Neste trabalho apresentamos condições suficientes para a existência de gráficos de curvatura média constante (CMC) com bordo em dois planos paralelos. Também são feitas estimativas para a altura de superfícies CMC com vetor normal orientado para fora limitadas por um cilindro ou horocilindro. / In this work we present su cient existence conditions for constant mean curvature (CMC) graphs with boundary in two parallel planes. We also make height estimates for outwards-oriented CMC surfaces bounded by a cylinder or horocylinder.

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京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19417号 / 情博第596号 / 新制||情||104(附属図書館) / 32442 / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 山本 章博, 教授 黒橋 禎夫, 教授 鹿島 久嗣, 准教授 Cuturi Marco / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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Nous nous intéressons à la comparaison de formes de courbes lisses prenant leurs valeurs dans une variété riemannienne M. Dans ce but, nous introduisons une métrique riemannienne invariante par reparamétrisations sur la variété de dimension infinie des immersions lisses dans M. L’équation géodésique est donnée et les géodésiques entre deux courbes sont construites par tir géodésique. La structure quotient induite par l’action du groupe des reparamétrisations sur l’espace des courbes est étudiée. À l’aide d’une décomposition canonique d’un chemin dans un fibré principal, nous proposons un algorithme qui construit la géodésique horizontale entre deux courbes et qui fournit un matching optimal. Dans un deuxième temps, nous introduisons une discrétisation de notre modèle qui est elle-même une structure riemannienne sur la variété de dimension finie Mn+1 des "courbes discrètes" définies par n + 1 points, où M est de courbure sectionnelle constante. Nous montrons la convergence du modèle discret vers le modèle continu, et nous étudions la géométrie induite. Des résultats de simulations dans la sphère, le plan et le demi-plan hyperbolique sont donnés. Enfin, nous donnons le contexte mathématique nécessaire à l’application de l’étude de formes dans une variété au traitement statistique du signal radar, où des signaux radars localement stationnaires sont représentés par des courbes dans le polydisque de Poincaré via la géométrie de l’information. / We are concerned with the comparison of the shapes of open smooth curves that take their values in a Riemannian manifold M. To this end, we introduce a reparameterization invariant Riemannian metric on the infinite-dimensional manifold of these curves, modeled by smooth immersions in M. We derive the geodesic equation and solve the boundary value problem using geodesic shooting. The quotient structure induced by the action of the reparametrization group on the space of curves is studied. Using a canonical decomposition of a path in a principal bundle, we propose an algorithm that computes the horizontal geodesic between two curves and yields an optimal matching. In a second step, restricting to base manifolds of constant sectional curvature, we introduce a detailed discretization of the Riemannian structure on the space of smooth curves, which is itself a Riemannian metric on the finite-dimensional manifold Mn+1 of "discrete curves" given by n + 1 points. We show the convergence of the discrete model to the continuous model, and study the induced geometry. We show results of simulations in the sphere, the plane, and the hyperbolic halfplane. Finally, we give the necessary framework to apply shape analysis of manifold-valued curves to radar signal processing, where locally stationary radar signals are represented by curves in the Poincaré polydisk using information geometry.

This work presents two novel methods of estimating the state of a dynamic system in a Kalman Filtering framework. The first is an application specific method for use with systems performing Visual Odometry in a mostly planar scene. Because a Visual Odometry method inherently provides relative information about the pose of a platform, we use this system as part of the time update in a Kalman Filtering framework, and develop a novel way to propagate the uncertainty of the pose through this time update method. Our initial results show that this method is able to reduce localization error significantly with respect to pure INS time update, limiting drift in our test system to around 30 meters for tens of seconds. The second key contribution of this work is the Manifold EKF, a generalized version of the Extended Kalman Filter which is explicitly designed to estimate manifold-valued states. This filter works for a large number of commonly useful manifolds, and may have applications to other manifolds as well. In our tests, the Manifold EKF demonstrated significant advantages in terms of consistency when compared to other filtering methods. We feel that these promising initial results merit further study of the Manifold EKF, related filters, and their properties.

