A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and s...

We present an exponential time integrator for the incompressible Navier-Stokes equation based on the exponential block Krylov (EBK) method [2]. The exact solution of the semi-discrete system, given by a discretization of the Navier--Stokes equation in the spatial dimensions, involves the matrix exponential function. So-called exponential integrators are usually directly based on the formal expr...

In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponentia...

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: instability tongues in Mathieu-type equations lose sharpness with addition of higher-frequency harmonics Mathieu potentials. Twenty years later, V. Arnold similar phenomenon on circle maps (and rediscovered result Levy). this paper we find third class object where type behavior takes place: area-preserving cylinder. l...

In this paper, the effect of the temperature change on the vibration frequency of mono-layer graphene sheet embedded in an elastic medium is studied. Using the nonlocal elasticity theory, the governing equations are derived for single-layered graphene sheets. Using Levy and Navier solutions, analytical frequency equations for single-layered graphene sheets are obtained. Using Levy solution, the...

We consider the numerical solution of one type integro-differential equation by a probability method based on fundamental martingale mixed Gaussian processes. As an application, we try to simulate estimation ruin with an unknown parameter driven not classical Levy process, but fractional Brownian motion.

In this paper we study k-noncrossing RNA structures with minimum arc-length 4 and at most k − 1 mutually crossing bonds. Let T k (n) denote the number of knoncrossing RNA structures with arc-length ≥ 4 over n vertices. We prove (a) a functional equation for the generating function ∑ n≥0 T [4] k (n)z n and (b) derive for k ≤ 9 the asymptotic formula T k (n) ∼ ck n +(k−1)/2) γ k . Furthermore we ...

In this paper we study k-noncrossing RNA structures with minimum arc-length 4 and at most k − 1 mutually crossing bonds. Let T [4] k (n) denote the number of k-noncrossing RNA structures with arc-length ≥ 4 over n vertices. We prove (a) a functional equation for the generating function P n≥0 T [4] k (n)z n and (b) derive for k ≤ 9 the asymptotic formula T [4] k (n) ∼ c k n −((k−1) 2 +(k−1)/2) γ...