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Approximate Isometries as an Eigenvalue Problem and Angular Momentum
Title
Approximate Isometries as an Eigenvalue Problem and Angular Momentum.
Creator
Wilder, Shawn M., Beetle, Christopher, Graduate College
Date Issued
2013-04-12
PURL
http://purl.flvc.org/fcla/dt/3361373
Subject Headings
Black holes (Astronomy), Eigenvalues
Format
Document (PDF)
Subspace detection and scale evolutionary eigendecomposition
Title
Subspace detection and scale evolutionary eigendecomposition.
Creator
Kyperountas, Spyros C., Florida Atlantic University, Erdol, Nurgun, College of Engineering and Computer Science, Department of Computer and Electrical Engineering and Computer Science
Abstract/Description
A measure of the potential of a receiver for detection is detectability. Detectability is a function of the signal and noise, and given any one of them the detectability is fixed. In addition, complete transforms of the signal and noise cannot change detectability. Throughout this work we show that "Subspace methods" as defined here can improve detectability in specific subspaces, resulting in improved Receiver Operating Curves (ROC) and thus better detection in arbitrary noise environments....
Show moreA measure of the potential of a receiver for detection is detectability. Detectability is a function of the signal and noise, and given any one of them the detectability is fixed. In addition, complete transforms of the signal and noise cannot change detectability. Throughout this work we show that "Subspace methods" as defined here can improve detectability in specific subspaces, resulting in improved Receiver Operating Curves (ROC) and thus better detection in arbitrary noise environments. Our method is tested and verified on various signals and noises, both simulated and real. The optimum detection of signals in noise requires the computation of noise eigenvalues and vectors (EVD). This process neither is a trivial one nor is it computationally cheap, especially for non-stationary noise and can result in numerical instabilities when the covariance matrix is large. This work addresses this problem and provides solutions that take advantage of the subspace structure through plane rotations to improve on existing algorithms for EVD by improving their convergence rate and reducing their computational expense for given thresholds.
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Date Issued
2001
PURL
http://purl.flvc.org/fcla/dt/11965
Subject Headings
Eigenvalues, Eigenvectors, Wavelets (Mathematics)
Format
Document (PDF)
Exploring the stability of an eigenvalue problem approximation technique used to define the angular momentum of almost spherical black holes
Title
Exploring the stability of an eigenvalue problem approximation technique used to define the angular momentum of almost spherical black holes.
Creator
Wilder, Shawn M., Beetle, Christopher, Graduate College
Date Issued
2011-04-08
PURL
http://purl.flvc.org/fcla/dt/3164804
Subject Headings
Eigenvalues, Black holes (Astronomy), Deformations (Mechanics)
Format
Document (PDF)
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems
Title
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems.
Creator
Wilder, Shawn M., Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the...
Show moreIn general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004235
Subject Headings
Boundary element methods, Boundary value problems, Differential equations, Elliptic -- Numerical solutions, Differential equations, Partial -- Numerical solutions, Eigenvalues, Spectral theory (Mathematics)
Format
Document (PDF)