Hilbert Space of Probability Density Functions Based on Aitchison Geometry

The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.     

关于微分熵的一类强偏差定理

利用关于乘积分布密度的相对熵和相对熵率的概念,建立了相依连续型随机变量序列关于参考微分熵的一类强偏差定理,证明中给出了将Laplace变换应用于微分熵强偏差定理的研究的一种途径．     

一致Lipschitz映射的遍历定理及其遍历分解

设(X,d)是一个可分的度量空间,Cu(X,d)是由全体一致连续函数所组成的C(X,d)的子空间,T是定义在X上的一致Lipschitz映射,那么对f∈Cu(X),1/n n∑k=1 Uk If在Cu(X)上收敛.从这个基本结果出发,利用Cu(X,d)的共扼空间的表示定理,得到了相空间的Yosida型遍历分解;利用空间的嵌入技术证明了非一致Lipschitz映射的大数法则.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.     

Existence of two solutions of m-point boundary value problem for second order dynamic equations on time scales

In this paper, by means of a new twin fixed-point theorem in a cone, the existence of at least two positive solutions of m-point boundary value problem for second order dynamic equations on time scales is considered.     

Limit theorems for partially hyperbolic systems

We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms. If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory.

     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

The Chaotic Mixmaster and the Suppression of Chaos in Scalar–Tensor Cosmologies

We show that there is a threshold in energy for the onset of chaos in cosmology for the Universe described as a dynamical system derived from the Einstein equations of General Relativity (GR). In the case of the mixmaster model (homogeneous and anisotropic cosmology with a Bianchi IX metric), the chaos occurs precisely at the prescribed necessary value H _vac=0 of the GR for the energy of the Universe while the system is found to be regular for H<0 and chaotic for H>0 with respect to its pure vacuum part. In the case of generalized scalar tensor theories within the Bianchi IX model, we show using the ADM formalism and a conformal transformation that the energy of the dynamical system as compared to vacuum lies below the zero energy threshold. The system is thus not exhibiting chaos and the conclusion still holds in the presence of ordinary matter as well. The suppression of chaos occurs in a similar way for stiff matter alone.     

关于任意随机变量序列泛函的强极限定理

本文在k是固定的正整数，{fn}是R^k 1上的Borel可测函数列时，得到了任意随机变量序列{Xrn≥0}的泛函{fn(Xn-k,…,Xn)}的强极限定理，它是Chung的关于独立随机变量序列的强大数律的推广，作为推论，得到了k重非齐次马尔科夫链的一类强极限定理．     

具有无限时滞的二阶微分方程解的存在性

§ 1.　IntroductionRecently ,thedifferentialequationswithdeviatingargumentswereusuallydiscussed(see[1 ],[4],[5 ]) .In [1 ],AGARWALRPandO’REGANDconsideredequationy″(t) =f(t,y(t) ,y(σ(t) ) ) ,　a.e .t∈ [0 ,1 ]y(t) =ψ(t) ,　　　　　　　　t∈ [-r ,0 ]y( 1 ) =a ,( )andtheydiscussedtheexistenceofatleastonesolutionforequation ( ) .Inthispaper ,weconsideramoregeneralequation-x″(t) =f(t ,xt) ,　t∈ [0 ,1 ]x(t) =ψ(t) ,　　　　t∈ ( -∞ ,0 ]x( 0 ) =x( 1 ) =0 ,( 1 .1 )andsomeexistencetheor…