dc.creator | Su, Yujian | |
dc.date.accessioned | 2020-08-23T15:44:14Z | |
dc.date.available | 2015-11-30 | |
dc.date.issued | 2015-11-30 | |
dc.identifier.uri | https://etd.library.vanderbilt.edu/etd-11122015-154408 | |
dc.identifier.uri | http://hdl.handle.net/1803/14500 | |
dc.description.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 $0<s<d$ (or $s=log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ 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 $f(0)=limlimits_{x o 0^+}f(x)$. | |
dc.format.mimetype | application/pdf | |
dc.subject | polarization | |
dc.subject | max-min problems | |
dc.subject | Epstein Hurwitz Zeta function | |
dc.subject | Ewald summation | |
dc.subject | periodic energy | |
dc.title | Discrete Minimal Energy on Flat Tori and Four-Point Maximal Polarization on S^2 | |
dc.type | dissertation | |
dc.contributor.committeeMember | Kirill Bolotin | |
dc.contributor.committeeMember | Akram Aldroubi | |
dc.contributor.committeeMember | Edward Saff | |
dc.contributor.committeeMember | Alexander Powell | |
dc.type.material | text | |
thesis.degree.name | PHD | |
thesis.degree.level | dissertation | |
thesis.degree.discipline | Mathematics | |
thesis.degree.grantor | Vanderbilt University | |
local.embargo.terms | 2015-11-30 | |
local.embargo.lift | 2015-11-30 | |
dc.contributor.committeeChair | Douglas Hardin | |