高斯随机变量序列的收敛性和抽象Wiener空间

研究Ｇａｕｓｓ随机变量序列的０－１律与相关的抽象Ｗｉｅｎｅｒ空间．主要结果如下：设｛Ｆｎ（ｗ）｝是概率空间（Ω，，Ｐ）上的Ｇａｕｓｓ随机变量序列．若对任何实数是高斯随机变量，则有收敛或１。     

非紧集上的变分原理

对紧度量空间(X，d)，T：X→X是连续映射，μ是遍历不变测度，我们考虑集合K，它是使得-logμ（Bn（x，ε））/n关于n以及ε的极限等于测度熵hμ(T)的那些X中的点所构成的集合．我们证明了变分原理：测度熵hμ(T)等于测度为1的集合的拓扑熵的下确界，事实上我们证到了测度熵hμ(T)就等于集合K的拓扑熵．     

城市供水管网连通性分析研究

结合图论中的遍历理论,给出了供水管网连通性分析和实现方法,并应用到管网模型拓扑结构检查中去,通过实例证明,该方法大大提高了建立准确的管网拓扑结构的工作效率.     

可列非齐次马氏链的绝对平均强遍历性质

研究了非齐次马氏链的绝对平均强遍历性,由于绝对平均强遍历性在马氏决策过程和信息论中有十分广泛的应用,许多学者在这方面做了大量的研究.得到了非齐次马氏链满足这种强遍历的两个充分条件.     

单瞬时态单边生灭Q过程的常返性和遍历性

设Q为单瞬时态单边生灭矩阵，该文利用单边生灭Q－矩阵的定性理论和单时态Q过程的分解定理得到了单边生灭Q－矩阵存在常返Q过程和遍历Q过程的充要条件，并在Q－矩阵满足定理条件时构造了单边生灭常返Q过程和遍历Q过程。     

混沌性或噪声信号引起的确定性响应

利用无效介质构成一个具有波的遍历特性的系统,初步探讨了随机信号的激励行为,研究表明,带有某种随机性的输入信号,同样能产生确定性的脉冲输出。并讨论了时间逆转效应和随机共振现象之间可能存在的关联。     

Eigenvalues, inequalities and ergodic theory

This paper surveys the main results obtained during the period 1992—1999 on three aspects mentioned in the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e. the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (§ 1). Here, a probabilistic method — coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger’s method which comes from Riemannian geometry. This consists of § 2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented (§ 3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.     

一类极小系统的动力性状

引进了真弱几乎周期点极小系统这一新概念,即一个含有真的弱几乎周期点而不存在真子系统具有此性质的系统,并证明了该系统是拓扑传递,具有满测度中心.结果表明:该系统是Takens-Ruelle混沌的,强遍历和完全遍历敏感的.     

SIMPLE COMPUTING OF THE CUSTOMER LIFETIME VALUE:A FIXED LOCAL-OPTIMAL POLICY APPROACH

In this paper,we present a new method for finding a fixed local-optimal policy for computing the customer lifetime value.The method is developed for a class of ergodic controllable finite Markov chains.We propose an approach based on a non-converging state-value function that fluctuates（increases and decreases） between states of the dynamic process.We prove that it is possible to represent that function in a recursive format using a one-step-ahead fixed-optimal policy.Then,we provide an analytical formula for the numerical realization of the fixed local-optimal strategy.We also present a second approach based on linear programming,to solve the same problem,that implement the c-variable method for making the problem computationally tractable.At the end,we show that these two approaches are related：after a finite number of iterations our proposed approach converges to same result as the linear programming method.We also present a non-traditional approach for ergodicity verification.The validity of the proposed methods is successfully demonstrated theoretically and,by simulated credit-card marketing experiments computing the customer lifetime value for both an optimization and a game theory approach.     

Banach空间中的非Lipschitzian一般半群的非线性遍历定理（英文）

设G为半群,C为具FrEchet可微范数的一致凸Banach空间X的非空有界闭凸子集.（■）={T_t：t∈G}为C上到自身的渐近非扩张型半群,且F（■）非空.在本文中,我们证明了：对■的任一殆轨道u（·）,■co{u（ts）,t∈G}∩F（S）至多为单点集.进一步,对x∈C,∩_（s∈G）co{T_（ts）x,t∈G}∩F（■）非空当且仅当存在C到F（■）上非扩张压缩P,使得对任意t∈G,PT_t=T_tP=P,Px∈co{T_tx,t∈G}.这一结果不仅推广了许多已知结果,而且说明它们中的一些关键条件是不必要的.