Une mosaïque de Poisson-Voronoï est une partition aléatoire de l'espace euclidien en polyèdres, appelés cellules, obtenue à partir d'un ensemble aléatoire discret de points appelés germes. A chaque germe correspond une cellule, qui est l'ensemble des points de l'espace qui sont plus proches de ce germes que des autres germes. Ces modèles sont souvent utilisées dans divers domaines tels que la biologie, les télécommunications, l'astronomie, etc. Les caractéristiques de ces mosaïques et des cellules associées ont été largement étudiées dans l'espace euclidien mais les travaux sur les mosaïques de Voronoï dans un cadre non-euclidien sont rares.Dans cette thèse, on étend la définition de mosaïque de Voronoï à une variétériemannienne de dimension finie et on s'intéresse aux caractéristiques des cellules associées. Plus précisément, on mesure dans un premier temps l'influence que peut avoir la géométrie locale de la variété, c'est-à-dire les courbures sur les caractéristiques moyennes d'une cellule, comme son volume ou son nombre de sommets, en calculant des développements asymptotiques des ces caractéristiques moyennes à grande intensité. Dans un deuxième temps, on s'interroge sur la possibilité de retrouver la géométrie locale de la variété à partir des caractéristiques combinatoires de la mosaïque sur la variété. En particulier, on établit desthéorèmes limites, quand l'intensité du processus des germes tend vers l'infini, pour le nombre de sommets de la mosaïque dans une fenêtre, ce qui permet de construire un estimateur de la courbure et d'en donner quelques propriétés.Les principaux résultats de cette thèse reposent sur la combinaison de méthodesprobabilistes et de techniques issues de la géométrie différentielle. / A Poisson-Voronoi tessellation is a random partition of the Euclidean space intopolytopes, called cells, obtained from a discrete set of points called germs. To each germ corresponds a cell which is the set of the points of the space which are closer to this germ than to the other germs. These models are often used in several domains such as biology, telecommunication, astronomy, etc. The caracteristics of these tessellations and cells have been widely studied in the Euclidean space but only a few works concerns non-Euclidean Voronoi tessellation. In this thesis, we extend the definition of Poisson-Voronoi tessellation to a Riemannian manifold with finite dimension and we study the caracteristics of the associated cells. More precisely, we first measure the influence of the local geometry of the manifold, namely the curvatures, on the caracteristics of the cells, e.g. the mean volume or the mean number of vertices. Second, we aim to recover the local geometry of the manifold from the combinatorial properties of the tessellation on the manifolds. In particular, we establish limit theorems for the number of vertices of the tessellation, when the intensity of the process of the germs tends to infinity. This leads to the construction of an estimator of the curvature of the manifold and makes it possible to derive some properties of it. The main results of this thesis relies on the combination of stochastic methods and techniques from the differential geometry theory.

Cette thèse porte sur l'existence de champs browniens fractionnaires indexés par des variétés riemanniennes. Ces objets héritent des propriétés qui font le succès du mouvement brownien fractionnaire classique (H-autosimilarité des trajectoires ajustable, accroissements stationnaires), mais autorisent à considérer des applications où les données sont portées par un espace qui peut par exemple être courbé ou troué. L'existence de ces champs n'est assurée que lorsque la quantité 2H est inférieure à l'indice fractionnaire de la variété, qui n'est connu que dans un petit nombre d'exemples. Dans un premier temps nous donnons une condition nécessaire pour l'existence de champ brownien fractionnaire. Dans le cas du champ brownien (correspondant à H=1/2) indexé par des variétés qui ont des géodésiques fermées minimales, cette condition s'avère très contraignante : nous donnons des résultats de non-existence dans ce cadre, et montrons notamment qu'il n'existe pas de champ brownien indexé par une variété compacte non simplement connexe. La condition nécessaire donne également une preuve courte d'un fait attendu qui est la non-dégénérescence du champ brownien indexé par les espaces hyperboliques réels. Dans un second temps nous montrons que l'indice fractionnaire du cylindre est nul, ce qui constitue un exemple totalement dégénéré. Nous en déduisons que l'indice fractionnaire d'un espace métrique n'est pas continu par rapport à la convergence de Gromov-Hausdorff. Nous généralisons ce résultat sur le cylindre à un produit cartésien qui possède une géodésique fermée minimale, et donnons une majoration de l'indice fractionnaire de surfaces asymptotiquement proches du cylindre au voisinage d'une géodésique fermée minimale. / The aim of the thesis is the study of the existence of fractional Brownian fields indexed by Riemannian manifolds. Those fields inherit key properties of the classical fractional Brownian motion (sample paths with self-similarity of adjustable parameter H, stationary increments), while allowing to consider applications with data indexed by a space which can be for example curved or with a hole. The existence of those fields is only insured when the quantity 2H is inferior or equal to the fractional index of the manifold, which is known only in a few cases. In a first part we give a necessary condition for the fractional Brownian field to exist. In the case of the Brownian field (corresponding to H=1/2) indexed by a manifold with minimal closed geodesics this condition happens to be very restrictive. We give several nonexistence results in this situation. In particular we show that there exists no Brownian field indexed by a nonsimply connected compact manifold. Our necessary condition also gives a short proof of an expected result: we prove the nondegeneracy of fractional Brownian fields indexed by the real hyperbolic spaces. In a second part we show that the fractional index of the cylinder is null, which gives a totally degenerate case. We deduce from this result that the fractional index of a metric space is noncontinuous with respect to the Gromov-Hausdorff convergence. We generalise this result about the cylinder to a Cartesian product with a closed minimal geodesic. Furthermore we give a bound of the fractional index of surfaces asymptotically close to the cylinder in the neighbourhood of a closed minimal geodesic.

