Type of Document	Dissertation
Author	Su, Yujian
URN	etd-11122015-154408
Title	Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2
Degree	PhD
Department	Mathematics
Advisory Committee	Advisor Name Title Douglas Hardin Committee Chair Akram Aldroubi Committee Member Alexander Powell Committee Member Edward Saff Committee Member Kirill Bolotin Committee Member
Keywords	polarization max-min problems Epstein Hurwitz Zeta function Ewald summation periodic energy
Date of Defense	2015-11-04
Availability	unrestricted
Abstract Let $Lambda$ be a lattice in $R^d$ with positive co-volume. Among $Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $mathcal{E}_{s,Lambda}(N)$. While the dominant term in the asymptotic expansion of $mathcal{E}_{s,Lambda}(N)$ as $N$ goes to infinity in the long range case that $00$ they are of the form $C_{s,d}|Lambda|^{-s/d}N^{1+s/d}$ and $-frac{2}{d}Nlog N+left(C_{log,d}-2zeta'_{Lambda}(0) ight)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $Lambda$. We also solve the $4$-point maximal polarization problem on $S^2$. We prove that the vertices of a regular tetrahedron on $S^2$ maximize the minimum of discrete potentials on $S^2$ whenever the potential is of the form $sumlimits_{k=1}^{4}f(|x-x_k|^2)$, where $f:[0,4] ightarrow[0,infty]$ is non-increasing and strictly convex with $f(0)=limlimits_{x o 0^+}f(x)$.
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