On étudie le groupe modulaire G du plan privé d'un ensemble de Cantor et les classes de Brouwer du groupe modulaire du plan privé de Z. Ces objets apparaîssent naturellement en dynamique topologique sur les surfaces. Dans le premier chapitre, on s'intéresse au groupe G et à son action sur le graphe des rayons, qui est un analogue déni par Danny Calegari du complexe des courbes pour le plan privé d'un ensemble de Cantor. En particulier, on montre que ce graphe est de diamètre infini et hyperbolique. On utilise ensuite l'action de G sur ce graphe hyperbolique pour exhiber un quasi-morphisme non trivial explicite sur G et pour montrer que le deuxième groupe de cohomologie bornée de G est dedimension infinie. Enfin, on donne un exemple d'un élément hyperbolique de G dont la longueur stable des commutateurs est nulle. Dans le second chapitre, on développe de nouveaux outils pour la théorie de Brouwer homotopique. En particulier, on décrit un ensemble canonique de droites de réduction, l'ensemble des murs, qui sépare le plan en zones de translation maximales et en zones irréductibles. On se restreint ensuite au cas des classes de Brouwer relativement à quatre orbites, et on les décrit explicitement en ajoutant au diagramme de Handel et à l'ensemble des murs un emmêlement, qui est essentiellement une classe d'isotopie de courbes sur le cylindre privé de deux points. / We study the mapping class group G of the complement of a Cantor set in the plane and the Brouwer mapping classes of the mapping class group of the complement of Z in the plane. These objects arise naturally in topological dynamics on surfaces. In the first chapter, we study the group G and its action on the ray graph, which is the analog dened by Danny Calegari of the complex of curves for the complement of a Cantor set in the plane. In particular, we show that this graph has infinite diameter and is hyperbolic. We use the action of G on this graph to find an explicit non trivial quasimorphism on G and to show that this group has infinite dimensional second bounded cohomology. We give an example of a hyperbolic element of G with vanishing stable commutator length. In the second chapter, we give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set, the set of "walls", which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relatively to four orbits and describe them explicitly by adding to Handel's diagram and to the set of walls a "tangle", which is essentially an isotopy class of simple closed curves in the cylinder minus two points.

