On the distribution of the length of the longest increasing subsequence of random permutations

The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of .

     

The relation of the d-orthogonal polynomials to the Appell polynomials

We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.

A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.     

Approximating dominant singular triplets of large sparse matrices via modified moments

A procedure for determining a few of the largest singular values and corresponding singular vectors of large sparse matrices is presented. Equivalent eigensystems are solved using a technique originally proposed by Golub and Kent based on the computation of modified moments. The asynchronicity in the computations of moments and eigenvalues makes this method attractive for parallel implementations on a network of workstations. Although no obvious relationship between modified moments and the corresponding eigenvectors is known to exist, a scheme to approximate both eigenvalues and eigenvectors (and subsequently singular values and singular vectors) has been produced. This scheme exploits both modified moments in conjunction with the Chebyshev semi-iterative method and deflation techniques to produce approximate eigenpairs of the equivalent sparse eigensystems. The performance of an ANSI-C implementation of this scheme on a network of UNIX workstations and a 256-processor Cray T3D is presented.This research was supported in part by the National Science Foundation under grant numbers NSF-ASC-92-03004 and NSF-ASC-94-11394.     

On a modification of the Clenshaw-Curtis quadrature formula

In this paper we discuss a modification of the Clenshaw-Curtis quadrature formula. It is shown that for integrals, where the integrand may be expanded in a sufficiently rapid convergent Chebyshev series, we may split the sequence of calculated approximations into two sequences, one which approximates the integral from above and one which approximates it from below. Thus, at any step during the calculation we obtain both upper and lower bounds for the true value of the integral.Work performed while the author was working as a visiting scientist at CERN/Geneve.     

Weight Function Approach to q-Difference Equations for the q-Hypergeometric Polynomials

In this paper we define a new algebra generated by the difference operators D _q and D _q-1 with two analytic functions (x) and (x). Also, we define an operator M = J ₁ J ₂–J ₃ J ₄ s.t. all q-hypergeometric orthogonal polynomials Y _n(x), x cos(), are eigenfunctions of the operator M with eigenvalues _q [n] _q . The choice of (x) and (x) depend on the weight function of Y _n (x).     

Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.     

B-Splines with Arbitrary Connection Matrices

We consider a space of Chebyshev splines whose left and right derivatives satisfy linear constraints that are given by arbitrary nonsingular connection matrices. We show that for almost all knot sequences such spline spaces have basis functions whose support is equal to the support of the ordinary B-splines with the same knots. Consequently, there are knot insertion and evaluation algorithms analogous to de Boors algorithm for ordinary splines.     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

Local behaviour of polynomials

In this paper we study the local behaviour of a trigonometric polynomial around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in . The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form are also proved.

     

A characterization of the Hamiltonian

The classical Hamiltonian ) of the very classical quantum harmonic oscillator, which is regarded as a germ of the most of what comes about in quantum mechanics, can be sublimed to an abstract operator in a separable Hilbert space. Having this done one may ask for a condition which would allow it to be identified among operators of a suitable class. This class is that corresponding to three diagonal matrices and the property which makes the action successful is a kind of diagonal invariance (up to change of basis) within the class in question.