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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2017.tde-11072017-170308
Document
Author
Full name
Patricia Tempesta
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Manoel, Miriam Garcia (President)
Baptistelli, Patricia Hernandes
Garcia, Ronaldo Alves
Labouriau, Isabel Salgado
Tari, Farid
Title in English
Simmetries in binary differential equations
Keywords in English
Binary differential equation
Compact Lie group
Equivariant quadratic 1-form
Representation theory
Symmetry
Abstract in English
The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it.
Title in Portuguese
Simetrias em equações diferenciais binárias
Keywords in Portuguese
1-forma quadratica equivariante
Equação diferencial binária
Grupo de Lie compacto
Simetria
Teoria de representação
Abstract in Portuguese
O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.
 
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Publishing Date
2017-07-12
 
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