Applications of harmonic analysis to discrete geometry
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://www.teses.usp.br/teses/disponiveis/45/45134/tde-28042022-161312/ |
Resumo: | Harmonic analysis is the analysis of function spaces under the action of some group. In this project we consider applications of Harmonic analysis on Euclidean space, via the group action of translations, and applications of Harmonic analysis on the sphere, via the orthogonal group action. While the analysis on Euclidean space leads to the classical Fourier analysis and operations such as the Fourier transform, representation theory allows us to see the action of the orthogonal group with the same lens, in such a way that to functions of positive type correspond invariant and positive kernels in the sphere and to the Fourier inversion formula corresponds the decomposition of a spherical function into spherical harmonics. In this thesis we apply these elements to three different geometrical problems. In the first project we use semidefinite programming to bound the maximum number of equiangular lines with a fixed common angle in the Euclidean space and we show how this bound relates to previously known bounds for spherical codes and to independent sets in graphs. In the second project we consider the counting of integer points in dilates of a rational polytope P and use the development of the Fourier transform of a polytope via Stokes formula to determine a formula for the second-order Ehrhart coefficient, namely the coefficient of t^(d-2) in | tP intersection Z^d|. In the third project we consider again the Fourier transform of a polytope and use its development via Brion\'s theorem to show that it does not contain circles in its null set. Fourier analysis, polytopes, lattice sums, packing, equiangular lines, semidefinite programming bounds, spherical harmonics, Ehrhart quasi-polynomials. |
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Applications of harmonic analysis to discrete geometryAplicações de análise harmônica em geometria discretaAnálise de FourierEhrhart quasi-polynomialsEmpacotamentosEquiangular linesFourier analysisHarmônicos esféricosLattice sumsLimitantes de programação semidefinidaPackingPolitoposPolytopesQuasi-polinômios de EhrhartRetas equiangularesSemidefinite programming boundsSomas em reticuladosSpherical harmonicsHarmonic analysis is the analysis of function spaces under the action of some group. In this project we consider applications of Harmonic analysis on Euclidean space, via the group action of translations, and applications of Harmonic analysis on the sphere, via the orthogonal group action. While the analysis on Euclidean space leads to the classical Fourier analysis and operations such as the Fourier transform, representation theory allows us to see the action of the orthogonal group with the same lens, in such a way that to functions of positive type correspond invariant and positive kernels in the sphere and to the Fourier inversion formula corresponds the decomposition of a spherical function into spherical harmonics. In this thesis we apply these elements to three different geometrical problems. In the first project we use semidefinite programming to bound the maximum number of equiangular lines with a fixed common angle in the Euclidean space and we show how this bound relates to previously known bounds for spherical codes and to independent sets in graphs. In the second project we consider the counting of integer points in dilates of a rational polytope P and use the development of the Fourier transform of a polytope via Stokes formula to determine a formula for the second-order Ehrhart coefficient, namely the coefficient of t^(d-2) in | tP intersection Z^d|. In the third project we consider again the Fourier transform of a polytope and use its development via Brion\'s theorem to show that it does not contain circles in its null set. Fourier analysis, polytopes, lattice sums, packing, equiangular lines, semidefinite programming bounds, spherical harmonics, Ehrhart quasi-polynomials.Análise harmônica é a análise de espaços de funções sob a ação de algum grupo. Neste projeto consideramos aplicações de análise harmônica no espaço Euclideano, via a ação de translação, e aplicações de análise harmônica na esfera, via a ação do grupo ortogonal. Enquanto a análise no espaço Euclideano leva à análise de Fourier clássica e a operações tais como a transformada de Fourier, a teoria das representações nos permite ver a ação do grupo ortogonal sob um mesmo ponto de vista. Às funções de tipo positivo correspondem os núcleos positivos e invariantes na esfera e à fórmula de inversão de Fourier corresponde a decomposição de uma função esférica em harmônicos esféricos. Nesta tese aplicamos esses elementos em três problemas geométricos distintos. No primeiro projeto, usamos programação semidefinida para limitar o número máximo de retas equiangulares com um ângulo em comum fixo e mostramos como esse limitante se relaciona com limitantes conhecidos para códigos esféricos e para o número de independência de grafos. No segundo projeto consideramos a