Ladrilhamentos reticulados de Z^n por esferas de Lee

Detalhes bibliográficos
Autor(a) principal: Ribeiro, Roberta Alves do Nascimento [UNIFESP]
Data de Publicação: 2024
Tipo de documento: Dissertação
Idioma: por
Título da fonte: Repositório Institucional da UNIFESP
Texto Completo: https://hdl.handle.net/11600/70840
Resumo: O objetivo deste trabalho é o estudo de ladrilhamentos reticulados de Z^n por esferas de Lee. Investigaremos uma nova abordagem algébrica sobre esse problema, que é um caso especial da conjectura de Golomb–Welch. Utilizando esse novo método, é possível demonstrar a não existência de ladrilhamentos reticulados de Z^n por esferas de Lee com o mesmo raio r = 2 para infinitos valores da dimensão n. Tal método utiliza conceitos como os anéis de grupo e o grupo de caracteres, que conjuntamente oferecem um ambiente propício para uma nova abordagem utilizando um resultado conhecido acerca dos ladrilhamentos reticulados. Neste estudo, damos ênfase a dois artigos: ``Perfect codes in the Lee metric and the packing of polyominoes'', de Solomon W. Golomb e Lloyd R. Welch, que apresenta a conjectura e enuncia alguns fatos envolvendo ladrilhamentos por esferas de Lee e ``On the nonexistence of lattice tilings of Z^nby Lee spheres'', de Tao Zhang e Yue Zhou, que soluciona alguns casos particulares da conjectura para r=2.
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spelling http://lattes.cnpq.br/6312308130945210Ribeiro, Roberta Alves do Nascimento [UNIFESP]http://lattes.cnpq.br/7182535594249914Jorge, Grasiele Cristiane [UNIFESP]São José dos Campos, SP2024-03-14T15:50:40Z2024-03-14T15:50:40Z2024-02-15O objetivo deste trabalho é o estudo de ladrilhamentos reticulados de Z^n por esferas de Lee. Investigaremos uma nova abordagem algébrica sobre esse problema, que é um caso especial da conjectura de Golomb–Welch. Utilizando esse novo método, é possível demonstrar a não existência de ladrilhamentos reticulados de Z^n por esferas de Lee com o mesmo raio r = 2 para infinitos valores da dimensão n. Tal método utiliza conceitos como os anéis de grupo e o grupo de caracteres, que conjuntamente oferecem um ambiente propício para uma nova abordagem utilizando um resultado conhecido acerca dos ladrilhamentos reticulados. Neste estudo, damos ênfase a dois artigos: ``Perfect codes in the Lee metric and the packing of polyominoes'', de Solomon W. Golomb e Lloyd R. Welch, que apresenta a conjectura e enuncia alguns fatos envolvendo ladrilhamentos por esferas de Lee e ``On the nonexistence of lattice tilings of Z^nby Lee spheres'', de Tao Zhang e Yue Zhou, que soluciona alguns casos particulares da conjectura para r=2.The aim of this work is to explore lattice tilings of Z^n by Lee spheres. We will examine a novel algebraic approach to this problem, a specific instance of the Golomb-Welch conjecture. Using this new method, it is possible to illustrate the absence of lattice tilings of Z^n by Lee spheres with the same radius r = 2 for infinitely many values of the dimension n. This method incorporates concepts such as group rings and character groups, providing a favorable environment for a fresh perspective utilizing a known result concerning lattice tilings. In this study, we highlight two articles: "Perfect codes in the Lee metric and the packing of polyominoes" by Solomon W. Golomb and Lloyd R. Welch, which introduces the conjecture and outlines some facts related to tilings by Lee spheres, and "On the nonexistence of lattice tilings of Z^n by Lee spheres" by Tao Zhang and Yue Zhou, which resolves certain specific cases of the conjecture for r=2.