L'objectif du présent texte est de tenter de construire un modèle de l'acquisition des concepts mathématiques sans l'aide du langage en s'inspirant des thèses intuitionnistes de L.E.J. Brouwer et en les appliquant à des théories plus modernes de l'acquisition et de la représentation des concepts mathématiques, notamment, la théorie du sens des nombres de Stanislas Dehaene. Pour ce faire, nous initierons le lecteur à la pensée de Brouwer dans les deux premiers chapitres et développerons dans le troisième chapitre une nouvelle analyse de l'Intuition Primordiale de Brouwer dans laquelle il est possible d'identifier chaque élément impliqué dans l'acquisition des concepts mathématiques chez Brouwer et le rôle joué par chacun. Le chapitre quatre exposera la théorie de Dehaene selon laquelle nos capacités mathématiques sont le résultat de deux systèmes cognitifs de base, soit le système de répertoire d'objets et le système de représentation approximative de la numérosité. Nous tenterons ainsi d'améliorer le modèle de l'ontogenèse de Dehaene ainsi que les interprétations de Suzanne Carey, celles-ci soulignant le besoin de faire appel à d'autres systèmes de base incluant un système de représentation de l'ordre linéaire ainsi que le langage. Nous présenterons une hypothèse alternative en remarquant qu'une représentation approximative de la numérosité peut implicitement contenir une forme d'ordre et de relation de successeur capable de justifier une partie de l'apprentissage des listes de numéros chez les enfants. Aussi, nous tenterons d'expliquer le développement de concepts mathématiques plus avancés en nous basant sur la théorie de l'abstraction de Lawrence Barsalou, selon laquelle l'abstraction est une interprétation dynamique faite par des systèmes de symboles perceptuels. Nous pourrons alors expliquer la manière dont le système de répertoire d'objets réussit à développer une composante numérique en venant interpréter les représentations du système approximatif. Nous suggérerons enfin l'hypothèse que les concepts mathématiques puissent être aptes à se développer à mesure que ce système de répertoire d'objets produit des représentations structurées dans lesquelles les numérosités se voient regroupées de manière analogue au développement des nombres par l'IP de Brouwer. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Intuitionnisme, Abstraction, Concepts mathématiques, Fondements des mathématiques.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-18Bitstream added on 2014-06-13T20:16:07Z : No. of bitstreams: 1 000753998.pdf: 885291 bytes, checksum: e06a634b31c2012fc0b6d5e72ec13aa3 (MD5) / Este trabalho é espelhado no livro “Teoria do Índice” [1] de Daciberg Lima Gonçalves e José Carlos de Souza Kiihl, publicado em 1983 no 14o Colóquio Brasileiro de Matemática pelo IMPA. Para a leitura deste trabalho é necessário uma familiaridade prévia com Topologia Algébrica, na qual indicamos [2] e [3] para consulta. Inicialmente apresentaremos alguns pré-requisitos algébricos e topológicos necessários para o desenvolvimento do trabalho e a seguir estudaremos: pontos fixos de aplicações contínuas de X em X, em que X é um espaço topológico; Grau de Brouwer de aplicações contínuas de Sn em Sn (ou respectivamente (Bn+1; Sn) em (Bn+1; Sn)); Grau Local de uma aplicação contínua f de V em Sn em torno de um ponto Q 2 Sn, em que V Sn é um aberto e f1(Q) é um compacto e Índices dos Pontos Fixos de uma aplicação contínua de V em Sn, em que V Rn é um aberto / This work is based on the book titled “Teoria do Índice” [1] by Daciberg Lima Gonçalves and José Carlos de Souza Kiihl , published in 1983 in the 14o Brazilian Math Colloquium held by IMPA . In order to perform the reading of this work, a basic acquaintance from the algebraic topology is needed, on which we can indicate the following [2] and [3] references. Firstly, for the development of the work, some previous necessary algebraic and topological requirements are shown and the next topics will be studied: fixed points of continuous maps from X to X, where X is a topological space, Brouwer’s degree of continuous maps from Sn to Sn ( or respectively (Bn+1; Sn) to (Bn+1; Sn)), Local Degree of continuous maps from V to Sn around a point Q 2 Sn, where V Sn is an open set and f1(Q) is a compact set and Fixed Points Index of continuous maps from V to Sn, where V Rn is an open set

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Made available in DSpace on 2014-06-11T19:28:20Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-06-24Bitstream added on 2014-06-13T20:57:54Z : No. of bitstreams: 1 franco_lj_me_ia.pdf: 1858255 bytes, checksum: 7073124a61dd78b45b49278de09a7755 (MD5) / O presente trabalho disserta sobre hibridismos em música, sobretudo na obra violonística do cubano Leo Brouwer. Foi feita uma discussão de caráter geral sobre os diversos usos de hibridismos presentes em algumas músicas de compositores latinoamericanos. Para tanto foram abordados alguns autores dos chamados estudos culturais, por tratarem especificamente desse tema, ainda que usem outros termos ou palavras. Com isso pretende-se dar aos intérpretes de violão subsídios baseados na diversidade cultural, trazendo a interpretação como um campo de possibilidades. Por fim, a pesquisa apresentou elementos diversificados presentes na música de Brouwer, o que a caracteriza como uma prática híbrida / This dissertation is about hybridity in music, focusing on the work of the Cuban guitarist Leo Brouwer. The various ways hybridity is present in some songs of Latin American composers were discussed here. Therefore, some authors from what is called cultural studies were addressed, by specifically refer to this topic, albeit using other terms or words. This is intended to provide subsidies to guitar interpreters based on cultural diversity, bringing interpretation as a field of possibilities. Lastly, the research presented several elements found in Brouwer’s music that characterizes it as a hybrid practice

