## Files in this item

FilesDescriptionFormat

application/pdf

LIU-DISSERTATION-2017.pdf (853kB)
(no description provided)PDF

## Description

 Title: Asymptotically optimal shapes for counting lattice points and eigenvalues Author(s): Liu, Shiya Director of Research: Laugesen, Richard S. Doctoral Committee Chair(s): DeVille, Lee Doctoral Committee Member(s): Kirr, Eduard; Li, Xiaochun Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Lattice points, planar convex domain, p-ellipse, spectral optimization, Laplacian, Dirichlet eigenvalues, Neumann eigenvalues, Translated lattice, Schrödinger eigenvalues, harmonic oscillator. Abstract: In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches $1$ as the radius" approaches infinity. In particular, the result implies when $11$ (concave) and also when \$0 Issue Date: 2017-07-11 Type: Thesis URI: http://hdl.handle.net/2142/98364 Rights Information: Copyright 2017 Shiya Liu Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
﻿