Type of Document Dissertation Author Calef, Matthew Thomas Author's Email Address mcalef@alumni.uchicago.edu URN etd-07152009-162919 Title Theoretical and Computational Investigations of Minimal Energy Problems Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Douglas Hardin Committee Chair Akram Aldroubi Committee Member Ed Saff Committee Member Marcus Mendenhall Committee Member Mark Ellingham Committee Member Keywords

- potential theory; normalized d-energy; density
Date of Defense 2009-06-25 Availability unrestricted AbstractLet A be a d-dimensional compact subset of p-dimensional Euclidean space. For s in (0,d) the Riesz s-equilibrium measure is the unique Borel probability measure that minimizes the Riesz s-energy over the set of all Borel probability measures supported on A. In this paper we show that if A is a strictly self-similar d-fractal or a strongly Hausdorff d-rectifiable set, then the s-equilibrium measures converge in the weak-star topology to normalized Hausdorff measure restricted to A as s approaches d from below.

Additionally we describe numerical experiments on the 2-sphere involving discrete energies mediated by the Riesz s-kernel. These experiments provide approximate discrete minimal energies for N=20,...,200 and s=0,...,3 where, in the case s=0, the Riesz kernel becomes the logarithmic kernel. These experiments corroborate several conjectures regarding the asymptotic expansion as N goes go infinity of the minimal N-point energies. Further, the number of stable configurations observed as a function of N and s is reported. Finally, two algorithms used in this experiment are presented. The first minimizes the effect of roundoff error when computing sums of many terms, the second uses graph theory to speed the identification of isometries between collections of on the 2-sphere.

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