Submitted by Alison Vanceto (alison-vanceto@hotmail.com) on 2016-10-14T13:23:56Z No. of bitstreams: 1 DissRFG.pdf: 920064 bytes, checksum: c3dc1a69ce4522535a44ff78ac0ecec6 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:10:57Z (GMT) No. of bitstreams: 1 DissRFG.pdf: 920064 bytes, checksum: c3dc1a69ce4522535a44ff78ac0ecec6 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:11:05Z (GMT) No. of bitstreams: 1 DissRFG.pdf: 920064 bytes, checksum: c3dc1a69ce4522535a44ff78ac0ecec6 (MD5) / Made available in DSpace on 2016-10-21T13:11:13Z (GMT). No. of bitstreams: 1 DissRFG.pdf: 920064 bytes, checksum: c3dc1a69ce4522535a44ff78ac0ecec6 (MD5) Previous issue date: 2015-08-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this work, we will show an important tool of nonlinear analysis, which has great applicability in partial differential equations: the topological degree theory. We will construct the topological degree in finite and infinite dimensions and show its main properties. Through this theory we will prove existence of solutions for two nonlinear elliptic problems with Dirichlet's boundary conditions, which were studied in [8]. To make topological degree be applicable to such problems, it will be of great importance obtain a-priori estimatives for possible solutions of these problems. To this end, we'll use inequalities of Hardy-Sobolev's type. / Neste trabalho, vamos apresentar uma importante ferramenta da análise não linear, que tem grande aplicabilidade em equações diferenciais parciais: a teoria do grau topológico. Construiremos o grau topológico em dimensões finita e infinita e apresentaremos suas principais propriedades. Através dessa teoria, vamos provar a existência de soluções de dois problemas elípticos não lineares com condição de fronteira de Dirichlet, os quais foram estudados em [8]. Para que a técnica do grau topológico torne-se aplicável a tais problemas, ser a de grande importância a obtenção de estimativas a priori para as possíveis soluções destes problemas. Para tanto, usaremos desigualdades do tipo Hardy-Sobolev.

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[pt] Esse trabalho visa demonstrar os lemas de Sperner e aplicá-los nasdemonstrações do teorema de Monsky em Q2 e do teorema do ponto fixo deBrouwer em R2. Além disso, relatamos como esses lemas foram abordados com alunos da educação básica tendo como ferramenta educacional jogos de tabuleiro. / [en] This work aims to prove the Sperner s Lemmas and to apply them in proving the Monsky s Theorem in Q2 and the Brouwer fixed point Theorem in R2. Moreover, we report how these lemmas were addressed with students in basic education using board games as educational tools.

Leo Brouwer's Sonata Mitología de las Aguas No. I for flute and guitar, first published in 2017, has taken its place as an important twenty-first-century addition to the flute and guitar duo repertory. I provide a brief historical context for the work, followed by preparation guides for guitar alone and duo passages. My preparation guides include exercises and rehearsal strategies, focusing on those passages of the work that include controlled indeterminacy. The study of indeterminacy in music is unusual in the pedagogy of the classical guitarist; this leaves guitarists unprepared for dealing with pieces, especially chamber works, that use improvisation or aleatoric music as a primary element. I take a multifaceted approach to facilitate the realization of the indeterminate sections of the work; this includes demonstrations of my traditional music notation transcriptions and other rehearsal strategies and the application of music performance study systems by James Thurmond and Marcel Tabuteau. This document aims to provide guidance to creating an organic, natural aesthetic in the actualization of Brouwer's groundbreaking work.

This study examines phrase and larger formal structures in the first movement of Leo Brouwer’s Elegio de la Danza (1964), a work that draws on tonal and post-tonal traditions. By adapting key features of the tonal motive, as described by Douglass Green, and the tonal period, as proposed by Green and William Caplin, the model seeks to provide a tool for the discussion of phrases and larger forms in Brouwer’s work. An analysis of primary parameters, such as melody, harmony, and rhythm, provides the means to discuss how the composer articulates beginnings and endings of statements and responses, which are then grouped into antecedent and consequent phrases. These periods articulate large-scale sections, which outline a ternary formal design. Secondary parameters (dynamics, tempo markings, instrumental markings) further contribute to the identification of formal structures in Brouwer’s work.