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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


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