APPLIED MATH SEMINAR

Title: A random walk through numerical integration, Riesz configurations
and low discrepancy sequences on rectifiable sets.

Speaker: Steven B. Damelin, IMA and Georgia Southern University

When/where: Thursday, Oct. 20th, 4:15pm, AKW 200

Abstract:

The problem of distributing a large number of points uniformly on a
compact set is an interesting and difficult problem which has
attracted much research.

In this talk, we will focus on recent work of the author, Grabner, Mullen
and Maymeskul which deals with numerical integration, Riesz configurations
and low discrepancy sequences on rectifiable sets. In particular,
we will study certain arrangements of $N\geq 1$ points on sets $A$
from a class $A^d$ of $d$-dimensional compact sets embedded in $R^d'$,
$1\leq d\leq d^'$. As an example, we can take the $d$ dimensional
unit sphere $S^d$ realized as a subset of $R^{d+1}$.
We assume  that these points interact through a Riesz potential
$V=|\cdot|^{-s}$, where $s>0$
and $|\cdot|$ is the Euclidean distance in $R^d'$.

We will focus on the following ideas and new methods developed
in the case of the following items 1-3 below. Even in special cases such
as the sphere, most of what we develop below was until recently not known.

(1) The development of a numerical integration formula in terms of
Riesz energy which allows for discrepancy estimates on spheres for a large
class of smooth functions, typically Lipchitz of positive order.

(2) Estimates for separation and mesh norm of $0<s<d-1$ minimal extremal
configurations and associated biological scar defects.

(3) The existence of low discrepancy sequences such as nets built out of
bases of linear independent vectors with applications to
combinatorial designs and codes.

References: All papers can be found on my hompage:
http://www.ima.umn.edu/~damelin

-S.B. Damelin and P. Grabner, Numerical integration, energy and
asymptotic equidistribution on the sphere, Journal of Complexity,
19(2003), pp 231-246. (Postscript)  Corrigendum, Journal of Complexity,
(20)(2004), pp 883-884. Homepage: Number 32.
-S. B. Damelin and V. Maymeskul, On Point Energies, Separation
Radius and Mesh Norm for $s$-Extremal Configurations on Compact Sets in
$R^n$, to appear Journal of Complexity. Homepage: Number 34
- S.B. Damelin, G. Michalski, G. Mullen and D. Stone, On the number of
linearly independent binary vectors of fixed length with applications to
the existence of com