［3,P.335］提出如下问题：设P=（Pij（t）;i,j∈E,t≥0）是一个遍历转移函数,π=（πi;i∈E）是其平稳分布,何时存在常数v＞0和c＞0,使得成立？本文称满足（0.1）的P为多项式一致收敛的．对一类标准转移函数得到了它是多项式一致收敛的充要条件作为其应用,可以得到,对生灭q-矩阵,其最小Q-过程是多项式一致收敛的当且仅当R＝∞且S＜∞同时,还得到,当R＜∞,S＜∞时,其唯一的可配称诚实Q-过程是多项式一致收敛的。     

C半群的概率逼近问题

借助Pettis积分、随机过程、矩生成函数及算子值数学期望,给出了一般形式的C半群概率逼近指数公式、生成定理及其收敛速度的估计式,也从另一个角度得出C半群概率表示的Vonorovskaya型渐近公式.     

双连续n次积分C余弦函数的逼近定理

基于双连续半群概念,引入一致双连续半群序列概念,借助Laplace变换和Trotter-Kato定理,考察双连续n次积分C余弦函数与C-预解式之间的关系,得到逼近定理的稳定性条件,进而得出双连续n次积分C余弦函数逼近定理.从而对Banach空间强连续半群逼近定理和双连续半群逼近定理进行了推广,为相应抽象的Cauchy问题提供了解决方案.     

带移民和瞬态拯救的Markov分枝过程的存在唯一性与遍历性

考虑一类带有移民和瞬态拯救的Markov分枝过程. 证明了如果拯救速率可和, 则不存在过程. 在拯救速率不可和的情形下, 建立了过程存在性判别准则, 并给出了这一判别准则的一些便于验证的等价条件. 同时, 得到了过程唯一性判别准则; 证明了对给定的q-矩阵Q, 虽然存在无穷多个 Q-过程, 但不中断Q-过程却是唯一的. 给出不中断Q-过程的构造; 证明不中断Q-过程总是遍历的, 给出平稳分布的具体表达式.     

Relative modular convergence of positive linear operators

In this paper we define a new type of modular convergence by using the notion of the relatively uniform convergence. We prove a Korovkin-type approximation theorem via this type of convergence in modular spaces. Then, we construct an example such that our new approximation result works but its classical cases do not work.     

Markov Integrated Semigroups and their Applications to Continuous-Time Markov Chains

A Markov integrated semigroup G(t) is by definition a weaklystar differentiable and increasing contraction integrated semigroup on l _∞. We obtain a generation theorem for such semigroups and find that they are not integrated C ₀-semigroups unless the generators are bounded. To link up with the continuous-time Markov chains (CTMCs), we show that there exists a one-to-one relationship between Markov integrated semigroups and transition functions. This gives a clear probability explanation of G(t): it is just the mean transition time, and allows us to define and to investigate its q-matrix. For a given q-matrix Q, we give a criterion for the minimal Q-function to be a Feller-Reuter-Riley (FRR) transition function, this criterion gives an answer to a long-time question raised by Reuter and Riley (1972). This research was supported by the China Postdoctoral Science Foundation (No.2005038326).     

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.     

Siegmund Duality for Continuous Time Markov Chains on Z_+~d

For the continuous time Markov chain with transition function P(t) on Z_+~d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t).If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

随机单调Markov积分半群

讨论了Markov积分半群的单调性和转移函数的单调性的等价性,并得到最小的Q半群是单调的充要条件.     

Statistically Relatively Uniform Convergence of Positive Linear Operators

In this paper we define a new type of statistical convergence by using the notions of the natural density and the relatively uniform convergence. We study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of statistically relatively uniform convergence of sequences of positive linear operators.     

�лָ������ɷ�֧���̵�Feller����

首先, 当$Q$是一个拟单调的q矩阵的时候, 我们找出最小的$Q$函数是一个Feller的转移函数的准则. 然后我们把这个结论应用于生成分支q矩阵并得到相应的生成分支过程的Feller准则. 特别地, 设$\theta$是分支q矩阵中的非线性数, 总是存在一个分点$\theta_0$满足$1\leq\theta_0\leq2$或$\theta_0<+\infty$使得 生成分支过程是否是Feller的要依据$\theta<\theta_0$或者$\theta>\theta_0$.     

On the convergence of policy iteration for controlled diffusions

The convergence of an approximation scheme known as policy iteration has been demonstrated for controlled diffusions by Fleming, Puterman, and Bismut. In this paper, we show that this approximation scheme is equivalent to the Newton-Kantorovich iteration for solving the optimality equation and exploit this equivalence to obtain a new proof of convergence. Estimates of the rate of convergence of this procedure are also obtained.This research was partially supported by NRC Grant No. A-3609.     

Equi-statistical convergence of positive linear operators

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.     

Fluid approximation for generalized Jackson network with vacations

Using a bounding technique, we prove that the fluid model of generalized Jackson network (GJN) with vacations is the same as a GJN without vacations, which means that vacation mechanism does not affect the dynamic performance of GJN under fluid approximation. Furthermore, in order to present the impact of vacation on the performance of GJN, we show that exponential rate of convergence for fluid approximation only holds for large N, which is different from a GJN without vacations. The results on fluid approximation and convergence rate are embodied by the queue length, workload, and busy time processes.