During the last half century there has been a resurgence of interest in Monge's 18th century mass transportation problem, with most of the activity limited to continuous spaces. This thesis, consequently, develops techniques based on mass transportation for the purpose of obtaining tight concentration inequalities in a discrete setting. Such inequalities on n-fold products of graphs, equipped with product measures, have been well investigated using combinatorial and probabilistic techniques, the most notable being martingale techniques. The emphasis here, is instead on the analytic viewpoint, with the precise contribution being as follows. We prove that the modified log-Sobolev inequality implies the transportation inequality in the first systematic comparison of the modified log-Sobolev inequality, the Poincar inequality, the transportation inequality, and a new variance transportation inequality. The duality shown by Bobkov and Gtze of the transportation inequality and a generating function inequality is then utilized in finding the asymptotically correct value of the subgaussian constant of a cycle, regardless of the parity of the length of the cycle. This result tensorizes to give a tight concentration inequality on the discrete torus. It is interesting in light of the fact that the corresponding vertex isoperimetric problem has remained open in the case of the odd torus for a number of years. We also show that the class of bounded degree expander graphs provides an answer, in the affirmative, to the question of whether there exists an infinite family of graphs for which the spread constant and the subgaussian constant differ by an order of magnitude. Finally, a candidate notion of a discrete Ricci curvature for finite Markov chains is given in terms of the time decay of the Wasserstein distance of the chain to its stationarity. It can be interpreted as a notion arising naturally from a standard coupling of Markov chains. Because of its natural definition, ease of calculation, and tensoring property, we conclude that it deserves further investigation and development. Overall, the thesis demonstrates the utility of using the mass transportation problem in discrete isoperimetric and functional inequalities.

Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.

Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the well-studied problem of finding the complete set of torus obstructions.

Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the well-studied problem of finding the complete set of torus obstructions.

Este trabalho foi desenvolvido com o objetivo de estudar a aplicação, para sistema de frenagem veicular, de uma máquina elétrica sem escovas, de armadura toroidal, e fluxo magnético axial produzido por ímãs permanentes de terras raras, a chamada máquina Torus. A máquina foi construída no LMEAE e estudada inicialmente como motor em outro trabalho. Mas, para que se possa avaliar seu funcionamento em sistema de frenagem, o foco é do ponto de vista da máquina como gerador. São realizados testes dinâmicos e estáticos experimentalmente e modelo numérico pelo método dos elementos finitos com um formato de ímãs permanentes de seção setorial, possibilitando o comparativo com a versão anterior da máquina que empregou ímãs de seção quadrada. Mudanças físicas e no sistema de acionamento da máquina, e ensaios de frenagem dinâmica foram realizados. Modelagem analítica para indução magnética foi desenvolvida utilizando-se da técnica de Transformação Conforme. O trabalho busca apresentar as características da máquina e justificativas que demonstram o seu potencial de aplicabilidade em um subsistema veicular sob frenagem regenerativa e a capacidade de fornecimento de energia a um sistema de armazenamento com uma parte de energia cinética, ou seja, baterias e supercapacitores. / This work is carried out with the aim to study the application, by a vehicular braking system, of a brushless electrical machine with a toroidal armature core, and axial magnetic flux delivered by rare earth permanent magnets, the so-called Torus machine. The machine was built in the LMEAE, and previously studied as a motor by other work. However, in order to assess its performance in a braking system, the focus is the point of view of the machine as a generator. Static and dynamic tests are implemented as well as a numerical model by means of the finite element method, in order to compare the machine with sector poles permanent magnets and with square magnet poles. Physical changes and on the driving system of the machine, and dynamic braking tests are performed. The analytical modelling for the magnetic induction was developed using the technique of conformal transformation. The study aims to present the features of the machine and demonstrates its potential applicability to a vehicular subsystem under regenerative braking and the ability to supply an energy storage system with part of the kinetic energy, i.e. batteries and super capacitors.

