in this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an ivp of first order hybrid  random differential equations with the linear perturbation of second type. a comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential equ...

 In this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an IVP of first order hybrid  random differential equations with the linear perturbation of second type. A comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential eq...

We discuss inequalities and applications for percolation and randomcluster models. The relevant areas of methodology concern the following two types of inequality: inequalities involving the probability of a general increasing event, and certain differential inequalities involving the percolation probability. We summarise three areas of application of such inequalities, namely strict inequality...

Shannon’s entropy power inequality characterizes the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies. Since the pioneering work of Shannon, there has been a steady stream of results over the years, trying to understand the structure of Shannon’s entropy power inequality, as well as trying to develop similar entropy power i...

We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.

in a celebrated work by shao [13] several inequalities for negatively associated random variables were proved. in this paper we obtain some maximal inequalities for associated random variables. also we establish a maximal inequality for demimartingales which generalizes and improves the result of christofides [4].

This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.

By employing feedback control laws, the robust stochastic stability of distributed parameter large-scale control systems under random structural perturbations is investigated. The results are obtained by using vector Lyapunov-like functionals in the context of the block-comparison theorem, the theory of systems of differential inequalities, and decomposition-aggregation method. The complexity v...

in this article, we develop the distributed order fractional hybrid differential equations (dofhdes) with linear perturbations involving the fractional riemann-liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-lipschit...

Statistical Mechanics and Random Matrices 3 1. Introduction 6 2. Motivations 7 3. The different scales; typical results 12 Lecture 1. Wigner matrices and moments estimates 15 1. Wigner's theorem 16 2. Words in several independent Wigner matrices 23 3. Estimates on the largest eigenvalue of Wigner matrices 25 Lecture 2. Gaussian Wigner matrices and Fredholm determinants 27 1. Joint law of the ei...