We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the resu...

In this article using the inverse Laplace transform, we show analytical solutions for the generalized mass transfers with (and without) a chemical reaction. These transfers have been expressed as the Couette flow with the fractional derivative of the Caputo sense. Also, using the Hankel contour for the Bromwich's integral, the solutions are given in terms of the generalized Airy functions.

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

Riesz fractional derivatives of a function, Dα xf(x) (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy fun...

An analytical expression of an Airy-Gauss beam passing through a fractional Fourier transform (FRFT) system is derived. The normalized intensity distribution, phase distribution, centre of gravity, effective beam size, linear momentum, and kurtosis parameter of the Airy-Gauss beam are demonstrated in FRFT plane, respectively. The influence of the fractional order p on the normalized intensity d...

Light-sheet fluorescence microscopy (LSFM) is a kind of volumetric imaging methodology suited for long term living specimens at high temporal-spatial resolution. A single Airy beam (SAB) light-sheet can extend the field view benefiting from its non-diffracting nature, but cost out-of-focus background and low contrast caused by side lobes illumination. Here, we propose method to generate sort mu...

In a celebrated paper, Dyson shows that the spectrum of a n× n randomHermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n non-colliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuou...