Type of Document Dissertation Author Fitzpatrick, Justin Liam Author's Email Address email@example.com URN etd-08092010-093437 Title The Geometry of Optimal and Near-Optimal Riesz Energy Configurations Degree PhD Department Mathematics Advisory Committee
Advisor Name Title Douglas Hardin Committee Chair Edward B. Saff Committee Member Emmanuele DiBenedetto Committee Member Gieri Simonett Committee Member James Dickerson Committee Member Keywords
- Riesz energy
- geometric inequalities
- Voronoi diagrams
Date of Defense 2010-08-09 Availability unrestricted AbstractThis thesis discusses recent and classical results concerning the asymptotic properties (as N gets large) of ``ground state' configurations of N particles restricted to a compact set A of Hausdorff dimension d interacting through through an inverse power law 1/r^s for some s>0.
It has been observed that, as s becomes large, ground state configurations approach best-packing configurations on A. When d=2, it is generally believed that ground state configurations form a hexagonal lattice. This thesis aims to justify this belief in the case d=2 through the study of geometric inequalities for polygons. Specifically, it is shown that, when s is large, a normalized energy associated to interactions from particles that are ``nearest neighbors' to a fixed point in the configuration is minimized when the nearest neighbors form a regular polygon. This technique provides new lower bounds for the energy for 2-dimensional ground state configurations.
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