关于算术、对数、指数、几何平均不等式的概率证法

本文用概率方法证明了关于两个相异的正数的几何平均、对数平均、指数平均和算术平均的不等式关系 .     

Convexity and concavity of eigenvalue sums

It is well known that(H), the sum of the negative eigenvalues of a Hermitian matrixH, is a concave and increasing function ofH. In contrast to this, we prove that forA nonsingular Hermitian andP positive definite, the functionP(AP)=(P ^1/2 AP ^1/2) is convex and decreasing. Several other results of this nature are also proved.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

A variational approach to nonlinear and interacting diffusions

Abstract

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including nonhomogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter is also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.     

Eponential Decay To Stable States In Phase Transitions Via A Double Log–Transformation

Motivated by various Hamilton–Jacobi–Bellman equations arising in deteministic optimal control we will modify the concept of viscosity solution introduced by Crandall and Lions for convex (or concave) hamiltonians and semicontinuous solutions. We will see that we can dispense with the Crandall–Lions requirement that we touch the solution by test functions from both above and below and require only touching from one side, Which side depends on whether the solution is upper or lower semicontinuous and the hamiltonian is concave. The advantage of testing from only one side is that Semicontinuous solutions can only be touched from one side. It is shown that this is sufficient to characterize the solution.     

Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e ^− V in ℝ ⁿ are established. It is proved by using Prékopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures with p > 1 in ℝ ⁿ , and give out explicit estimates for their constants. This work is supported by NSFC (No. 10721091), 973-Project (No.2006CB805901) and DFMEC (NO. 20070027007).     

Quantum Information Scrambling in Non-Markovian Open Quantum System

In this paper, we investigate the dynamics of a spin chain whose two end spins interact with two independent non-Markovian baths by using the non-Markovian quantum state diffusion (QSD) equation approach. Specifically, two issues about information scrambling in an open quantum system are addressed. The first issue is that tripartite mutual information (TMI) can quantify information scrambling properly via its negative value in a closed system, whether it is still suitable to indicate information scrambling in an open quantum system. We find that negative TMI is not a suitable quantifier of information scrambling in an open quantum system in some cases, while negative tripartite logarithmic negativity (TLN) is an appropriate one. The second one is that up to now almost all information scrambling in open quantum systems reported were focus on a Markovian environment, while the effect of a non-Markovian environment on information scrambling is still elusive. Our results show that the memory effect of an environment will be beneficial to information scrambling. Moreover, it is found that the environment is generally detrimental for information scrambling in the long-term, while in some cases it will be helpful for information scrambling in the short-term.