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)grasiele.jorge@unifesp.br86 fhttps://hdl.handle.net/11600/70840porUniversidade Federal de São PauloCódigos perfeitosMétrica de LeeReticuladosLadrilhamentoConjectura de Golomb-WelchLadrilhamentos reticulados de Z^n por esferas de LeeLattice tilings of Z^n by Lee spheresinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UNIFESPinstname:Universidade Federal de São Paulo (UNIFESP)instacron:UNIFESPInstituto de Ciência e Tecnologia (ICT)47638727886Matemática Pura e AplicadaMatemática DiscretaCódigos Corretores de ErrosORIGINALDissertação - Roberta A N Ribeiro - PPGMAT.pdfDissertação 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InstitucionalPUBhttp://www.repositorio.unifesp.br/oai/requestopendoar:34652024-03-18T12:28:06Repositório Institucional da UNIFESP - Universidade Federal de São Paulo 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dc.title.none.fl_str_mv Ladrilhamentos reticulados de Z^n por esferas de Lee
dc.title.alternative.none.fl_str_mv Lattice tilings of Z^n by Lee spheres
title Ladrilhamentos reticulados de Z^n por esferas de Lee
spellingShingle Ladrilhamentos reticulados de Z^n por esferas de Lee
Ribeiro, Roberta Alves do Nascimento [UNIFESP]
Códigos perfeitos
Métrica de Lee
Reticulados
Ladrilhamento
Conjectura de Golomb-Welch
title_short Ladrilhamentos reticulados de Z^n por esferas de Lee
title_full Ladrilhamentos reticulados de Z^n por esferas de Lee
title_fullStr Ladrilhamentos reticulados de Z^n por esferas de Lee
title_full_unstemmed Ladrilhamentos reticulados de Z^n por esferas de Lee
title_sort Ladrilhamentos reticulados de Z^n por esferas de Lee
author Ribeiro, Roberta Alves do Nascimento [UNIFESP]
author_facet Ribeiro, Roberta Alves do Nascimento [UNIFESP]
author_role author
dc.contributor.advisorLattes.none.fl_str_mv http://lattes.cnpq.br/6312308130945210
dc.contributor.authorLattes.none.fl_str_mv http://lattes.cnpq.br/7182535594249914
dc.contributor.author.fl_str_mv Ribeiro, Roberta Alves do Nascimento [UNIFESP]
dc.contributor.advisor1.fl_str_mv Jorge, Grasiele Cristiane [UNIFESP]
contributor_str_mv Jorge, Grasiele Cristiane [UNIFESP]
dc.subject.por.fl_str_mv Códigos perfeitos
Métrica de Lee
Reticulados
Ladrilhamento
Conjectura de Golomb-Welch
topic Códigos perfeitos
Métrica de Lee
Reticulados
Ladrilhamento
Conjectura de Golomb-Welch
description O objetivo deste trabalho é o estudo de ladrilhamentos reticulados de Z^n por esferas de Lee. Investigaremos uma nova abordagem algébrica sobre esse problema, que é um caso especial da conjectura de Golomb–Welch. Utilizando esse novo método, é possível demonstrar a não existência de ladrilhamentos reticulados de Z^n por esferas de Lee com o mesmo raio r = 2 para infinitos valores da dimensão n. Tal método utiliza conceitos como os anéis de grupo e o grupo de caracteres, que conjuntamente oferecem um ambiente propício para uma nova abordagem utilizando um resultado conhecido acerca dos ladrilhamentos reticulados. Neste estudo, damos ênfase a dois artigos: ``Perfect codes in the Lee metric and the packing of polyominoes'', de Solomon W. Golomb e Lloyd R. Welch, que apresenta a conjectura e enuncia alguns fatos envolvendo ladrilhamentos por esferas de Lee e ``On the nonexistence of lattice tilings of Z^nby Lee spheres'', de Tao Zhang e Yue Zhou, que soluciona alguns casos particulares da conjectura para r=2.
publishDate 2024
dc.date.accessioned.fl_str_mv 2024-03-14T15:50:40Z
dc.date.available.fl_str_mv 2024-03-14T15:50:40Z
dc.date.issued.fl_str_mv 2024-02-15
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
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url https://hdl.handle.net/11600/70840
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dc.format.none.fl_str_mv 86 f
dc.coverage.spatial.none.fl_str_mv São José dos Campos, SP
dc.publisher.none.fl_str_mv Universidade Federal de São Paulo
publisher.none.fl_str_mv Universidade Federal de São Paulo
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instname:Universidade Federal de São Paulo (UNIFESP)
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