Licenciatura em Matemática (Sede)

URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/24

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APP - Artigo Publicado em Periódico
TAE - Trabalho Apresentado em Evento
TCC - Trabalho de Conclusão de Curso

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A compacidade em alguns universos topológicos
(2021-07-13) Lima, Alexandre César Bispo; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/4592972030162451
This work aims to study and establish relationships between compact sets and topology, emphasizing the compactness of the unitary closed ball in different contexts. For this, initially we dealt with topological spaces in Chapter 1, we developed basic concepts and tools that will be useful until we get to the central theme of compactness. Then, in Chapter 2, we focus the study on the more particular environment of metric spaces, where we develop the concepts and results in order to end the chapter with a compactness characterization of the unitary closed ball of space. Finally, in Chapter 3, we study how weak and weak* topologies, drawing as a final result the famous Banach- Alaoglu-Bourbaki Theorem, which tells us that the unitary closed ball in the topological dual of a space of Banach is weak* compact.
Espaços métricos: continuidade, completude e compacidade
(2021-02-19) Oliveira, Alessandra Arcanjo Lisboa de; Araújo, Yane Lísley Ramos; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/6642941380570085; http://lattes.cnpq.br/2572639684291501
This work has as main objective to study continuity, completeness and compactness in the theory of metric spaces. A metric space is a non-empty set in which the notion of distance between its elements is well defined. The present study is interesting because the results presented here generalize some of the results observed in the theory of continuity and compactness in Euclidean spaces, Rn, with n [greater than or equal to] 1. Furthermore, these results are valid in more abstract spaces such as some sequence or function spaces, whose notion of distance escapes from intuition and entails intriguing facts, such as the fact that closed balls are not necessarily compact.
Um estudo sobre equações diferenciais ordinárias em dinâmica populacional
(2019-12-14) Franco, Mariana Pereira; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/1514122794309246
In this work we will emphasize a modeling by differential equations for the dynamics between two populations in a predation relation, which is known in the literature as Volterra’s Predator-Prey Model. With this purpose, we will present necessary mathematical tools for a proper problem analysis: a brief study of the solution methods for some ordinary differential equations, the results that underlie them and some aplications; and notions of stability of singularities of autonomous systems.
Uma introdução aos espaços de Lebesgue: completude, separabilidade e reflexibilidade
(2022-06-09) Wanderley, Lucas Rodrigues; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/9012501383942232
Through this work, I aim to study the properties of separability, reflexivity, completeness, and duality of the spaces Lp (X, Σ, μ) with 1 ≤ p ≤ ∞. For this study, in Chapter 1, we will address preliminary concepts that will serve as a basis in demonstrating future results, highlighting the concept of completeness of a metric space and some of its characteristics. Following this, in Chapter 2, we will discuss what a separable and reflexive space entails, as well as present some of their main properties. Lastly, and not least importantly, for the construction of this work, we will present the study carried out on Lebesgue spaces, aiming to verify the properties of completeness, separability, and reflexivity.
Pontos fixos em espaços métricos completos e o Teorema de Picard
(2023-09-21) Lima, Ana Catarine Freitas de; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/8761735112729494
This work aims to deepen the study of complete metric spaces, focusing especially on the analysis of fixed points. Our intention is to demonstrate Banach’s Fixed Point Theorem and, subsequently, apply this theory to ordinary differential equations through Picard’s Theorem. To achieve this objective, we will begin by addressing the fundamental concepts of metric spaces, with an emphasis on understanding the basic elements , offering examples and introducing topological concepts, as well as the notion of continuity. We will conduct the study until we reach the definition of complete metric spaces, and then analyze the notion of fixed point. Finally, we will demonstrate the main theorem, which establishes the existence and uniqueness of solutions to initial value problems in ordinary differential equations.
O teorema da função inversa e sua relação com as superfícies regulares
(2023-09-18) Bezerra, Bruna Vitória Borges; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/7230765885728286
This work’s main objective is to study and establish relationships between the inverse function theorem, which is presented in the Euclidean space Rn, with regular surfaces, in the context of Differential Geometry. We will show how a result arising from the context of mathematical Analysis can serve as a basis for introducing one of the main objects of study in Differential Geometry. For this construction, we will initially address basic concepts involving the topology of the Euclidean space Rn, which will be present throughout the text. Next, we will present the fundamental notions and results about continuity and differentiability in the n-dimensional Euclidean space and finally, we will introduce regular surfaces together with some relevant results to establish a natural and expected relationship with the topology of Euclidean spaces and o Inverse function theorem.