关于树指标非齐次马氏链的广义熵遍历定理

主要研究了树指标非齐次马氏链的广义熵遍历定理.首先证明了树指标非齐次马氏链上的二元函数延迟平均的强极限定理.然后得到了树指标非齐次马氏链上状态出现延迟频率的强大数定律,以及树指标非齐次马氏链的广义熵遍历定理.作为推论,推广了一些已有结果.同时,证明了局部有限无穷树树指标有限状态随机过程广义熵密度的一致可积性.     

齐次树上三次循环树指标马氏链的强极限定理

首先给出齐次树上三次循环树指标马氏链的定义,利用构造鞅的方法,研究齐次树上三次循环树指标马氏链的强极限定理,并给出其状态及状态序偶发生频率的强大数定律.     

关于树指标马氏链强极限定理的若干研究

树指标随机过程是近年来概率论的研究方向之一,已引起了概率论、物理学、计算机等学科的广泛关注.树指标马氏链是一类重要的树指标随机过程.近年来,作者与其合作者利用研究概率论强极限定理的新方法,在树指标马氏链的强极限定理方面得出一系列研究成果,其中包括树指标马氏链的若干强大数定律和Shannon-McMillan定理.本文系统地介绍了这些工作,特别地,详细介绍了作者在树指标马氏链的强大数定律和Shannon-McMillan方面的工作,并结合一个定理的证明,介绍所用的方法.     

树指标马氏链的等价定义

国内外关于树指标随机过程的研究已经取得了一定的成果.Benjamini和Peres首先给出了树指标马氏链的定义.Berger和叶中行研究了齐次树图上平稳随机场熵率的存在性.杨卫国与刘文研究了树上马氏场的强大数定律与渐近均分性.杨卫国又研究了一般树指标马氏链的强大数定律.为了以后更有效的研究树指标随机过程的一系列相关问题,本文在分析研究前人成果的基础上,给出了树指标马氏链的等价定义,并用数学归纳法证明了其等价性.     

二叉树分枝马氏链的强大数定律和Shannon-McMillan定理

首先研究二叉树有限状态分枝马氏链的随机序偶出现频率的强大数定律,之后研究二叉树有限状态分枝马氏链函数的强大数定律,作为推论得到二叉树有限状态分枝马氏链的ShannonMcMillan定理.     

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本文利用构造鞅的方法, 研究了Cayley树图上奇偶马氏链场的强极限定理, 给出了Cayley树图上奇偶马氏链场关于状态和状态序偶出现频率的强大数定律, 推广了一个已知结果.     

关于非齐次树上m重连续状态马氏链的一个强大数定律

强大数定律一直是国际概率论界研究的中心课题之一.通过构造适当的非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,研究给出了一个关于树指标m重连续状态马氏链的强大数定律.     

关于渐近循环马氏链泛函的强大数定律

研究了一类非齐次马氏链———渐近循环马氏链泛函的强大数定律,首先引出了渐近循环马氏链的概念,然后给出了若干引理.利用了渐近循环马氏链关于状态序偶出现频率的强大数定理给出并证明了关于渐近循环马氏链泛函的强大数定律,所得定理作为推论可得到已有的结果.     

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本文的目的是要研究Cayley树图上奇偶马氏链场的渐近均分割性\bd 首先我们给出Cayley树 图上奇偶马氏链场关于状态和状态序偶出现频率的强大数定律, 然后证明其具有a.e.收敛性 的渐近均分割性\bd     

马氏链集间转移的一类强大数定律

本文研究了马氏链从一个状态子集到另一个状态子集的转移概率的极限性质.利用Doob鞅收敛定理,获得了任意随机序列的强大数定律、马氏链泛函的强大数定律和强遍历定理.推广了马氏链传统转移概率的极限性质和强极限定理.     

Existence and concentration of ground states of coupled nonlinear Schrödinger equations

In this paper we consider the existence and concentration of ground states of coupled nonlinear Schrödinger equations with trap potentials. When the interaction between two states is repulsive, we prove the existence of ground states. Then concentration phenomenon of these ground states is studied as the small perturbed parameter (Planck constant) approaches zero. Roughly speaking, we prove that components of the ground states concentrate at the unique global minimum points of their potentials. Moreover, we prove the existence of ground states when the interaction is attractive.     

Dynamical systems which realize given random bi-sequences of points on their orbits

A dynamical system consists of a phase space of possible states, together with an evolution rule that determines all future states and all past states given a state at any particular moment. In this paper, we show that for any countable random infinite bi-sequences of states of some phase space, there exists an evolution rule in C⁰-topology which realizes precisely the given sequences of states on their orbits and satisfies some regular conditions on the times to realize the states.     

一类多量子比特完全可分态的可分表示及应用

可分量子态能够写成乘积态的凸组合.通过可分量子态的可分表达式可以更好地理解量子态的数学结构.给出了一类多量子比特完全可分量子态的可分分解表示.此外,通过该类可分态的完全可分表示得到了一类阶数为2k的正交矩阵.     

Action as function of period for ground states of semilinear elliptic equations

The ground states of semilinear elliptic equations are considered. We make the conjecture that the set of ground states forms a finite-dimensional manifold. We analyse this conjecture indirectly. More precisely, we investigate a restriction of the associated functional of action on the set of ground states. On this approach some new finite-dimensional properties of the functional of action and of the ground states are deduced. Received November 1998     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

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??In this paper, various concepts of states for Markov chains in random environments are introduced into the research field, and the relationships between these states are discussed. Especially, a partition of the state space is given. Also, the properties of those states are investigated. At last, some examples are given to illustrate the rationality of those states.     

Calculation of Berry's phase in squeezed states

A simple method based on generalized coherent states is proposed for calculation of Berry's phase. In this paper we calculate Berry's phase for a translated oscillator in standard coherent states as well as Berry's phase in squeezed states and spin coherent states, i.e., coherent states for the SU(1, 1) and SU(2) groups, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 51–59, 1988.     

Privileges of First-Person Reference and of Third-Person Reference

It is a widely held view that persons have privileged knowledge about their own minds, although numerous different views on what this privilege exactly consists of exist. One possible way of interpreting it is to claim that persons can refer to their own mental states in a privileged way. I will argue that this view has to be extended. Our common-sense view about reference to mental states implies that besides privileges of first-person reference to one's own mental states, there also exist privileges of third-person reference to the mental states of others: Other persons can refer to all of the mental states of a person in a way that the person cannot. In a next step, I will explain that persons can take two perspectives towards their own mental states: a first-person perspective and a third-person perspective. I will conclude that the possibilities of first-person reference from a third-person perspective are limited.     

一个趋化性模型非常数平衡解的存在性

对带两个趋化性参数的趋化性模型平衡解的存在性问题进行研究.在参数满足特定的条件下,应用局部分岔理论得到非常数平衡解的局部分岔结构,从而证明了该趋化性模型存在无穷多个非常数正平衡解.     

Asymptotics of Information Entropy for the Two-Dimensional Analog of the Relativistic Hydrogen Atom in the Kozlov-Nikishin Model

For the two-dimensional Lorentz-invariant model of the hydrogen atom, we obtain wave functions of bound states in coordinate representation and, for nonexcited (in time) states, also in momentum representation. For such states, the short-wave asymptotics of the information entropy is studied.