Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

Gaussian measure of sections of convex bodies

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.     

On the existence of multiresolution analysis for framelets

We show that a compactly supported tight framelet comes from an MRA if the intersection of all dyadic dilations of the space of negative dilates, which is defined as the shift-invariant space generated by the negative scales of a framelet, is trivial. We also construct examples of (non-tight) framelets, which are arbitrarily close to tight frame framelets, such that the corresponding space of negative dilates is equal to the entire space L ²ℝ.The first author was partially supported by the NSF grant DMS–0441817     

单个生成元Walsh p-进制平移不变空间伸缩的交与并

p-进制MRA与GMRA是构造L~2(R_+)中小波框架的重要工具. L~2(R+)中嵌套子空间序列交集为{0},并集为L~2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L~2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L~2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p~j supp(■φ)=R+,其中■为定义在L~2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L~2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".     

Quasi-interpolation by thin-plate splines on a square

Quasi-interpolation is one method of generating approximations from a space of translates of dilates of a single function ψ. This method has been applied widely to approximation by radial basis functions. However, such analysis has most often been performed in the setting of an infinite uniform grid of centers. In this paper we develop general error bounds for approximation by quasiinterpolation on ann-cube. The quasi-interpolant analyzed involves a finite number, growing ash ^?n , of translates of dilates of the function ψ, and a bounded number of edge functions. The centers of the translates of dilates of ψ form a uniformly spaced grid within the cube. These error bounds are then applied to approximation by thin-plate splines on a square. The result is an O(ω(f, [-1,1]²,h)) error bound for approximation by thin-plate splines supplemented with eight arctan functions.     

A Weyl type formula for Fourier spectra and frames

We prove qualitative and quantitative results concerning the asymptotic density in dilates of centered convex bodies of the frequency vectors of orthogonal exponential bases and frames associated to bounded domains in Euclidean space.

     

The Canonical Dual Frame of a Wavelet Frame

In this paper we show that there exist wavelet frames that have nice dual wavelet frames, but for which the canonical dual frame does not consist of wavelets, i.e., cannot be generated by the translates and dilates of a single function.     

An elliptic equation with perturbed p-Laplace operator

We consider an elliptic boundary value problem with perturbed p-Laplacian, p>1. The perturbation consists of a linear operator that dilates or compresses the argument of a function and is close to the identity. A weak existence theorem is obtained.     

Lipschitz Linearization of the Maximal Hyperbolic Cross Multiplier

Journal of Fourier Analysis and Applications - We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition...     

Direct and inverse problems in additive number theory and in non-abelian group theory

We obtain new direct and inverse results for Minkowski sums of dilates and we apply them to solve certain direct and inverse problems in Baumslag–Solitar groups, assuming appropriate small doubling properties.     

On Existence of the Even Lp Gaussian Minkowski Problem for p > n∗

This paper concerns the even Lp Gaussian Minkowski problem in n-dimensional Euclidean space R n . The existence of the solution to the even Lp Guassian Minkowski problem for p > n is obtained.     

Ehrhart polynomials of cyclic polytopes

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In (Coefficients and roots of Ehrhart polynomials, preprint), the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d=2. In our article, we prove the conjecture for any dimension.     

Wavelet Decompositions of Nonrefinable Shift Invariant Spaces

The motivation for this work is a recently constructed family of generators of shift invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of dilates of the shift invariant space that are well suited for applications.     

Sums of Dilates in Ordered Groups

We address the ‘sums of dilates’ problem by looking for non-trivial lower bounds on sumsets of the form k·X + l·X, where k and l are non-zero integers and X is a subset of a possibly non-abelian group G (written additively). In particular, we investigate the extension of some results so far known only for the integers to the context of torsion-free or linearly orderable groups, either abelian or not.     

光子结构的猜想及其数学仿真

通过研究光的波粒二象性,对光子结构进行了研究并提出了自建光子模型.自建光子模型具有以下特性:①光子与原子核存在引力作用;②光子的电场与磁场交替变换使其具备了相位效应,且光强由光子数与光子相位和两个因素决定.在此基础上,通过借鉴天体物理中关于二体运动的三种轨迹,利用其中结论对大量原子核的引力作用进行概率分析,使用中心极限定理提出出射角度基本符合高斯分布的假设.接着对光子的相位叠加效应进行了分析.最后,对大量光子进行模拟仿真,并将仿真结果与单缝衍射、双缝干涉、多缝干涉现象等实验现象进行了对比.     

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

On the value of Gaussian sums

Gauss' original proof for the value of Gaussian sums relies on a summation of Gaussian polynomials. It is demonstrated how his proof can be simplified by using a different summation of Gaussian polynomials.     

Compactly Supported Frames for Decomposition Spaces

In this article we study a construction of compactly supported frame expansions for decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces. This is done by showing that finite linear combinations of shifts and dilates of a single function with sufficient decay in both direct and frequency space can constitute a frame for Triebel-Lizorkin type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to Triebel-Lizorkin type spaces and the associated modulation spaces. Next, we prove that two function systems which are sufficiently close have an almost diagonal “change of frame coefficient” matrix. Finally, we approximate to an arbitrary degree an already known frame for Triebel-Lizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both direct and frequency space.     

Singularity among selfsimilar Gaussian random fields with different scaling parameters and others

Abstract

It is shown in this paper that the probability measures generated by selfsimilar Gaussian random fields are mutually singular, whenever they have different scaling parameters. So are those generated from a selfsimilar Gaussian random field and a stationary Gaussian random field. Certain conditions are also given for the singularity of the probability measures generated from two Gaussian random fields whose covariance functions are Schoenberg–Lévy kernels, and for those from stationary Gaussian random fields with spectral densities.