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-11T09:02:46Z (GMT). No. of bitstreams: 1 Cavalcante_RodrigoGusmao_D.pdf: 1972914 bytes, checksum: 6cdc127efe5cbfcb242fe471138f03f4 (MD5) Previous issue date: 2007 / Resumo: Em geral, o espaço EucIidiano é utilizado no projeto e na análise de desempenho da maior parte dos sistemas de comunicações atuais. Nesta tese, verificamos que o modelo de um sistema de comunicação não necessariamente está restrito ao espaço Euclidiano, mas sim a uma variedade Riemanniana. Com isso, os sistemas de comunicaçoes podun ser analisados em um contexto mais geral, no qual constatamos que a curvatura do espaço influencia em seus desempenhos. Corno exemplo, estudamos a curvatura de meios ópticos e propomos novos perfis de guias de ondas, fibras ópticas e lentes de interesse prático. Além disso, caracterizamos a curvatura de modulações não-lineares (twisted) e verificamos que o valor máximo permitido para a energia média do ruído está relacionada ao valor da curvatura da modulação. Neste contexto, as moclulações associadas a superfícies mínimas apresentaram bons desempenhos, pois tais modulações são pontos críticos do erro quadrático médio. Mostramos também que o espaço de sinais possui métrica induzida da superfície associada à modulação. Com isso, foi possível demonstrar que os espaços de sinais com curvatura negativa são os que apresentam melhor desempenho segundo a probabilidade média de erro. Dessa forma, alguns exemplos de constelações de sinais geometricamente uniformes foram construídos e analisados em variedades Riemannianas. Finalizamos este trabalho notando que na maioria das vezes que o espaço hiperbólico é utilizado nos blocos ele um sistema ele comunicações, o desempenho desse sistema tende a se aproximar do ponto ótimo de operação / Abstract: ln general, the Euclidian space is used in the design and performance analysis in most of the current communication systems. ln this thesis, we note that the model of a communication system is not necessarily restricted to the Euclidian space, more precisely, the model can be linked to Riemannian manifolds. Thus, the communication systems could be analyed in a broader context, in which the curvature of space influence on their performance. As an example, we studied the curvature of optical medium and propose new profiles of waveguides, optical fibers and lenses of practical interest. Moreover, we have characterized the curvature of twisted modulations and found that the maximum value allowed for the average energy of noise is related to the value of the curvature of the modulation. ln this context, the modulation associatecl with minimal surfaces showed good performance, because these modulations are critical points of minimum the mean-square error. VVe show that the signal space has induced metric associated with surface of the modulation. Thus, we have shown that the signal space with negative curvature is the space where the average error probability decrease a function of the curvature. Thus, some examples of geometrically uniform signal constellations were constructecl and analyzed on Riemannian manifolds. Finally we note that most of the time that hyperbolic space is considered in blocks of a communication system, then the performance of this system tends to be closer to the optimum point of operation / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Benedito Leandro Neto.pdf: 688417 bytes, checksum: c1ac127d257e0a8d59d30de577413351 (MD5) Previous issue date: 2011-09-02 / In this work, we started with an historical study of Ricci Solitons showing that they, often, arise as a auto-similar solution for the Ricci flow. It was demonstrated, then, some basic concepts of Riemannian Geometry and a formal definition of a Ricci Solitons. To conclude the work, it was presented a study analysis of the [6] article, establishing , among other results, two theorems: the first one, an estimation for the potential function of a Gradient Shrinking Ricci Solitons, complete non-compact, and, the second one, an estimation for the volume of a Gradient Shrinking Ricci Solitons, complete non-compact. / Nesse trabalho, nós começamos com um levantamento histórico sobre os Ricci Sólitons, mostrando que, muitas vezes, eles surgem como solução auto-similar do fluxo de Ricci. Em seguida, introduzimos alguns conceitos básicos de geometria Riemanniana e definimos formalmente um Rici Sóliton. Concluimos o trabalho com um estudo aprofundado do artigo [6], do qual mostramos, dentre outros resultados, dois teoremas: uma estimativa para a função potencial de um Ricci Sóliton Gradiente Shrinking, completo e não-compacto e uma estimativa superior para o volume de um Ricci Sóliton Gradiente Shrinking, completo e não-compacto.