Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 1980 / Vita. / Bibliography: leaves 50-51. / by Jeanne Duflot. / Ph. D. / Ph. D. Massachusetts Institute of Technology, Department of Mathematics

The notation of t/i numbers is used to describe knots and links on the double torus in two different ways, as a step toward the eventual classification of double-torus links.

Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths. / Libyan Cultural Affairs

Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator. / QC 20101108

Active Galactic Nuclei (AGN) are compact regions in the centers of some galaxies. They emit significantly in the whole range of the electromagnetic spectrum and show variability at different timescales. Observational evidence suggests the presence of a dusty torus obscuring the central radiation source of AGN. According to the Unified Model the observed general properties of AGN emission can be understood on the basis of orientation of this torus toward an observer. Two main types of AGN are distinguished: Type 1, with detected emission from the inner torus cavity viewed pole-on, and Type 2, viewed through the obscuring torus. There are numerous attempts in the past decade to model the emission from the torus, considering a homogeneous distribution of dust. However, important problems in explaining the observations still remain unsolved: it is hard to suppress the 10 m emission feature of silicate dust for a pole-on view and at the same time produce an absorption feature for an edge-on viewed torus; despite the huge optical depths inferred from X-ray observations of Type 2 sources, the observed absorption feature is shallow. Unlike observations, models of homogeneous tori with large optical depths always produce deep absorption feature. While it is realized that dust contained in clumps would resolve these issues, modeling of a clumpy medium poses a serious computational challenge. We are the first to incorporate clumps in our model of a dusty torus and to successfully explain the infrared emission from AGN. We model two types of clouds: directly illuminated by the AGN and diffusely heated by other clouds. We calculate the emission of the first type as angle-averaged emission from a dusty slab. The second type of clouds is modeled as dusty spheres embedded in the radiation field of the directly heated clouds. The radiative transfer problem for a dusty slab and externally heated sphere is solved exactly with our code DUSTY. The overall emission of the torus is found by integration over the spatial distribution of clouds. We find a very good agreement of our model results with observations. Comparison with them can constrain the physical conditions in the AGN dusty tori.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Atualmente, existe uma tendÃncia mundial para a busca de novas fontes de energia com a menor agressÃo possÃvel ao meio ambiente e ao homem, principalmente com vistas à geraÃÃo de eletricidade. No Brasil, de acordo com o banco de informaÃÃo de geraÃÃo da AgÃncia Nacional de Energia ElÃtrica - ANEEL, a geraÃÃo por fonte eÃlica jà totaliza cerca de 2.788 MW de potÃncia em operaÃÃo. AlÃm dos jà instalados, 163 parques eÃlicos estÃo em construÃÃo e mais 127 foram outorgados pelo governo, nos diversos estados brasileiros. Este cenÃrio, favorÃvel ao desenvolvimento de energia eÃlica, leva a diversos trabalhos na Ãrea, relacionados tanto à previsÃo de vento, como a novas mÃquinas elÃtricas para aerogeradores. Considerando esta tendÃncia, a presente dissertaÃÃo visa o projeto de um gerador elÃtrico, com potÃncia de 50 kW, utilizando-se ÃmÃs permanentes de NeodÃmio-Ferro-Boro, para aplicaÃÃo na geraÃÃo de eletricidade a partir da energia eÃlica. A topologia escolhida para o gerador à a da mÃquina de Torus, caracterizada por ser simples, compacta e de alta eficiÃncia. A modelagem 3d eletromagnÃtica para anÃlises dos campos no gerador elÃtrico à feita atravÃs do software comercial COMSOL MultiphysicsÂ, que utiliza o MÃtodo de Elementos Finitos (MEF) para discretizar o domÃnio estudado. / Today, there is a worldwide trend in the search for new sources of energy with the lowest possible environmental pollution and the minimum hazard for the human health, mainly due to the generation of electricity. In Brazil, according to the information database for electricity generation from AgÃncia Nacional de Energia ElÃtrica - ANEEL, the energy production through wind power totalizes about 2.788 MW of power in operation now. Additionally, 163 wind farms are under construction and 127 more were allowed by the government for all the Brazilian states. This scenery, favorable to the development of wind energy, leads to several works of research in the area, related to both wind forecast and the new electric machines for wind turbines. Considering such tendency, this dissertation aims to project an electric generator with the power of 50kW, using Neodymium-Iron-Boron permanent magnets, to be exploited in generating electricity from wind energy. The chosen topology for the generator is that of the Torus machine, characterized by being simple and compact as well as by having a high efficiency. The electromagnetic modeling for analysis of the electric field generator is made by employing the commercial software COMSOL MultiphysicsÂ, which uses the Finite Element Method (FEM) to discretize the studied domain.

Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of automorphisms on the covering space is equivalent to the original surface.

Quantum systems with a finite Hilbert space, where position x and momen- tum p take values in Z(d) (integers modulo d), are studied. An analytic representation of finite quantum systems is considered. Quantum states are represented by analytic functions on a torus. This function has exactly d zeros, which define uniquely the quantum state. The analytic function of a state can be constructed using its zeros. As the system evolves in time, the d zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, for various Hamiltonians, are given. In addition, for given paths ³n(t) of the d zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum systems are considered. Special cases where M of the zeros follow the same path are also studied, and general ideas are demonstrated with several ex- amples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It is evidenced within the study that a small perturbation of the initial values of the zeros splits a path with multiplicity M into M different paths.

