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Classical and Computational Analysis / Analyse classique et quantative
(Peter Borwein, Organizer)

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FRANCOIS BERGERON, Université du Québec à Montréal, Montréal, Québec  H3C 3P8
Diagonally harmonic polynomials

Harmonic polynomial for the symmetric group in two set of n variables are introduced as common zeros of all homogeneous diagonally symmetric differential operators. The linear span of these polynomials is of dimension (n+1)^n-1, and we can give an explicit description of its character for the diagonal action of S_n. Its study sugests many nice refinements and generalizations of classical harmonic polynomials.

JON BORWEIN, Simon Fraser University, Burnaby, British Columbia  V5A 1S6
Multi-variable sinc integrals and volumes of polyhedra

Using classical Fourier transform techniques, we establish inequalities for integrals of the form

ó õ ¥ 0  n Õ k = 0  sin(akx) akx  dx.

We then give quite striking closed form evaluations of such integrals and finish by discussing various extensions and applications. We also describe our related numeric and symbolic computational work.

The firat part of this work is available at www.cecm.sfu.ca/preprints in:

David Borwein and Jonathan M. Borwein, Some remarkable properties of sinc and related integrals. CECM 142(1999), preprint; The Ramanujan J., in press.

while later work is joint with B. J. Mares as well.

LEN BOS, Department of Mathematics, University of Calgary, Calgary Alberta  T2N 1N4
Fekete type points for radial basis interpolation

Suppose that f:[0,¥)® R and that x₁,... x_n are distinct points in R^d. A Radial Basis interpolant based on f is a simple function of the form s(x) = å_k a_k f(|x-x_k|) where the a_k are real and chosen so that s(x_j) = z_j, 1 £ j £ n, for given values of z_j. By Fekete type points we mean those which maximize the determinant of the associated matrix [f(|x_i-x_j|)] over some compact region. In this talk we will discuss the univariate version of this problem and show that for a large class of functions, f, the Fekete points are asymptotically uniformly distributed.

JACQUE CARETTE, Maple
The relationship between ``function'' and ``expression''

For at least 2 centuries now mathematicians have routinely written down expressions such as sin(2*Pi*x) + arctan(y) and called them functions. This intuitive definition of function was in fact so pervasive that in older textbooks one can even find ``proofs'' that all functions are (essentially) continuous. Later, when the modern definition of function was widely adopted, it was realized that in fact most functions are actually totally discontinuous. With the modern age, and more specifically, with the widespread use of Computer Algebra systems, this issue is about to become current again: computationally speaking, the only function class that is easily ``computable'' is the one defined by expressions!

In this talk we will examine what it could mean to treat an expression as a function and, more specifically, different ways to give a semantic/functional interpretation to expressions with an eye towards exceptional cases such as x/x, x*sin(1/x) and diff(Ö[((x²))],x). In particular, we will consider pointwise, local and global interpretations, and show that all of them lead to undesirable behaviour in certain circumstances.

ROB CORLESS, Ontario Research Centre for Computer Algebra, Western Ontario
Derivatives

A computer program to take derivatives of expressions was the first ``computer algebra'' program ever written; indeed, that program predates FORTRAN. Every computer algebra system written since, for forty years, has included facilities to take derivatives. What on earth is there new to say about derivatives and computer algebra?

This talk looks at cases where humans are (at this time, still) better than computer algebra systems; at fractional derivatives; and at n-th derivatives. Surprisingly, there are still some things we don't know how to get a computer to do. This talk includes joint work with M. Benghorbal.

MARK GEISBRECT, University of Western Ontario
Fast algorithms for integer matrices

The ability to compute quickly with integer matrices is at the heart of many algorithms in computational algebra, number theory, and group theory. Moreover, the inherent complexity of these problem is fundamental to our understanding of the most basic mathematical computations. In this talk, I will describe a number of new and quantifiably faster algorithms for computing determinants, solving Diophantine linear systems, and finding Smith forms and Frobenius forms of large and often very sparse integer matrices.

IVO KLEMES, Department of Mathematics and Statistics,, McGill University, Montréal, Québec  H3A 2K6
Finite Toeplitz matrices and sharp Littlewood conjectures

The sharp Littlewood Conjecture states that for fixed N ³ 1, if D(z) = 1+z+ z²+¼+z^N-1 then on the unit circle |z| = 1, ||D||₁ is the minimum of ||f||₁ for f of the form f(z) = c₀+c₁ z^n₁+¼+ c_N-1z^n_N-1 with |c_k| = 1, and more generally that ||D||_p is the min/max of ||f||_p for fixed p Î [0,2]/[2,¥]. We prove this for the special case f(z) = 1 ±z±z² ±¼±z^N-1 and p Î [0,4], by first proving stronger results for the eigenvalues of finite sections of the Toeplitz matrices of |D|² and |f|², in particular for their Schatten p-norms. We also state several conjectures to the effect that these stronger results should hold for the general case of f. The approach is motivated by the uncertainty principle and two theorems of Szegö.

ADRIAN LEWIS, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario  N2L 3G1
Optimizing the stability of a matrix

Perhaps the most basic problem in control theory is ``static output feedback stabilization'': given matrices A, B and C, find a matrix K so that the square matrix A+BKC has all its eigenvalues in the left half plane. Recognizing when this is possible is a major open problem. More generally, varying K to move the eigenvalues to the left improves the stability of the associated dynamical system, and so is of great practical interest. Optimal choices of K often lead to multiple eigenvalues, and hence severe nonsmoothness, modeling difficulties, and numerical challenges. I will discuss theory and computational techniques.

IGOR PRITSKER, Oklahoma State University, Stillwater, Oklahoma  74078, USA
Bounds for the integer Chebyshev constant

We study the problem of minimizing the supremum norm by polynomials with integer coefficients. Introducing methods of the weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of the integer Chebyshev polynomials, and a lower bound for the integer Chebyshev constant.

TOMAS RANSFORD, Département de mathématiques et de statistique, Université Laval, Québec  G1K 7P4
Countability via capacity

Let K be a compact subset of C, and let c denote logarithmic capacity. We prove that (c(L) = 0Þc(K+L) = 0) if and only if K is countable. As an application, we obtain a short proof of the scarcity theorem for countable analytic multifunctions. This is joint work with Norm Levenberg, Jérémie Rostand and Zbigniew Slodkowski.

TONY THOMPSON, Dalhousie University, Halifax, Nova Scotia  B3H 3J5
Geometric means of ellipsoids

In the usual proof of the uniqueness of the Löwner-John ellipsoid circumscribed (resp. inscribed) to a convex body an arithmetic (resp. harmonic) mean is used. Given two ellipsoids (both centred at the origin) one may take both averages to produce two new ellipsoids. Evidently, this process can be iterated to give two sequences that converge to a common limiting ellipsoid which may be viewed as the geometric mean of the two original ellipsoids. Indeed, if one regards the process as being applied to positive definite operators A and B and if AB = BA then it yields the usual (AB)^1/2. If the operators do not commute, then it yields a generalized geometric mean which may be of independent interest.

Firey and, more recently, Lutwak have also considered p-means of convex bodies by considering averages of their support functions. In this context, the arithmetic and harmonic means first referred to correspond to the case p = ±2.