Current Search: Ray, Shannon (x)
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Title
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Embedding convex polyhedral metrics using the adiabatic isometric mapping (AIM) algorithm.
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Creator
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Ray, Shannon, Graduate College
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Abstract/Description
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Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly positive Gaussian curvature at each vertex could be isometrically embedded in three-dimensional Euclidean space as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms that realize the Alexandrov Embedding in polynomial time. Following his proof, we produced the adiabatic isometric mapping AIM algorithm. The AIM algorithm is approximately quadratic in time...
Show moreAlexandrov proved that any simplicial complex homeomorphic to a sphere with strictly positive Gaussian curvature at each vertex could be isometrically embedded in three-dimensional Euclidean space as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms that realize the Alexandrov Embedding in polynomial time. Following his proof, we produced the adiabatic isometric mapping AIM algorithm. The AIM algorithm is approximately quadratic in time and reproduces edge lengths up to arbitrary accuracy.
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Date Issued
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2014
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PURL
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http://purl.flvc.org/fau/fd/FA00005163
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Format
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Document (PDF)
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Title
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Quasi-local energy of rotating black hole spacetimes and isometric embeddings of 2-surfaces in Euclidean 3-space.
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Creator
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Ray, Shannon, Miller, Warner A., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
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Abstract/Description
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One of the most fundamental problems in classical general relativity is the measure of e↵ective mass of a pure gravitational field. The principle of equivalence prohibits a purely local measure of this mass. This thesis critically examines the most recent quasi-local measure by Wang and Yau for a maximally rotating black hole spacetime. In particular, it examines a family of spacelike 2-surfaces with constant radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which,...
Show moreOne of the most fundamental problems in classical general relativity is the measure of e↵ective mass of a pure gravitational field. The principle of equivalence prohibits a purely local measure of this mass. This thesis critically examines the most recent quasi-local measure by Wang and Yau for a maximally rotating black hole spacetime. In particular, it examines a family of spacelike 2-surfaces with constant radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which, the Wang and Yau quasi-local energy has yet to be explored. In this region, the results of this thesis indicate that the Wang and Yau quasi-local energy yields complex values and is essentially equivalent to the previously defined Brown and York quasi-local energy. However, an application of their quasi-local mass is suggested in a dynamical setting, which can potentially give new and meaningful measures. In supporting this thesis, the development of a novel adiabatic isometric mapping algorithm is included. Its purpose is to provide the isometric embedding of convex 2-surfaces with spherical topology into Euclidean 3-space necessary for completing the calculation of quasilocal energy in numerical relativity codes. The innovation of this algorithm is the guided adiabatic pull- back routine. This uses Ricci flow and Newtons method to give isometric embeddings of piecewise simplicial 2-manifolds, which allows the algorithm to provide accuracy of the edge lengths up to a user set tolerance.
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Date Issued
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2017
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PURL
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http://purl.flvc.org/fau/fd/FA00004865, http://purl.flvc.org/fau/fd/FA00004865
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Subject Headings
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Gravitational fields., General relativity (Physics), Newton-Raphson method., Ricci flow.
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Format
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Document (PDF)
