The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

The aim of this thesis is Russell's analysis of Peano arithmetic. This analysis was presented by Russell and A. N. Whitehead in the book Prin- cipia Mathematica and then in Russell's book Introduction to Mathema- tical Philosophy which provided this approach in a more accessible form. The thesis focuses on Russell's critique of original Peano axioms and his effort to use only logical definitions instead of axioms. Another goal of the thesis is Russell's theory of classes and substitution of classes by propositional functions. Furthermore, the type theory for propositional functions is introduced and explained. All is converted into present-day logical notation. Moreover, the non-standard models of Russell's Peano arithmetic are studied. Finally, there are two particular arithmetic exam- ples illustrating the purposes of the thesis. Key words Bertrand Russell, Peano Arithmetic, classes, propositional funkction

Στην παρούσα διπλωματική εργασία παρουσιάζεται μια μελέτη τυπωμένων κεραιών ευκλείδειας, αλλά και φράκταλ, κυρίως, γεωμετρίας, για ασύρματες τερματικές συσκευές. Συγκεκριμένα μελετώνται για πρώτη φορά η φράκταλ καμπύλη του Hilbert, του Peano καθώς και η ευκλείδεια γεωμετρία σχήματος Μαιάνδρου, ως γεωμετρία για κεραία σε μια τυπωμένη διάταξη, τέτοια ώστε να χρησιμοποιείται το επίπεδο γείωσης των ασύρματων συσκευών σαν ένα τμήμα του ακτινοβολούμενου συστήματος. Στα πλαίσια της διπλωματικής εξετάζεται η ικανότητα σμίκρυνσης και παραγωγής πολυσυχνοτικής συμπεριφοράς των παραπάνω γεωμετριών, αν χρησιμοποιηθούν ως κεραίες, καθώς και τα χαρακτηριστικά εμπέδησης και ακτινοβολίας τους. Επιπλέον διερευνάται ο ρόλος του επιπέδου γείωσης στην ακτινοβολία του συστήματος, η κατανομή του ρεύματος στα διάφορα μέρη της διάταξης καθώς και η μεταβολή των διαγραμμάτων ακτινοβολίας των κεραιών καθώς αλλάζει η συχνότητα συντονισμού της κάθε κεραίας. Επίσης εξετάζεται ο αριθμός των μπαντών που συντονίζεται η κάθε κεραία ,το κατά πόσο τα χαρακτηριστικά ακτινοβολίας και εμπέδησης παραμένουν αμετάβλητα σε κάθε μπάντα, αλλά και αν η κάθε μπάντα χρησιμοποιείται από διάφορες εφαρμογές, όπως η κινητή τηλεφωνία η τα ασύρματα δίκτυα. Στη συνέχεια οι διατάξεις που μελετώνται συγκρίνονται μεταξύ τους και προτείνονται αυτές που υπερέχουν σε κάθε περίπτωση. Τέλος παρουσιάζονται τα συμπεράσματα αλλά και μελλοντικές κατευθύνσεις. Επιπλέον, όσον αφορά το θεωρητικό μέρος της διπλωματικής, παρουσιάζεται συνοπτικά η έννοια της φράκταλ γεωμετρίας και γιατί αυτή προτείνεται ως γεωμετρία για κεραία. Αναφέρονται τα βασικότερα στοιχεία της θεωρίας μικρών κεραιών και συνοψίζονται οι σημαντικότερες δημοσιεύσεις πάνω στις κεραίες Hilbert και Peano. Οι μελέτες αφορούν διάφορα μεγέθη κεραιών σε διάφορες μπάντες λειτουργίας και σε διαφορετικές διατάξεις, δίνοντας μια σαφή εικόνα για την ικανότητα σμίκρυνσης των κεραιών αυτών ,την λειτουργία τους ως ακτινοβολητές αλλά κυρίως για το εάν και πόσο υπερέχουν έναντι απλών ευκλείδειων διατάξεων. / -

Determining the homotopy type of one-dimensional Peano continua has been an open question of some interest. We give a complete invariant of the homotopy type of such continua, which consists of a pair of subspaces together with a relative homology group. Along the way, we describe reduced forms for one-dimensional Peano continua.

In this work, we investigated the construction of natural, integer, rational, real, complex, quaternion and Octonion numbers. More precisely, the set of real numbers was achieved by applying two methods: Dedekind Cuts and Equivalence Classes of Cauchy Sequences. Our study is only based on using Peano Axioms, which are directly related to the natural numbers, in order to get the basic properties satis ed by these numbers. In addition, we carefully proved the elementary results involving real numbers. This process in question was developed constructively throughout of the concepts of the integer and rational numbers. Next, we show that it is possible to establish the existence of complex numbers along with their more usual arithmetic properties. Finally, we nish each chapter of our work showing some possible applications in each set worked. / No presente trabalhos, investigamos, cuidadosamente, a construção do números Naturais, inteiros, Racionais, Reais e Complexos. Sendo que, o conjunto dos números reais foi obtido através dos conhecidos métodos: Cortes de Dedekind e Classes de Equivalência por sequência de Cauchy. O estudo consistiu em utilizar os famosos Axiomas de Peano, ps quais estão relacionados aos números naturais, em ordem a obter as em conhecidas propriedades elementares, satisfeitas para todos esses números. E, a partir deste conhecimento, encontramos rigorosamente as provas dos resultados básicos envolvendo os números reais. Este processo em questão, foi desenvolvida de maneira construtiva através dos números inteiros e racionais. Em seguida, mostramos que é possível estabelecer a existência de números complexos, juntamente com suas propriedades aritméticas mais usuais. Por fim, terminamos cada capítulo do nosso trabalho, mostrando algumas possíveis aplicações em cada conjunto trabalhado.