Este trabalho foi desenvolvido com o objetivo de estudar a aplicação, para sistema de frenagem veicular, de uma máquina elétrica sem escovas, de armadura toroidal, e fluxo magnético axial produzido por ímãs permanentes de terras raras, a chamada máquina Torus. A máquina foi construída no LMEAE e estudada inicialmente como motor em outro trabalho. Mas, para que se possa avaliar seu funcionamento em sistema de frenagem, o foco é do ponto de vista da máquina como gerador. São realizados testes dinâmicos e estáticos experimentalmente e modelo numérico pelo método dos elementos finitos com um formato de ímãs permanentes de seção setorial, possibilitando o comparativo com a versão anterior da máquina que empregou ímãs de seção quadrada. Mudanças físicas e no sistema de acionamento da máquina, e ensaios de frenagem dinâmica foram realizados. Modelagem analítica para indução magnética foi desenvolvida utilizando-se da técnica de Transformação Conforme. O trabalho busca apresentar as características da máquina e justificativas que demonstram o seu potencial de aplicabilidade em um subsistema veicular sob frenagem regenerativa e a capacidade de fornecimento de energia a um sistema de armazenamento com uma parte de energia cinética, ou seja, baterias e supercapacitores. / This work is carried out with the aim to study the application, by a vehicular braking system, of a brushless electrical machine with a toroidal armature core, and axial magnetic flux delivered by rare earth permanent magnets, the so-called Torus machine. The machine was built in the LMEAE, and previously studied as a motor by other work. However, in order to assess its performance in a braking system, the focus is the point of view of the machine as a generator. Static and dynamic tests are implemented as well as a numerical model by means of the finite element method, in order to compare the machine with sector poles permanent magnets and with square magnet poles. Physical changes and on the driving system of the machine, and dynamic braking tests are performed. The analytical modelling for the magnetic induction was developed using the technique of conformal transformation. The study aims to present the features of the machine and demonstrates its potential applicability to a vehicular subsystem under regenerative braking and the ability to supply an energy storage system with part of the kinetic energy, i.e. batteries and super capacitors.

Orientador: Sergio Antonio Tozoni / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T19:58:00Z (GMT). No. of bitstreams: 1 Stabile_RegisLeandroBraguim_D.pdf: 1552111 bytes, checksum: af2b74d1076ee2c6dd825049748fd3fd (MD5) Previous issue date: 2014 / Resumo: As teorias de n-larguras e de entropia foram introduzidas por Kolmogorov na década de 1930. Desde então, muitos trabalhos têm visado obter estimativas assintóticas para n-larguras e números de entropia de diferentes classes de conjuntos. Neste trabalho, investigamos n-larguras e números de entropia de operadores multiplicadores definidos sobre o toro d-dimensional. Na primeira parte, estabelecemos estimativas inferiores e superiores para n-larguras e números de entropia de operadores multiplicadores gerais. Na segunda parte, aplicamos estes resultados para operadores multiplicadores específicos, associados a conjuntos de funções finitamente e infinitamente diferenciáveis sobre o toro. Em particular, demonstramos que as estimativas obtidas são exatas em termos de ordem em diversas situações / Abstract: The theories of n-widths and entropy were introduced by Kolmogorov in the 1930s. Since then, many works aims to find estimates for n-widths and entropy numbers of different classes of sets. In this work, we investigate n-widths and entropy numbers of multiplier operators defined on the d-dimensional torus. In the first part, upper and lower bounds are established for n-widths and entropy numbers of general multiplier operators. In the second part, we apply these results to specific multiplier operators, associated with sets of finitely and infinitely differentiable functions on the torus. In particular, we prove that, the estimates obtained are order sharp in various situations / Doutorado / Matematica / Doutor em Matemática

