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 AbstractLet $Lambda$ 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 $0~~0$ 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 $f(0)=limlimits_{x o 0^+}f(x)$.

Files

Filename Size Approximate Download Time (Hours:Minutes:Seconds)

28.8 Modem 56K Modem ISDN (64 Kb) ISDN (128 Kb) Higher-speed Access Su.pdf397.73 Kb 00:01:50 00:00:56 00:00:49 00:00:24 00:00:02

Browse All Available ETDs by
( Author |
Department )

If you have more questions or technical problems, please Contact LITS.