关于随机选择的一个强极限定理

本文利用似然比和区间剖分法研究相依Ｎ值随机变量序列选择的极限性质,得到关于赌博方法的一个强极限定理的若干推广。     

Relativization of dynamical properties

In the past twenty years,great achievements have been made by many researchers in the studies of chaotic behavior and local entropy theory of dynamical systems.Most of the results have been generalized to the relative case in the sense of a given factor map.In this survey we offer an overview of these developments.     

Topologically Mixing and Hypercyclicity of Tuples

In this paper,we characterize conditions under which a tuple of bounded linear operators is topologically mixing.Also,we give conditions for a tuple to be hereditarily hypercyclic with respect to a tuple of syndetic sequences.     

Entropy points and applications

First notions of entropy point and uniform entropy point are introduced using Bowen's definition of topological entropy. Some basic properties of the notions are discussed. As an application it is shown that for any topological dynamical system there is a countable closed subset whose Bowen entropy is equal to the entropy of the original system.

Then notions of C-entropy point are introduced along the line of entropy tuple both in topological and measure-theoretical settings. It is shown that each C-entropy point is an entropy point, and the set of C-entropy points is the union of sets of C-entropy points for all invariant measures.

     

A posteriori joint detection of a recurring tuple of reference fragments in a quasi-periodic sequence

The problem of joint detection of a recurring tuple of reference fragments in a noisy numerical quasi-periodic sequence is solved in the framework of the a posteriori (off-line) approach. It is assumed that (i) the total number of fragments in the sequence is known, (ii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iii) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is shown that the problem consists of testing a set of simple hypotheses about the mean of a random Gaussian vector. A specific feature of the problem is that the cardinality of the set grows exponentially as the vector dimension (i.e., the length of the observed sequence) and the number of fragments in the sequence increase. It is established that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that maximize a special auxiliary objective function with linear inequality constraints. It is shown that this function is maximized by solving the basic extremum problem. It is proved that this problem is solvable in polynomial time. An exact algorithm for its solution is substantiated that underlies an algorithm guaranteeing optimal (maximum-likelihood) detection of a recurring tuple of reference fragments. The results of numerical simulation demonstrate the noise stability of the detection algorithm.     

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tuple TΦn is (-1)n time wrapping number of Φn around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown that C*(Ω)C*(Bn) for every Poincare domain Ω in Cn(n≠2). It is also noticed that C*(Ω)C*(B2) if and only if the Poincaré conjecture is true for Ω.     

Local entropy theory for a countable discrete amenable group action

The local properties of entropy for a countable discrete amenable group action are studied. For such an action, a local variational principle for a given finite open cover is established, from which the variational relation between the topological and measure-theoretic entropy tuples is deduced. While doing this it is shown that two kinds of measure-theoretic entropy for finite Borel covers coincide. Moreover, two special classes of such an action: systems with uniformly positive entropy and completely positive entropy are investigated.     

投影算子组的联合谱

本文研究Hilbert空间H上投影算子组的联合谱.首先通过计算给出正则投影对的联合谱,进而给出一般的投影算子对的联合谱.本文还对两个投影算子的和与差的可逆性给出一些等价刻画.特别地,当P和Q为正则投影对时,本文通过计算算子组[I, P, Q]的联合谱来给出σ(P+Q)和σ(P-Q)的具体刻画.反过来,本文证明两类具有特定形式的复数集分别是Hilbert空间上正则投影对的和与差的谱.本文也给出一般的投影算子对的和与差的谱.最后,本文计算特定条件下的3个投影算子组[P, Q, R]的联合谱.     

Taylor spectral invariance for crisscross commuting pairs on Banach spaces

A theorem on the commuting property of Taylor's spectrum for crisscross commuting pairs is proved in this paper.