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Title page for ETD etd-11122015-154408


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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