Title

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

Document Type

Presentation

Presentation Date

2010

Conference Name

Optimal Configurations on the Sphere and Other Manifolds

Conference Location

Nashville, TN, USA

Publication Date

2010

Peer Review

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

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