Accessory parameters in conformal mapping: exploiting isomonodromic tau functions
Autor(a) principal: | |
---|---|
Data de Publicação: | 2018 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/33450 |
Resumo: | Conformal mappings are important mathematical tools in some applied contexts, e.g. electrostatics and classical fluid dynamics. In order to construct a conformal mapping from a canonical simply connected region to the interior of a circular arc polygon with more than three vertices, the accessory parameter problem arises: In general, the mapping is a solution of a differential equation with unknown parameters which hinder its direct integration. Such parameters can be obtained through approximation techniques with relative small computational effort unless the target domain has an elongated aspect, causing the well known difficulty – the ‘crowding’ phenomenon – to emerge. In this thesis, in the pursuit of calculating the accessory parameters as a Riemann-Hilbert problem, we determine them in terms of isomonodromic tau functions and show how to extract the monodromy information from the geometry of the target domain. We also verify that the tau functions satisfy Toda equations, and this leads to the determination that pre-images of vertex positions are zeros of associated tau functions. We investigate the special case of circular arc quadrilaterals first and in more detail. The isomonodromic tau function then is related to the Painlevé VI transcendent and to certain correlation functions in conformal field theory, yielding asymptotic expansions for the tau function in terms of the monodromy data. We use these expansions to present explicit examples and discuss why the ‘crowding’ phenomenon is not a hindrance for the new framework. In addition, since Schwarz-Christoffel mappings to polygons emerge as a limit when the curvature of the circular arcs goes to zero, we reproduce the well known result for the aspect ratio of rectangles as a function of the accessory parameter. Here, the tau function assumes a closed form in terms of Jacobi theta functions – the Picard solution. Moreover, we use tau function asymptotic expansions to calculate the conformal modules of some trapezoids and find perfect agreement with the literature. We conclude with the investigation of mappings to circular arc polygons with any number of sides, and we comment on the numerical implementation for these cases. |
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SILVA, Tiago Anselmo dahttp://lattes.cnpq.br/0539134718084964http://lattes.cnpq.br/8859998369703134CUNHA, Bruno Geraldo Carneiro da2019-09-20T20:48:05Z2019-09-20T20:48:05Z2018-09-13https://repositorio.ufpe.br/handle/123456789/33450Conformal mappings are important mathematical tools in some applied contexts, e.g. electrostatics and classical fluid dynamics. In order to construct a conformal mapping from a canonical simply connected region to the interior of a circular arc polygon with more than three vertices, the accessory parameter problem arises: In general, the mapping is a solution of a differential equation with unknown parameters which hinder its direct integration. Such parameters can be obtained through approximation techniques with relative small computational effort unless the target domain has an elongated aspect, causing the well known difficulty – the ‘crowding’ phenomenon – to emerge. In this thesis, in the pursuit of calculating the accessory parameters as a Riemann-Hilbert problem, we determine them in terms of isomonodromic tau functions and show how to extract the monodromy information from the geometry of the target domain. We also verify that the tau functions satisfy Toda equations, and this leads to the determination that pre-images of vertex positions are zeros of associated tau functions. We investigate the special case of circular arc quadrilaterals first and in more detail. The isomonodromic tau function then is related to the Painlevé VI transcendent and to certain correlation functions in conformal field theory, yielding asymptotic expansions for the tau function in terms of the monodromy data. We use these expansions to present explicit examples and discuss why the ‘crowding’ phenomenon is not a hindrance for the new framework. In addition, since Schwarz-Christoffel mappings to polygons emerge as a limit when the curvature of the circular arcs goes to zero, we reproduce the well known result for the aspect ratio of rectangles as a function of the accessory parameter. Here, the tau function assumes a closed form in terms of Jacobi theta functions – the Picard solution. Moreover, we use tau function asymptotic expansions to calculate the conformal modules of some trapezoids and find perfect agreement with the literature. We conclude with the investigation of mappings to circular arc polygons with any number of sides, and we comment on the numerical implementation for these cases.CAPESCNPqMapas conformes são ferramentas matemáticas importantes em alguns contextos aplicados, e.g. eletrostática e dinâmica de fluidos clássicos. Ao se tentar construir um mapa conforme de uma região simplesmente conexa canônica para o interior de um polígono de arcos circulares com mais de três vértices, surge o problema dos parâmetros acessórios: Em geral, o mapa é uma solução de uma equação diferencial com parâmetros desconhecidos que dificultam a integração da equação, mas podem ser obtidos por técnicas de aproximação. Nesta tese, em busca de calcular os parâmetros acessórios como um problema de Riemann-Hilbert, nós os determinamos em termos de funções tau isomonodrômicas e mostramos como extrair informações sobre a monodromia a partir da geometria do domínio alvo. Também verificamos que as funções tau satisfazem equações de Toda, e isto permite a determinação de que as pré-imagens das posições dos vértices são zeros de funções tau associadas. Nós investigamos o caso especial dos quadriláteros de arcos de círculo primeiro e mais detalhadamente. Nesta situação, a função tau isomonodrômica é relacionada ao sexto transcendente de Painlevé e a certas funções de correlação em teoria de campos conformes, produzindo expansões assintóticas para a função tau