In this thesis, we develop a feedback-invariant theory of local controllability for affine distributions. We begin by developing an unexplored notion in control theory that we call proper small-time local controllability (PSTLC). The notion of PSTLC is developed for an abstraction of the well-known notion of a control-affine system, which we call an affine system. Associated to every affine system is an affine distribution, an adaptation of the notion of a distribution. Roughly speaking, an affine distribution is PSTLC if the local behaviour of every affine system that locally approximates the affine distribution is locally controllable in the standard sense. We prove that, under a regularity condition, the PSTLC property can be characterized by studying control-affine systems. The main object that we use to study PSTLC is a cone of high-order tangent vectors, or variations, and these are defined using the vector fields of the affine system. To better understand these variations, we study how they depend on the jets of the vector fields by studying the Taylor expansion of a composition of flows. Some connections are made between labeled rooted trees and the coefficients appearing in the Taylor expansion of a composition of flows. Also, a relation between variations and the formal Campbell-Baker-Hausdorff formula is established. After deriving some algebraic properties of variations, we define a variational cone for an affine system and relate it to the local controllability problem. We then study the notion of neutralizable variations and give a method for constructing subspaces of variations. Finally, using the tools developed to study variations, we consider two important classes of systems: driftless and homogeneous systems. For both classes, we are able to characterize the PSTLC property. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-01-11 20:11:45.466

On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-10-26T19:04:54Z No. of bitstreams: 1 DissFLC.pdf: 1637955 bytes, checksum: b94dbdba30fb59739948038e7cc356cf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:30:42Z (GMT) No. of bitstreams: 1 DissFLC.pdf: 1637955 bytes, checksum: b94dbdba30fb59739948038e7cc356cf (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:30:48Z (GMT) No. of bitstreams: 1 DissFLC.pdf: 1637955 bytes, checksum: b94dbdba30fb59739948038e7cc356cf (MD5) / Made available in DSpace on 2016-11-08T18:30:55Z (GMT). No. of bitstreams: 1 DissFLC.pdf: 1637955 bytes, checksum: b94dbdba30fb59739948038e7cc356cf (MD5) Previous issue date: 2016-09-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / We introduce some geometric experiments to the students, and particularly the handling of right-angled triangles, with the clear purpose of inserting, firming and emphasizing the importance, historical and practical, of the concept and meaning of the geometric tangent in an entire acute angle in triangle. Later, the students built a rudimentary theodolite that was used to measure vertical angles of tops inaccessible objects from the ground and consequently the estimate of their heights. The prospect objects were trees, poles, masts, antennas, spotlights, among other more familiar to the students of the ninth grade in elementary school. / Neste trabalho apresentamos aos alunos alguns experimentos geométricos e, sobretudo, a manipulação de triângulos retângulos, com o claro objetivo de inserir, fixar e ressaltar a importância, histórica e prática, do conceito e do significado geométrico da tangente de um ângulo agudo. Posteriormente, os alunos construíram um teodolito rudimentar que foi utilizado para medir ângulos verticais de topos de objetos inacessíveis em relação ao solo e consequentemente o cálculo de suas alturas. Os objetos explorados foram árvores, postes, mastros, antenas, refletores, entre outros mais familiares para os estudantes do nono ano do Ensino Fundamental.

Made available in DSpace on 2015-05-15T11:46:16Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 578974 bytes, checksum: e512f47deae8cd03667ae8e7c2143b34 (MD5) Previous issue date: 2013-05-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The notions of connection and covariant derivative has its origin in the field of Riemannian geometry , where there is no distinction between them. In fact, in this study we found that these notions are equivalent if we consider modules over K-algebras of finite type. We also show that the existence of connections implies the existence of covariant derivative. The main goal of this study is to determine which modules admit connections. We easily verified that the projective modules admit connections. In fact, they form an affine space. But we also display a module that is not projective and has connection. Later, inspired by Swan's theorem, we explore in a straightforward way modules formed by sections of the tangent bundle of some surfaces in 3-dimensional real space. Finally, we study the notion of connection introduced by Alain Connes in modules over K-algebras not necessarily commutative. And we find in that context that the modules that have connection are exactly the projectives modules. / As noções de conexão e derivada covariante tem sua origem na área de geometria riemanniana, onde não existe distinção entre elas. De fato, nós verificamos neste trabalho, que estas noções são equivalentes se considerarmos módulos sobre K-álgebras comutativas de tipo finito. Também mostramos que a existência de conexões implica na existência de derivada covariante. O objetivo central deste trabalho é determinar que módulos admitem conexão. Verificamos facilmente que os módulos projetivos admitem conexões. De fato, elas formam um espaço afim. Mas também exibimos um módulo não projetivo que possui conexão. Posteriormente, inspirados pelo teorema de Swan, exploramos de maneira direta os módulos formados pelas seções do fibrado tangente de algumas superfícies no espaço 3- dimensional real. Por fim, estudamos a noção de conexão introduzida por Alain Connes em módulos sobre K-álgebras não necessariamente comutativas. E verificamos nesse contexto que os módulo que admitem conexão são exatamente os módulos projetivos.