High performance parallel and distributed computing systems are used to solve large, complex, and data parallel scientific applications that require enormous computational power. Data parallel workloads which require performing similar operations on different data objects, are present in a large number of scientific applications, such as N-body simulations and Monte Carlo simulations, and are expressed in the form of loops. Data parallel workloads that lack precedence constraints are called arbitrarily divisible workloads, and are amenable to easy parallelization. Load imbalance that arise from various sources such as application, algorithmic, and systemic characteristics during the execution of scientific applications degrades performance. Scheduling of arbitrarily divisible workloads to address load imbalance in order to obtain better utilization of computing resources is a major area of research. Divisible load theory (DLT) and dynamic loop scheduling (DLS) algorithms are two algorithmic approaches employed in the scheduling of arbitrarily divisible workloads. Despite sharing the same goal of achieving load balancing, the two approaches are fundamentally different. Divisible load theory algorithms are linear, deterministic and platform dependent, whereas dynamic loop scheduling algorithms are probabilistic and platform agnostic. Divisible load theory algorithms have been traditionally used for performance prediction in environments characterized by known or expected variation in the system characteristics at runtime. Dynamic loop scheduling algorithms are designed to simultaneously address all the sources of load imbalance that stochastically arise at runtime from application, algorithmic, and systemic characteristics. In this dissertation, an analysis and performance evaluation of DLT and DLS algorithms are presented in the form of a scalability study and a robustness investigation. The effect of network topology on their performance is studied. A hybrid scheduling approach is also proposed that integrates DLT and DLS algorithms. The hybrid approach combines the strength of DLT and DLS algorithms and improves the performance of the scientific applications running in large scale parallel and distributed computing environments, and delivers performance superior to that which can be obtained by applying DLT algorithms in isolation. The range of conditions for which the hybrid approach is useful is also identified and discussed.

Inflatable structures have been a subject of renewed interest in recent years for space applications such as communication antennas, solar thermal propulsion, and entry/landing systems. This is because inflatable structures are very lightweight and on-orbit deployable. In addition, they have high strength-to-mass ratio and require minimal stowage volume, which makes them especially suitable for cost-effective large space structures. An inflated toroidal structure (torus) is often used there in order to provide structural support. For these structures to be effective, their vibration must be controlled while keeping the weight low. Piezoelectric materials have become strong candidates for actuator and sensor applications in the active vibration control of such structures due to their lightweight, conformability to the host structure, and distributed nature. In this study, our main focus is to understand the dynamic characteristics of an inflatable torus and to control its vibration using piezoelectric actuators and sensors. The first part of this study is concerned with theoretical formulations. We use Sanders' shell theory to derive the governing equations of motion for a shell subjected to pressure. To take into account the prestress effects of internal pressure, we use geometric nonlinearity, and to model the follower action of pressure force, we consider the work done by internal pressure during the vibration of the shell. These equations are then specialized to obtain approximate equations presented by previous researchers. We extend this analytical formulation to derive the equivalent forces due to piezoelectric actuators in unimorph and bimorph configurations and include their mass and stiffness effects in the governing equations. A sensor equation is also developed for the shell. The actuator and sensor equations are then written in terms of modal displacements and velocities so as to evaluate their interactions with different vibratory modes. In the second part, we focus on numerical studies related to an inflated torus. At first, we perform a free vibration analysis of the inflated torus using Galerkin's method. We study how different parameters (aspect ratio, internal pressure, and wall-thickness) of the inflated torus affect the natural frequencies and mode shapes of the inflated torus. We compare the results obtained from the theory used in this research with the results from different approximate theories and commercial finite element codes. The results suggest that the use of an accurate shell theory and pressure effect is very important for the vibration analysis of an inflated torus. Next, the modal behaviors of piezoelectric actuator and sensor are analyzed. A detailed study is done in order to understand how the size and location of actuator and sensor affect the modal forces, the modal sensing constants, and the overall performance for all the considered modes. In order to determine the optimal locations and sizes of actuators and sensors, we use a genetic algorithm. Natural frequencies and mode shapes are calculated considering the passive effects of actuators and sensors. Finally, we attempt the vibration control of the inflated torus using the optimally designed actuators and sensors and sliding mode controller/observer. The numerical simulations show that piezoelectric actuators and sensors can be used in the vibration control of an inflatable torus. The robustness properties of the controller and observer against the parameter uncertainty and disturbances are verified. / Ph. D.

There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.