em termos dos dados de monodromia. Nós usamos essas expansões para apresentar exemplos explícitos e discutir por que o fenômeno da aglomeração, que é uma dificuldade para outros métodos e ocorre quando o domínio alvo apresenta um aspecto alongado, não é um empecilho para a nova abordagem. Adicionalmente, como mapas de Schwarz-Christoffel emergem como um limite em que a curvatura dos arcos de círculo vai para zero, nós reproduzimos o resultado conhecido para a razão de aspecto de retângulos em função do parâmetro acessório. Aqui, a função tau assume uma forma fechada em termos de funções theta de Jacobi – a solução de Picard. Além disso, usamos expansões assintóticas da função tau para calcular módulos conformes de trapezóides e encontramos uma concordância perfeita com a literatura. Concluímos com a investigação dos mapas para polígonos de arcos circulares com qualquer número de lados, e comentamos a respeito da implementação numérica para estes casos.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em FisicaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessFísica teóricaMapa ConformeParâmetros AcessóriosDeformação IsomonodrômicaAccessory parameters in conformal mapping: exploiting isomonodromic tau functionsinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPETHUMBNAILTESE Tiago Anselmo da Silva.pdf.jpgTESE Tiago Anselmo da Silva.pdf.jpgGenerated Thumbnailimage/jpeg1213https://repositorio.ufpe.br/bitstream/123456789/33450/5/TESE%20Tiago%20Anselmo%20da%20Silva.pdf.jpga8138a122181efd368cbf802e38601efMD55ORIGINALTESE Tiago Anselmo da Silva.pdfTESE Tiago Anselmo da Silva.pdfapplication/pdf9427388https://repositorio.ufpe.br/bitstream/123456789/33450/1/TESE%20Tiago%20Anselmo%20da%20Silva.pdf027d1a991157ec60946bac22348dfd1aMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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dc.title.pt_BR.fl_str_mv |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
title |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
spellingShingle |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions SILVA, Tiago Anselmo da Física teórica Mapa Conforme Parâmetros Acessórios Deformação Isomonodrômica |
title_short |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
title_full |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
title_fullStr |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
title_full_unstemmed |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
title_sort |
Accessory parameters in conformal mapping: exploiting isomonodromic tau functions |
author |
SILVA, Tiago Anselmo da |
author_facet |
SILVA, Tiago Anselmo da |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/0539134718084964 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/8859998369703134 |
dc.contributor.author.fl_str_mv |
SILVA, Tiago Anselmo da |
dc.contributor.advisor1.fl_str_mv |
CUNHA, Bruno Geraldo Carneiro da |
contributor_str_mv |
CUNHA, Bruno Geraldo Carneiro da |
dc.subject.por.fl_str_mv |
Física teórica Mapa Conforme Parâmetros Acessórios Deformação Isomonodrômica |
topic |
Física teórica Mapa Conforme Parâmetros Acessórios Deformação Isomonodrômica |
description |
Conformal mappings are important mathematical tools in some applied contexts, e.g. electrostatics and classical fluid dynamics. In order to construct a conformal mapping from a canonical simply connected region to the interior of a circular arc polygon with more than three vertices, the accessory parameter problem arises: In general, the mapping is a solution of a differential equation with unknown parameters which hinder its direct integration. Such parameters can be obtained through approximation techniques with relative small computational effort unless the target domain has an elongated aspect, causing the well known difficulty – the ‘crowding’ phenomenon – to emerge. In this thesis, in the pursuit of calculating the accessory parameters as a Riemann-Hilbert problem, we determine them in terms of isomonodromic tau functions and show how to extract the monodromy information from the geometry of the target domain. We also verify that the tau functions satisfy Toda equations, and this leads to the determination that pre-images of vertex positions are zeros of associated tau functions. We investigate the special case of circular arc quadrilaterals first and in more detail. The isomonodromic tau function then is related to the Painlevé VI transcendent and to certain correlation functions in conformal field theory, yielding asymptotic expansions for the tau function in terms of the monodromy data. We use these expansions to present explicit examples and discuss why the ‘crowding’ phenomenon is not a hindrance for the new framework. In addition, since Schwarz-Christoffel mappings to polygons emerge as a limit when the curvature of the circular arcs goes to zero, we reproduce the well known result for the aspect ratio of rectangles as a function of the accessory parameter. Here, the tau function assumes a closed form in terms of Jacobi theta functions – the Picard solution. Moreover, we use tau function asymptotic expansions to calculate the conformal modules of some trapezoids and find perfect agreement with the literature. We conclude with the investigation of mappings to circular arc polygons with any number of sides, and we comment on the numerical implementation for these cases. |
publishDate |
2018 |
dc.date.issued.fl_str_mv |
2018-09-13 |
dc.date.accessioned.fl_str_mv |
2019-09-20T20:48:05Z |
dc.date.available.fl_str_mv |
2019-09-20T20:48:05Z |
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://repositorio.ufpe.br/handle/123456789/33450 |
url |
https://repositorio.ufpe.br/handle/123456789/33450 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Fisica |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.source.none.fl_str_mv |
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UFPE |
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UFPE |
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Repositório Institucional da UFPE |
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Repositório Institucional da UFPE |
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