## Mathematical Sciences Faculty Presentations

#### Title

Separation Results for Optimal Riesz Energy Points on the Sphere and Axis-supported External Fields

Presentation

2010

#### Conference Name

Optimal Configurations on the Sphere and Other Manifolds

#### Conference Location

Nashville, TN, USA

2010

Invited

#### Abstract

Let $\omega_N := {x_1, x_2, ... , x_N}$ be an optimal configuration on the unit sphere minimizing the Riesz energy. Considering the discrete and continuous minimum energy problems associated with the external field induced by a fixed point, the following separation result for $d-2 K_{s,d}/N^{1/d}$, $K_{s,d} :=(2B(d/2,1/2)/B(d/2; (d-s)/2)^{1/d}$, where B(x,y) denotes the Beta function. Motivated by this we extend the continuous minimal energy problem to axis-supported external fields. An interesting phenomenon occurs for $s = d-2$. Finally, we discuss some generalizations of the method utilized in the separation result above.

#### Keywords

Optimal configurations, Riesz Energy, Separation constants

#### Disciplines

Analysis | Mathematics

COinS

#### Link to Original Published Item

http://www.math.vanderbilt.edu/~optimal2010/abstracts.pdf