Este trabalho apresenta a construção dos conjuntos dos números naturais, inteiros, racionais e reais, buscando contemplar uma mediação entre alunos e professores do ensino médio que possa contribuir em uma abordagem facilitadora para o processo de ensino e aprendizagem. A construção dos conjuntos numéricos é feita de modo progressivo, apresentando leis e propriedades que definem cada um deles. Os capítulos apresentam teoremas que são provados de modo que o leitor possa conseguir, efetivamente, estabelecer um elo entre a teoria matemática e suas abstrações iniciais inerentes aos estudantes em formação. / This work presents the construction of the sets of natural, integer, rational and real numbers, aiming to contemplate a mediation between high school students and teachers that can contribute to an easy approach to the teaching and learning processes. The construction of the numerical sets is done progressively presenting laws and properties that define each one of them. The chapters present theorems that are proven so that the reader can effectively establish a link between mathematical theory and its initial abstractions inherent in the students in formation.

En este trabajo, introducimos los conjuntos densificables, una nueva clase de subconjuntos de espacios métricos en los que problemas de optimización global e integración múltiple se pueden reducir a problemas unidimensionales, resolviendo el mismo problema sobre ciertas curvas (llamadas curvas alfa-densas). Para realizar el estudio de los conjuntos densificables en espacios métricos, introducimos las nociones de densificador, pseudo-densificabilidad, aproximabilidad por caminos y aproximabilidad numerable por caminos, que proporcionan propiedades topológicas y métricas de dichos conjuntos. Estos conceptos han permitido caracterizar la clase de subconjuntos densificables de los espacios euclídeos con interior no vacío. Extendemos el concepto de densificabilidad a espacios topológicos en general, introduciendo las nociones de densificabilidad simple, condicional, secuencial y topológica. De esta manera, problemas de optimización global pueden ser simplificados aun en ausencia de una métrica. Además, probamos que una de estas extensiones es óptima, en el sentido que ninguna condición más débil permite la mencionada simplificación utilizando una sucesión prefijada de curvas. Asimismo, comparamos la densificabilidad topológica con la extensión de la densificabilidad ya existente a subconjuntos de espacios vectoriales topológicos. Introducimos la noción de densificabilidad lineal, que combina ventajas de ambos conceptos. Finalmente, presentamos una aplicación de la teoría de curvas alfa-densas al cálculo de la dimensión logarítmica.

Orientador: Prof. Dr. Vinicius Cifú Lopes / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016. / Neste trabalho apresentaremos uma visão sobre os princípios das curvas planas. Iniciamos o desenvolvimento dos estudos com as cônicas: parábola, elipse e hipérbole que são aplicadas no Ensino Médio normalmente usando equações cartesianas. Abordaremos o assunto destas e outras curvas usando equações paramétricas, com intuito de mostrar a vantagem de utilizá-las. Abrangeremos em nossos estudos a catenária, a cicloide e a curva de Bézier, curvas as quais não são estudadas no Ensino Básico, mas poderiam ser apresentadas como um desafio motivador ao estudo da Matemática, explorando suas várias aplicações que acontecem de maneira natural em nosso cotidiano. Apresentaremos propriedades gerais das curvas como: continuidade, parametrização, comprimento de arco, curva suave, curvatura e outras, além de realizar a demonstração do teorema fundamental das curvas planas e para finalizar estudaremos uma curva exótica, conhecida como curva de preenchimento de espaço, construída pela primeira vez pelo matemático italiano Giuseppe Peano. / In this work we will present an insight into the principles of flat curves. We start with the conics: parabola, ellipse and hyperbole which are applied in high school usually using Cartesian equations. We will discuss those and other curves using parametric equations, in order to show the advantage of using them. We will cover in our studies the catenary, the cycloid and a Bézier curve, curves which are not studied in basic education, but could be presented as a challenging motivation to the study of Mathematics by exploring their various uses that happen naturally in our everyday lives. We will introduce general properties of curves as: continuity, parameterization, arc length, smooth curve, curvature and others, in addition to the proof of the fundamental theorem of plane curves, and finally we will study an exotic curve, known as space-filling curve, built for the first time by the Italian mathematician Giuseppe Peano.

