Higher-order harmonics of general limited diffraction Bessel beams

In this paper, we extensively study the higher-order harmonic generation of the general limited diffraction m-thorder Bessel beam. The analysis is based on successive approximations of the Khokhlov–Zabolotskaya–Kuznetsov(KZK)equation. Asymptotic expansions are presented for higher-order harmonic Bessel beams in near and far fields. The validity of asymptotic approximation is also analyzed. The higher-order harmonic of the Bessel beam with the lowest zero-order is taken as a special example.     

Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae

Procedures are described for the high-precision calculation of the modified Bessel function K (x), 0 < < 1, and the Airy function Ai(x), for positive arguments x, as pre-requisites for generating Gaussian quadrature rules having these functions as weight function.     

On Inversion Of Bessel Potentials Associated With The Laplace–Bessel Differential Operator

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.     

Series solutions of coupled differential equations with one regular singular point

We consider two linear second-order ordinary differential equations. r=0 is a regular singular point of these equations. Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solution, because the differential equations are coupled.

In this paper, we present an extended Method of Frobenius on a coupled system of two ordinary differential equations. These equations come from the micropolar theory, which is one of the three kinds of the new 3M physics.     

Reciprocity for multirestricted Stirling numbers

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.     

Wavelets associated with Hankel transform and their Weyl transforms

The Hankel transform is an important transform. In this paper, we study thewavelets associated with the Hankel transform, then define the Weyl transform of thewavelets. We give criteria of its boundedness and compactness on the L~p-spaces.     

Diffusions, exit time moments and Weierstrass theorems

Let be a one-dimensional diffusion with infinitesimal generator given by the operator where is a smooth, positive real-valued function and the ratio of and is a constant. Given a compact interval, we prove a Weierstrass-type theorem for the exit time moments of and their corresponding (naturally weighted) first derivatives, and we provide an algorithm that produces uniform approximations of arbitrary continuous functions by exit time moments. We investigate analogues of these results in higher-dimensional Euclidean spaces. We give expansions for several families of special functions in terms of exit time moments.

     

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X_t⁽⁾,t0) be the BESQ process starting at x. We are interested in large deviations as for the family {^–1X_t⁽⁾,tT}, – or, more generally, for the family of squared radial OU process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.Mathematics Subject Classification (2000): 60F10, 60J60     

Maximal Inequalities for CIR Processes

Let X be a Cox—Ingersoll—Ross (CIR) process given by