Licenciatura em Matemática (Sede)
URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/24
Siglas das Coleções:
APP - Artigo Publicado em Periódico
TAE - Trabalho Apresentado em Evento
TCC - Trabalho de Conclusão de Curso
Navegar
8 resultados
Resultados da Pesquisa
Item O floco de neve de Koch e suas propriedades: funções contínuas sem derivada em ponto algum(2024-07-31) Santos, Vivian Maria dos; Clemente, Rodrigo Genuino; http://lattes.cnpq.br/4351609162717260; http://lattes.cnpq.br/0771390443429539In this work, we present the existence of real continuous functions that have no derivative at any point. For this, we use the function developed by the mathematician Helge von Koch as an example, demonstrating that this function is continuous at all points but not differentiable at any point. We show how this curve is constructed and discuss itsproperties. To highlight these facts, many constructions of such functions are based on infinite series of functions. Therefore, we introduce some fundamental concepts and results from Mathematical Analysis, specifically, Sequences and Series of functions, which allow us to investigate the continuity and differentiability properties. Finally, we will comment on an interesting result that reveals that the set of these functions constitutes a dense and residual set in the complete metric space, meaning that these functions exist abundantly. The proof of this statement is based on Baire’s Theorem, which generally states that any countable union of thin sets is so small that its complement is dense.Item 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/9012501383942232Through 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.Item 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/8761735112729494This 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.Item Introdução à compressão fractal de imagens através de sistemas de funções iteradas(2023-05-12) Silva, Maria Fernanda Pires da; Silva, Tarciana Maria Santos da; http://lattes.cnpq.br/1650180237175460; http://lattes.cnpq.br/4722608617162314The study object of this work is the fractal image compression method through systems of iterated functions. This technique consists of describing, through affine transformations, fractals that have a special characteristic: self-similarity. To understand this method of compression, we make a brief explanation about fractal geometry, start a study on linear transformations and define affine transformations in the plane. Then, we focus on the concepts of Metric Spaces necessary for understanding Banach’s Fixed Point Theorem, which is the key for the application of systems of iterated functions in the construction of self-similar fractals. We present the Hausdorff distance, as it is used in the compression of real images that have little or no similarity and, finally, we show the application in practice by building two very important fractals: the Sierpinski Triangle and the Sierpinski Carpet.Item Um estudo sobre completude e compacidade em espaços métricos(2019-12-18) Silva, Hugo Henryque Coelho e; Araújo, Yane Lísley Ramos; http://lattes.cnpq.br/6642941380570085; http://lattes.cnpq.br/1324983852661350In this work we will present a study about the theory of complete and compact metric spaces. Initially, we will cover some basic concepts related to the theory of metric spaces, continuity and sequences in metric spaces. Next, we will list a motivation for the study of the theory of complete metric spaces, some of their properties and valid results in these spaces, such as Baire’s theorem and Banach’s fixed point theorem as well as some of its applications. Finally, we will present a study about the theory of compact metric spaces, addressing its general properties and some important results of the mathematical analysis that are valid in these spaces, as we can mention Riesz theorem and Ascoli-Arzelá theorem.Item 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/4592972030162451This 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.Item 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/2572639684291501This 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.Item O Teorema de Baire e suas aplicações na Análise Funcional(2019-07-29) Albuquerque, Anne Caroline Carneiro de; Silva, Clessius; http://lattes.cnpq.br/2401078773322406; http://lattes.cnpq.br/5028414803312146In this work, we will present the applications of Baire’s Theorem to Functional Analysis, more specifically to the classic results of this area, Banach Steinhaus Theorem, Open Application Theorem and Closed Graph Theorem.