Random matrix models provide a phenomenological description of vast variety physical phenomena. Prominent examples include the eigenvalue statistics quantum (chaotic) systems, which are characterized by spectral form factor (SFF). Here, we calculate SFF unitary ensembles infinite order with weight function satisfying assumptions Szeg"{o}’s limit theorem. We then consider parameter-dependent cri...

We rst give a brief survey of some aspects of orthogonal polynomials. The three-term recurrence relation gives a tridiagonal matrix and the corresponding Ja-cobi operator gives useful information about the orthogonalizing measure and the asymptotic behavior of the zeros of the orthogonal polynomials. The Toda lattice and other similar dynamical systems (Langmuir lattice or Kac-Van Moerbeke latt...

Let w() be a positive weight function on the interval ;) and associate the positive deenite inner product on the unit circle of the complex plane by hf; gi w = 1 2 R f(e ii)g(e ii)w()d. For a sequence of points f k g 1 k=1 included in a compact subset of the open unit disk, we consider the orthogonal rational functions (ORF) f k g 1 k=0 that are obtained by orthogonalization of the sequence f1;...

‎Universally prestarlike functions of order $alphaleq 1$ in the‎ ‎slit domain $Lambda=mathbb{C}setminus [1,infty)$ have been‎ ‎recently introduced by S‎. ‎Ruscheweyh.This notion generalizes the‎ ‎corresponding one for functions in the unit disk $Delta$ (and‎ ‎other circular domains in $mathbb{C}$)‎. ‎In this paper‎, ‎we obtain‎ ‎the Fekete-Szegö coefficient functional for transforms of such‎ ‎f...

Denote by Pn =1 a f (x ) the Gaussian quadrature rule for the integral R 1 1 f (x) dx. We give a simple explicit expression for the \variance" Pn =1 a2 . The method can be used to obtain similar results for the Lobatto rule. 1 1. The result A quadrature rule (on [ 1; 1]) is a functional Qn on C [ 1; 1] of the form Qn [f ] = n X =1 a f (x ) ; a 2 R (1) 1 x1 < x2 < : : : < xn 1 : The \variance of...

In order to numerically invert Laplace transforms to calculate probability distributions in queueing and related models, we need to be able to calculate the Laplace transform values. In many cases the desired Laplace transform values (e.g., of a waiting-time distribution) can be computed when the Laplace transform values of component probability density functions (pdf's) (e.g., of a service-tim...

This work deals with the H1 condition numbers and the distribution of the ~ N;Msingular values of the preconditioned operators f~ 1 N;M WN;M ÂN;Mg. ÂN;M is the matrix representation of the Legendre Spectral Collocation discretization of the elliptic operator A de ned by Au := u + a1ux + a2uy + a0u in (the unit square) with boundary conditions: u = 0 on 0; @u @ A = u on 1. ~ N;M is the sti ness ...