In this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

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...

let  be a sequence of arbitrary random variables with  and , for every  and  be an array of real numbers. we will obtain two maximal inequalities for partial sums and weighted sums of random variables and also, we will prove complete convergence for weighted sums , under some conditions on  and sequence .

in this paper, we extend some famous maximal inequalities and obtain strong laws of largenumbers for arbitrary random variables by use of these inequalities and martingale techniques.

In this paper our primary concern is with the establishment of weighted Hardy inequalities in L(Ω) and Rellich inequalities in L(Ω) depending upon the distance to the boundary of domains Ω ⊂ R with a finite diameter D(Ω). Improved constants are presented in most cases.

The role of inequalities in information theory is reviewed and the relationship of these inequalities to inequalities in other branches of mathematics is developed. Index Terms -Information inequalities, entropy power, Fisher information, uncertainty principles. I. PREFACE:~NEQUALITIES ININFORMATIONTHEORY I NEQUALITIES in information theory have been driven by a desire to solve communication th...

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d + 1 dimensions. A so-ca...