Estudaremos as singularidades genéricas de superfécies regradas em R3. O objetivo do trabalho é mostrar que as singularidades genéricas que ocorrem no conjunto das superfícies regradas são as mesmas que ocorrem no conjunto das aplicações diferenciáveis de R2 em R3, enquanto que as singularidades genéricas das superfícies desenvolvíveis, que formam um subconjunto das superfícies regradas, são mais degeneradas. / We study generic singularities of ruled surfaces in R3. In this work we show that generic singularities appearing in the set of ruled surfaces are the same that occur in the set of map germs from R2 to R3, while the generic singularities of developable surfaces are more degenerate.

Orientadores: Margarida Pinheiro Mello, Maria Aparecida Diniz Ehrhardt / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T15:49:27Z (GMT). No. of bitstreams: 1 Ferreira_DaianeGoncalves_M.pdf: 1441315 bytes, checksum: 1196d8b21c6254dbdd0e0d68266fa707 (MD5) Previous issue date: 2013 / Resumo: Métodos de Otimização de terceira ordem, embora de longa tradição, eram considerados, até passado recente, impraticáveis, devido à taxa com que o esforço computacional cresce em função da dimensão do problema. Avanços no desenvolvimento de estruturas de dados, rotinas que trabalham com estas estruturas e a exploração da esparsidade de grande parte dos problemas encontrados na prática já permitem implementações destes métodos que podem torná-los competitivos com métodos de segunda ordem. O objeto desta dissertação é a apresentação do método de Halley, um método de terceira ordem, sua implementação em MATLAB e a realização de testes computacionais, visando uma comparação empírica de sua eficiência frente ao método de Newton, o método de segunda ordem mais empregado na atualidade / Abstract: Higher order optimization methods, though of long-standing tradition, until recently have been deemed impractical, due to the rate of increase of the computational effort as a function of the size of the problem. Advances in the development of data structures, routines that work with these structures and the use of the sparsity of a vast range of practical problems have led to implementations of these methods that are competitive with second order methods. The object of this dissertation is the study of Halley's method, a thirdorder method, the development of a MATLAB implementation thereof and its testing, aiming at an empirical comparison of its efficiency against that of Newton's method, the second-order method most widely used today / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

Neste trabalho estudamos as aplicações do método do Gradiente Espectral Projetado (SPG) em Meteorologia nos campos de previsibilidade, estabilidade e sensibilidade. Inicialmente revisamos os Vetores Singulares Lineares (LSVs) e em seguida apresentamos a teoria das Perturbações Condicionais Não-Lineares Ótimas (CNOPs). Enquanto os métodos clássicos estão baseados no Modelo Tangente Linear, as CNOPs são uma formulação do mesmo problema baseado em Programação Não-Linear. As CNOPs são descritas na literatura como responsáveis por melhorias em relação aos métodos anteriores. Finalmente analisamos três exemplos de aplicação do método à problemas de previsibilidade, estabilidade e sensibilidade. / A revision about applications of Spectral Projected Gradient (SPG) in meteorology is done in the fields of predictability, stability and sensitivity. Initially we review about Linear Singular Vectos (LSVs) and we present the Conditional Nonlinear Optimal perturbations (CNOPs). While the classic methods are based on the Tangent Linear Model, CNOPs are another formulation of the problem based on Nonlinear Programming. CNOPs are described in bibliography as responsible by better results than older methods. Finally we analyze three applications in predictability, stability and sensibility.

No description available.

Le but de cette thèse est d'étudier les trois problèmes suivantes : 1) On s'intéresse à la stabilité des cônes normaux et des sous-différentiels via deux types de convergence d'ensembles et de fonctions : La convergence au sens de Mosco et celle d'Attouch-Wets. Les résultats obtenus peuvent être vus comme une extension du théorème d'Attouch aux fonctions non nécessairement convexes sur des espaces de Banach localement uniformément convexes. 2) Pour une bornologie β donnée sur un espace de Banach X, on étudie la validité de la formule suivante (…). Ici Tβ(C; x) et Tc(C; x) désignent le β -cône tangent et le cône tangent de Clarke à C en x. On montre que si, X x X est ∂β-« trusted » alors cette formule est valable pour tout ensemble fermé non vide C ⊂ X et x ∈ C. Cette classe d'espaces contient les espaces ayant une norme équivalent β-différentiable, etplus généralement les espaces possédant une fonction "bosse" lipschitzienne et β-différentiable). Comme conséquence, on obtient que pour la bornologie de Fréchet, cette formule caractérise les espaces d'Asplund. 3) On examine la convexité des ensembles de Chebyshev. Il est bien connu que, dans un espace normé réflexif ayant la propriété Kadec-Klee, tout ensemble de Chebyshev faiblement fermé est convexe. On démontre que la condition de faible fermeture peut être remplacée par la fermeture faible locale, c'est-à-dire pour tout x ∈ C il existe ∈ > 0 tel que C ∩ B(x, ε) est faiblement fermé. On montre aussi que la propriété Kadec-Klee n'est plus exigée lorsque l'ensemble de Chebyshev est représenté comme une union d'ensembles convexes fermés. / The aim of this thesis is to study the following three problems: 1) We are concerned with the behavior of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch-Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. 2) For a given bornology β on a Banach space X we are interested in the validity of the following "lim inf" formula (…).Here Tβ(C; x) and Tc(C; x) denote the β-tangent cone and the Clarke tangent cone to C at x. We proved that it holds true for every closed set C ⊂ X and any x ∈ C, provided that the space X x X is ∂β-trusted. The trustworthiness includes spaces with an equivalent β-differentiable norm or more generally with a Lipschitz β-differentiable bump function. As a consequence, we show that for the Fréchet bornology, this "lim inf" formula characterizes in fact the Asplund property of X. 3) We investigate the convexity of Chebyshev sets. It is well known that in a smooth reflexive Banach space with the Kadec-Klee property every weakly closed Chebyshev subset is convex. We prove that the condition of the weak closedness can be replaced by the local weak closedness, that is, for any x ∈ C there is ∈ > 0 such that C ∩ B(x, ε) is weakly closed. We also prove that the Kadec-Klee property is not required when the Chebyshev set is represented by a finite union of closed convex sets.