contagem de pontos inteiros em dilatações de um politopo racional P e usamos o desenvolvimento da transformada de Fourier de um politopo pela fórmula de Stokes para determinar uma fórmula para o coeficiente de Ehrhart de segunda ordem, a saber o coeficiente de t^(d-2) em |tP interseção Z^d|. No terceiro projeto consideramos novamente a transformada de Fourier de um politopo e usamos seu desenvolvimento pelo teorema de Brion para mostrar que ela não possui círculos no seu conjunto nulo.Biblioteca Digitais de Teses e Dissertações da USPRobins, SinaiMachado, Fabrício Caluza2021-12-20info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/45/45134/tde-28042022-161312/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2022-05-09T20:02:41Zoai:teses.usp.br:tde-28042022-161312Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212022-05-09T20:02:41Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.none.fl_str_mv |
Applications of harmonic analysis to discrete geometry Aplicações de análise harmônica em geometria discreta |
title |
Applications of harmonic analysis to discrete geometry |
spellingShingle |
Applications of harmonic analysis to discrete geometry Machado, Fabrício Caluza Análise de Fourier Ehrhart quasi-polynomials Empacotamentos Equiangular lines Fourier analysis Harmônicos esféricos Lattice sums Limitantes de programação semidefinida Packing Politopos Polytopes Quasi-polinômios de Ehrhart Retas equiangulares Semidefinite programming bounds Somas em reticulados Spherical harmonics |
title_short |
Applications of harmonic analysis to discrete geometry |
title_full |
Applications of harmonic analysis to discrete geometry |
title_fullStr |
Applications of harmonic analysis to discrete geometry |
title_full_unstemmed |
Applications of harmonic analysis to discrete geometry |
title_sort |
Applications of harmonic analysis to discrete geometry |
author |
Machado, Fabrício Caluza |
author_facet |
Machado, Fabrício Caluza |
author_role |
author |
dc.contributor.none.fl_str_mv |
Robins, Sinai |
dc.contributor.author.fl_str_mv |
Machado, Fabrício Caluza |
dc.subject.por.fl_str_mv |
Análise de Fourier Ehrhart quasi-polynomials Empacotamentos Equiangular lines Fourier analysis Harmônicos esféricos Lattice sums Limitantes de programação semidefinida Packing Politopos Polytopes Quasi-polinômios de Ehrhart Retas equiangulares Semidefinite programming bounds Somas em reticulados Spherical harmonics |
topic |
Análise de Fourier Ehrhart quasi-polynomials Empacotamentos Equiangular lines Fourier analysis Harmônicos esféricos Lattice sums Limitantes de programação semidefinida Packing Politopos Polytopes Quasi-polinômios de Ehrhart Retas equiangulares Semidefinite programming bounds Somas em reticulados Spherical harmonics |
description |
Harmonic analysis is the analysis of function spaces under the action of some group. In this project we consider applications of Harmonic analysis on Euclidean space, via the group action of translations, and applications of Harmonic analysis on the sphere, via the orthogonal group action. While the analysis on Euclidean space leads to the classical Fourier analysis and operations such as the Fourier transform, representation theory allows us to see the action of the orthogonal group with the same lens, in such a way that to functions of positive type correspond invariant and positive kernels in the sphere and to the Fourier inversion formula corresponds the decomposition of a spherical function into spherical harmonics. In this thesis we apply these elements to three different geometrical problems. In the first project we use semidefinite programming to bound the maximum number of equiangular lines with a fixed common angle in the Euclidean space and we show how this bound relates to previously known bounds for spherical codes and to independent sets in graphs. In the second project we consider the counting of integer points in dilates of a rational polytope P and use the development of the Fourier transform of a polytope via Stokes formula to determine a formula for the second-order Ehrhart coefficient, namely the coefficient of t^(d-2) in | tP intersection Z^d|. In the third project we consider again the Fourier transform of a polytope and use its development via Brion\'s theorem to show that it does not contain circles in its null set. Fourier analysis, polytopes, lattice sums, packing, equiangular lines, semidefinite programming bounds, spherical harmonics, Ehrhart quasi-polynomials. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-12-20 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://www.teses.usp.br/teses/disponiveis/45/45134/tde-28042022-161312/ |
url |
https://www.teses.usp.br/teses/disponiveis/45/45134/tde-28042022-161312/ |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
|
dc.rights.driver.fl_str_mv |
Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Liberar o conteúdo para acesso público. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.coverage.none.fl_str_mv |
|
dc.publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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1809091148044566528 |