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A geometric method for eigenvalue problems with low-rank perturbations

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation...

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Detalles Bibliográficos
Autores principales: Anastasio, Thomas J., Barreiro, Andrea K., Bronski, Jared C.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society Publishing 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5627089/
https://www.ncbi.nlm.nih.gov/pubmed/28989749
http://dx.doi.org/10.1098/rsos.170390
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author Anastasio, Thomas J.
Barreiro, Andrea K.
Bronski, Jared C.
author_facet Anastasio, Thomas J.
Barreiro, Andrea K.
Bronski, Jared C.
author_sort Anastasio, Thomas J.
collection PubMed
description We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22, 239–254. (doi:10.1007/s10827-006-0010-x)), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48, 249–264. (doi:10.1093/imamat/48.3.249)).
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spelling pubmed-56270892017-10-08 A geometric method for eigenvalue problems with low-rank perturbations Anastasio, Thomas J. Barreiro, Andrea K. Bronski, Jared C. R Soc Open Sci Mathematics We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22, 239–254. (doi:10.1007/s10827-006-0010-x)), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48, 249–264. (doi:10.1093/imamat/48.3.249)). The Royal Society Publishing 2017-09-27 /pmc/articles/PMC5627089/ /pubmed/28989749 http://dx.doi.org/10.1098/rsos.170390 Text en © 2017 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Mathematics
Anastasio, Thomas J.
Barreiro, Andrea K.
Bronski, Jared C.
A geometric method for eigenvalue problems with low-rank perturbations
title A geometric method for eigenvalue problems with low-rank perturbations
title_full A geometric method for eigenvalue problems with low-rank perturbations
title_fullStr A geometric method for eigenvalue problems with low-rank perturbations
title_full_unstemmed A geometric method for eigenvalue problems with low-rank perturbations
title_short A geometric method for eigenvalue problems with low-rank perturbations
title_sort geometric method for eigenvalue problems with low-rank perturbations
topic Mathematics
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5627089/
https://www.ncbi.nlm.nih.gov/pubmed/28989749
http://dx.doi.org/10.1098/rsos.170390
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