<p>Inom många tillämpningar används konforma avbildningar för att transformera tvådimensionella områden till områden med enklare utseende. Ett exempel på ett sådant område är en kanal av varierande tjocklek begränsad av en kontinuerligt deriverbar kurva. I de tillämpningar som har motiverat detta arbete, är det viktigt att dessa egenskaper bevaras i det område en approximativ konform avbildning producerar, men det är också viktigt att begränsningskurvans riktning kan kontrolleras, särkilt i kanalens båda ändar.</p><p>Denna avhandling behandlar tre olika metoder för att numeriskt konstruera konforma avbildningar mellan ett enkelt standardområde, företrädesvis det övre halvplanet eller enhetscirkeln, och ett område begränsat av en kontinuerligt deriverbar kurva, där begränsningskurvans riktning kan kontrolleras, exakt eller approximativt.</p><p>Den första metoden är en utveckling av en idé, först beskriven av Peter Henrici, där en modifierad Schwarz-Christoffel-avbildning avbildar det övre halvplanet konformt på en polygon med rundade hörn.</p><p>Med utgångspunkt i denna idé skapas en algoritm för att konstruera avbildningar på godtyckliga områden med släta randkurvor.</p><p>Den andra metoden bygger också den på Schwarz-Christoffel-avbildningen, och utnyttjar det faktum att om enhetscirkeln eller halvplanet avbildas på en polygon kommer ett område Q i det inre av dessa, som till exempel en cirkel med centrum i origo och radie mindre än 1, eller ett område i övre halvplanet begränsat av två strålar, att avbildas på ett område R i det inre av polygonen begränsat av en slät kurva. Vi utvecklar en metod för att hitta ett polygonalt område P, utanför det Omega som man önskar att skapa en avbildning för, sådant att den Schwarz-Christoffel-avbildning som avbildar enhetscirkeln eller halvplanet på P, avbildar Q på Omega.</p><p>I båda dessa fall används tangentpolygoner för att numeriskt bestämma den önskade avbildningen.</p><p>Slutligen beskrivs en metod där en av Don Marshalls så kallade zipper-algoritmer används för att skapa en avbildning mellan det övre</p><p>halvplanet och en godtycklig kanal, begränsad av släta kurvor, som i båda ändar går mot oändligheten som räta parallella linjer.</p> / <p>In many applications, conformal mappings are used to transform two-dimensional regions into simpler ones. One such region for which conformal mappings are needed is a channel bounded by continuously differentiable curves. In the applications that have motivated this work, it is important that the region an approximate conformal mapping produces, has this property, but also that the direction of the curve can be controlled, especially in the ends of the channel.</p><p>This thesis treats three different methods for numerically constructing conformal mappings between the upper half-plane or unit circle and a region bounded by a continuously differentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately.</p><p>The first method is built on an idea by Peter Henrici, where a modified Schwarz-Christoffel mapping maps the upper half-plane conformally on a polygon with rounded corners. His idea is used in an algorithm by which mappings for arbitrary regions, bounded by smooth curves are constructed.</p><p>The second method uses the fact that a Schwarz-Christoffel mapping from the upper half-plane or unit circle to a polygon maps a region Q inside the half-plane or circle, for example a circle with radius less than 1 or a sector in the half--plane, on a region Omega inside the polygon bounded by a smooth curve. Given such a region Omega, we develop methods to find a suitable outer polygon and corresponding Schwarz-Christoffel mapping that gives a mapping from Q to Omega.</p><p>Both these methods use the concept of tangent polygons to numerically determine the coefficients in the mappings.</p><p>Finally, we use one of Don Marshall's zipper algorithms to construct conformal mappings from the upper half--plane to channels bounded by arbitrary smooth curves, with the additional property that they are parallel straight lines when approaching infinity.</p>