Matemática

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http://repositorio.ufes.br/handle/10/17495
Programa de Pós-Graduação em Matemática

Centro: CCE
Telefone: (27) 4009 2474
URL do programa: http://www.matematica.ufes.br/pos-graduacao/PPGMAT

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    A Transformada de Fourier para o Laplaciano Generalizado
    (Universidade Federal do Espírito Santo, 2024-03-05) Ramos Junior, Jomar Ferreira; Valentim, Fábio Júlio da Silva; https://orcid.org/0000-0003-2405-7696; http://lattes.cnpq.br/8745134398831488; https://orcid.org/0009-0006-1032-5169; http://lattes.cnpq.br/1800635453022041; Aranda, José Miguel Mendoza; https://orcid.org/; http://lattes.cnpq.br/8615067875072268; Silva, Jean Carlos da; https://orcid.org/; http://lattes.cnpq.br/9490078990099931
    This academic dissertation aims primarily to contribute to the enhancement of understanding of the Fourier Theory applied to the generalized Laplacian. The proposed methodology involves the construction of an orthonormal basis of eigenfunctions for the operator, based on the appropriate choice of Green’s functions. The central problem consists of finding the solution u(x) that satisfies certain boundary conditions for the equation Lu = f, using a series representation of the eigenfunctions of the operator L. The dissertation addresses fundamental aspects such as the definition of the domain of the generalized Laplacian, the analysis of Green’s functions and their applications in solving partial differential equations, as well as transformations for the generalized Laplacian. The interest in consolidating the Fourier Theory for the generalized Laplacian aims to provide a deeper understanding of the properties of this operator and its relation to Fourier Theory, establishing a foundation for future research, including more complex cases such as the differential operator in reverse order. This work represents a significant contribution to the understanding of the theory of the generalized Laplacian and its connections with Fourier Theory.
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    Sobre a geometria Lipschitz de polinômios quase-homogeneos
    (Universidade Federal do Espírito Santo, 2024-10-21) Aquino Neto, Gabriel da Macena de; Câmara, Leonardo Meireles; https://orcid.org/0000-0002-4637-8573; http://lattes.cnpq.br/9240898305551070; https://orcid.org/0009-0002-3889-6044; http://lattes.cnpq.br/0868237399371299; Silva, Thiago Filipe da; https://orcid.org/0000-0002-3152-0987; http://lattes.cnpq.br/5049713215002090; Fernandes, Alexandre César Gurgel ; https://orcid.org/0000-0001-7846-0312; http://lattes.cnpq.br/8791056897839415
    In this work, we will show how to determine, in a general context, whether two real quasi-homogeneous polynomials in two variables with weights ϖ = (p,q) are R-semialgebraically Lipschitz equivalent. Initially, we characterize the Lipschitz equivalence of real polynomial functions of one variable by comparing the values and also the multiplicities of the polynomial functions at their critical points. Sub sequently, under general conditions, we will reduce the problem of R-semialgebraic Lipschitz equivalence of quasi-homogeneous polynomials in two variables to the pro blem of Lipschitz equivalence of real polynomial functions of one variable. As an application of the theory developed throughout this dissertation, we will analyze the properties, in the context of R-semialgebraic Lipschitz equivalence, of a specific fa mily of quasi-homogeneous polynomials considered in [9, Henry and Parusinski], to show that the bi-Lipschitz equivalence of germs of analytic functions (R2,0) → (R,0) admits continuous moduli. Consequently, the R-semialgebraic Lipschitz equivalence of real quasi-homogeneous polynomials in two variables also admits continuous mo duli. Finally, we explore the possibility of simplifying the classification space of quasi-homogeneous polynomials.