Abstract We consider the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps in real separable Hilbert space. Under sufficient conditions, we obtain results by virtue theory strongly continuous cosine family bounded linear operators combined inequality technique Sadovskii fixed point theorem. Fi...

It is well-known that Kolmogorov’s inequality is the particular case bk = 1, for all k and n = 1 in (1.1). Afterwards this inequality was extended to real valued martingales (see [4]). Since then, this inequality has been studied by many authors. For the case of R-valued random variables, Sung [22] obtained the Hájek-Rényi inequality for the associated sequence. Liu et al. [19] considered the n...

We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on L2(R) of the form Uc := Fei(q(x)+cx), where q is a real polynomial of odd degree, c a real number, and F the Fourier transform. We show that Uc is a quantization of the classical Henon map fλ : R 2 → R (x, y) 7→ (y + q′(x) + c,−x), and show that ...

Two kinds of Bell-states diagonal (BSD) entanglement witnesses (EW) are constructed by using the algebra of Dirac γ matrices in the space-time of arbitrary dimension d, where the first kind can detect some BSD relativistic and non-relativistic m-partite multispinor bound entangled states in Hilbert space of dimension 2m⌊d/2⌋, including the bipartite Bell-type and iso-concurrence type states in ...

We introduce the concept of bi-square-mean almost automorphic functions and Stepanov-like square-mean pseudo almost automorphic functions for stochastic processes. Using the results, we also study the existence and uniqueness theorems for Stepanov-like square-mean pseudo almost automorphic mild solutions to a class of nonautonomous stochastic functional evolution equations in a real separable H...

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are Rn, Cn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2N, the Baire space NN, the infinite symmetric group S∞...

The Riemannian manifold structure of the classical (i.e. Einsteinian) spacetime is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The spaceH is associated with the space of states of a macroscopic testparticle in the universe. The spatial localization of state of the...

The Malliavin calculus is an infinite dimensional calculus on a Gaussian space, which is mainly applied to establish the regularity of the law of nonlinear functionals of the underlying Gaussian process. Suppose that H is a real separable Hilbert space with scalar product denoted by 〈·, ·〉H . The norm of an element h ∈ H will be denoted by ‖h‖H . Consider a Gaussian family of random variables W...

The Tau Method produces polynomial approximations of solutions of differential equations. The purpose of this paper is (i) to extend the recursive formulation of this method to general linear operator equations defined in a separable Hilbert space, and (ii) to develop an iterative refinement procedure which improves on the accuracy of Tau approximations. Applications to Fredholm integral equati...