




Classical and Computational Analysis / Analyse classique et
quantative (Peter Borwein, Organizer)
 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)^{n1}, 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
Multivariable 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(a_{k}x) a_{k}x

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 x_{1},... 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(xx_{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[((x^{2}))],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 nth
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+ z^{2}+¼+z^{N1} then on the unit circle z = 1,
D_{1} is the minimum of f_{1} for f of the form f(z) = c_{0}+c_{1} z^{n1}+¼+ c_{N1}z^{nN1} 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±z^{2} ±¼±z^{N1} 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
pnorms. 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 HilbertFekete 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 LöwnerJohn 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 pmeans 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.