Made available in DSpace on 2014-12-17T14:08:51Z (GMT). No. of bitstreams: 1 DanielTS_DISSERT.pdf: 2906320 bytes, checksum: 0844f1ce3a3b057671d92efe62faa050 (MD5) Previous issue date: 2012-04-06 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work, spoke about the importance of image compression for the industry, it is known that processing and image storage is always a challenge in petrobr?s to optimize the storage time and store a maximum number of images and data. We present an interactive system for processing and storing images in the wavelet domain and an interface for digital image processing. The proposal is based on the Peano function and wavelet transform in 1D. The storage system aims to optimize the computational space, both for storage and for transmission of images. Being necessary to the application of the Peano function to linearize the images and the 1D wavelet transform to decompose it. These applications allow you to extract relevant information for the storage of an image with a lower computational cost and with a very small margin of error when comparing the images, original and processed, ie, there is little loss of quality when applying the processing system presented . The results obtained from the information extracted from the images are displayed in a graphical interface. It is through the graphical user interface that the user uses the files to view and analyze the results of the programs directly on the computer screen without the worry of dealing with the source code. The graphical user interface, programs for image processing via Peano Function and Wavelet Transform 1D, were developed in Java language, allowing a direct exchange of information between them and the user / Neste trabalho, apresenta-se a import?ncia da compress?o de imagens para a ind?stria de petr?leo, sabe-se que o processamento e armazenamento de imagens ? sempre um desafio nas grandes empresas de petr?leo, para otimizar o tempo de armazenamento e armazenar um n?mero m?ximo de imagens e dados. ? exposto algumas ferramentas para o processamento e armazenamento de imagens no dom?nio wavelet. A proposta apresentada baseia-se na Fun??o de Peano e na transformada wavelet 1D. O sistema de armazenamento tem como objetivo a otimiza??o do espa?o computacional, tanto para o armazenamento como para transmiss?o de imagens. Sendo necess?rio para isso, a aplica??o da Fun??o de Peano para linearizar as imagens com m?xima concentra??o de pontos vizinhos e a transformada wavelet 1D para decomp?-la. Estas aplica??es permitem extrair informa??es relevantes para o armazenamento de uma imagem com um menor custo computacional e com uma margem de erro muito pequena quando se compara as imagens, original e processada, ou seja, h? pouca perda de qualidade ao aplicar o sistema de processamento apresentado. Os resultados obtidos a partir das informa??es extra?das das imagens s?o apresentados numa interface gr?fica. ? atrav?s da interface gr?fica que o usu?rio visualiza as imagens e analisa os resultados do programa diretamente na tela do computador sem a preocupa??o de lidar com os c?digos fontes. A interface gr?fica, os programas de processamento de imagens via Fun??o de Peano e a TransformadaWavelet 1D foram desenvolvidos em linguagem java, possibilitando uma troca direta de informa??es entre eles e o usu?rio

Made available in DSpace on 2015-05-15T11:46:22Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1916558 bytes, checksum: 2a74bd2ee59f2f8ed1aa50acfcc283c4 (MD5) Previous issue date: 2014-12-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is divided in two subjects. The first concerns about the Bohnenblust-Hille and Hardy- Littlewood multilinear inequalities. We obtain optimal and definitive generalizations for both inequalities. Moreover, the approach presented provides much simpler and straightforward proofs than the previous one known, and we are able to show that in most cases the exponents involved are optimal. The technique used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this thesis to improve the constants for vector-valued Bohnenblust-Hille type inequalities. The second subject has as starting point the existence of Peano spaces, that is, Haurdor spaces that are continuous image of the unit interval. From the point of view of lineability we analyze the set of continuous surjections from an arbitrary euclidean spaces on topological spaces that are, in some natural sense, covered by Peano spaces, and we conclude that large algebras are found within the families studied. We provide several optimal and definitive result on euclidean spaces, and, moreover, an optimal lineability result on those special topological vector spaces. / Este trabalho édividido em dois temas. O primeiro diz respeito às desigualdades multilineares de Bohnenblust-Hille e Hardy-Littlewood. Obtemos generalizações ótimas e definitivas para ambas desigualdades. Mais ainda, a abordagem apresentada fornece demonstrações mais simples e diretas do que as conhecidas anteriormente, além de sermos capazes de mostrar que os expoentes envolvidos são ótimos em varias situações. A técnica utilizada combina ferramentas probabilísticas e interpolativas; esta ultima e ainda usada para melhorar as estimativas das versões vetoriais da desigualdade de Bohnenblust-Hille. O segundo tema possui como ponto de partida a existência de espaços de Peano, ou seja, os espaços de Hausdor que são imagem contínua do intervalo unitário. Sob o ponto de vista da lineabilidade, analisamos o conjunto das sobrejecoes contínuas de um espaço euclidiano arbitrário em um espaço topológico que, de certa forma, e coberto por espaços de Peano, e concluímos que grandes álgebras são encontradas nas famílias estudadas. Fornecemos vários resultados ótimos e definitivos em espaços euclidianos, e, mais ainda, um resultado de lineabilidade ótimo naqueles espaços vetoriais topológicos especiais.