Um dos teoremas conhecidos de Poincaré afirma: Seja f um homeomorfismo do círculo que preserva orientação. Se p/q, com mdc(p, q) = 1, é o número de rotação de f, então f possui um ponto periódico de período q. Quando o conceito de número de rotação para um homeomorfismo do círculo é generalizado para um homeomorfismo f : T2 ? T2 homotópico à identidade, o resultado é um subconjunto convexo do plano R2, chamado conjunto de rotação e é denotado por ½(F) onde F é um levantamento de f. No caso que ½(F) tem interior não vazio, J. Franks obteve resultados análogos ao Teorema de Poincaré. Nesta dissertação estudamos um resultado análogo, obtido por Jonker e Zhang, quando ½(F) não tem interior. Mais precisamente: assumimos que ½(F) é um segmento de reta com inclinação irracional e mostramos que se 1 n(p1, p2) ? ½(F), com mdc(p1, p2, n) = 1, então f possui um ponto periódico de período n / One of the well know results of Poincaré state: Let f be an orientation preserving circle homeomorphism. If p/q, with mdc(p, q) = 1, is the rotation number of f, then there is a periodic point for f whose period is q. When the concept of rotations number, for orientation preserving circle homeomorphism, is generalized for torus homeomorphism f : T2 ? T2 that are homotopic to the identity, it results in a convex subset of R2, called rotation set and is denoted by ½(F) where F is a lifting of f. In the case that ½(F) has non-empty interior, J. Franks proved similar results to the Poincaré Theorem. In this work, when ½(F) has empty interior, we study an similar result obtained by Jonker and Zhang. More precisely: they suppose that the rotation set ½(F) is a line segment with irrational slope and demonstrate that if 1 n(p1, p2) ? ½(F), with mdc(p1, p2, n) = 1, then f has a periodic point of period n

O toro é uma excrescência óssea convexa, com crescimento lento, considerado de origem congênita, benigno, também denominado de exostose, com pouca significância clínica, sendo necessária sua remoção quando dificulta a confecção de prótese totais, parciais ou removíveis e a posição da língua, interferindo na fala ou ainda, dificultam a intubação cirúrgica. O objetivo deste trabalho foi analisar, por meio da eletromiografia, o padrão de comportamento (morfológico e funcional) da atividade dos músculos masseter e temporal nas condições posturais da mandíbula como o Repouso, Protrusão, Máxima Interscupidação Habitual (MIH), Apertamento Dental com Parafilme M®, Lateralidades Direita e Esquerda. O presente estudo também propôs analisar a eficiência dos ciclos mastigatórios e para tanto, foi utilizado a avaliação da mastigação habitual e não habitual por meio da integral da envoltória do sinal eletromiográfico obtido dos músculos masseter e temporal durante a mastigação habitual de uvas passas e amendoim e da mastigação não-habitual de Parafilme M®, considerando a presença ou não do toro mandibular. Diante dos resultados obtidos no presente trabalho foi possível concluir que, de um modo geral, houve aumento na atividade eletromiográfica nos músculos masseteres de indivíduos portadores de toro mandibular (Grupo Casos) quando comparado aos indivíduos do Grupo Controles nas condições clínicas de protrusão, lateralidades direita e esquerda, máxima intercuspidação habitual (MIH), apertamento dental com parafilme M®. Na mastigação com parafilme M®, mastigação com amendoins e mastigação com uvas passas verificou-se aumento da atividade para os músculos temporais e masseteres, evidenciando que a presença de toro mandibular está associada a alterações no padrão de ativação da musculatura mastigatória nas condições posturais da mandíbula e na mastigação. / The torus is a convex bony outgrowth, considered congenital, benign, also called exostosis, with little clinical significance, necessitating its removal difficult when the making of complete dentures, removable partial or tongue position and interfering speech or hinder surgical intubation. The aim of this study was to analyze, using electromyography, the default behavior (morphological and functional) of activity of the masseter and temporal muscles in conditions mandible posture as Home, protrusion, Maximum Interscupidação Habitual (MIH), Dental Clenching with Parafilm M®, lateral excursion (right and left). This study also proposed to analyze the efficiency of the mastigatory cycles. Therefore, we used the evaluation of usual mastication and unusual mastication through the envoltory integral of the EMG signal obtained from the masseter and temporalis muscles during mastication of raisins and peanuts and unusual chewing Parafilm M®, considering the presence or no in mandibular torus. Given the results obtained in this study it was concluded that, in general, increased the EMG activity in the masseter muscles of patients with mandibular torus (Group Case) when compared to individuals of the Group Controls in the mandible clinical conditions as protrusion, lateral excursion (right and left), maximum intercuspation (MIH), clenching with parafilm M®. There was an increase in EMG activity of the temporal and masseter muscles during mastication with parafilm M®, mastication with peanuts and mastication with raisins, showing that the presence of mandibular torus is associated with changes in the default activation of masticatory muscle in conditions mandible posture and chewing.