Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char. / This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-09T17:38:25Z No. of bitstreams: 1 JAMILLY LOURÊDO ROCHA - DISSERTAÇÃO PPGMAT 2014..pdf: 5707925 bytes, checksum: 8010cd451ac64c8a7fccc36a2f8313f6 (MD5) / Made available in DSpace on 2018-08-09T17:38:25Z (GMT). No. of bitstreams: 1 JAMILLY LOURÊDO ROCHA - DISSERTAÇÃO PPGMAT 2014..pdf: 5707925 bytes, checksum: 8010cd451ac64c8a7fccc36a2f8313f6 (MD5) Previous issue date: 2014-08 / Capes / Com uma aplicação adequada do conhecido princípio do máximo generalizado de Omori-Yau, obtemos resultados de rigidez com relação a hipersuperfícies imersas completascomcurvaturamédiadelimitadanoespaçohiperbólicoHn+1 (n+1)-dimensional. Em nossa abordagem exploramos a existência de uma dualidade natural entreHn+1 e a metade Hn+1 do espaço de SitterSn+11 , cujo modelo é chamado de steady state space. / As a suitable application of the well known generalized maximum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the (n+1)-dimensional hyperbolic spaceHn+1. In our approach, we explore the existence of a natural duality betweenHn+1 and the half Hn+1 of the de Sitter spaceSn+11 , which models the so-called steady state space.

In this work we prove two theorems that characterize the hypersurfaces in the unitary sphere of dimension n+1. The first result, obtained by H. Alencar and M. do Carmo, classifies hypersurfaces with constant mean curvature in the sphere. This result was published in April 1994 in Proceedings of The American Mathematical Society, volume 120, number 4 with the title Hypersurfaces with Constant Mean Curvature. The second result was obtained by Li Haizhong in the article Hypersurfaces with Constant Scalar Curvature in Space Forms, published in 1996 in the journal Mathematisch Annalen, volume 305. The theorem of Li Haizhong characterizes hypersurfaces with constant scalar curvature in the sphere. We prove the theorem of Li Haizhong using the results obtained by H. Alencar and M. do Carmo. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação apresentamos dois teoremas que caracterizam as hipersuperfícies na esfera unitária de dimensão n+1. O primeiro resultado, obtido por H. Alencar e M. do Carmo, classifica as hipersuperfícies com curvatura média constante na esfera. Este resultado foi publicado em abril de 1994 no Proceedings of The American Mathematical Society, volume 120, número 4 com o título Hypersurfaces With Constant Mean Curvature.O segundo resultado provado nesta dissertação foi obtido por Li Haizhong no artigo Hypersurfaces With Constant Scalar Curvature in Spaces Forms, publicado em 1996 no Mathematische Annalen, volume 305. O Teorema de Li Haizhong caracteriza as hipersuperfícies com curvatura escalar constante na esfera. Demonstraremos o Teorema de Li Haizhong utilizando os resultados obtidos por H. Alencar e M. do Carmo.

The goal of this dissertation is to present a R. Reilly's demonstration of the theorem of Alexandrov . The theorem states that The only compact hypersurfaces, conected, of constant mean curvature, immersed in Euclidean space are spheres. The theorem of Alexandrov was proved by A. D. Alexandrov in the article Uniqueness Theorems for Surfaces in the Large V, published in 1958 by Vestnik Leningrad University, volume 13, number 19, pages 5 to 8. In his demonstration, Alexandrov used the famous Principle of tangency, introduced by him in that article. In the year 1962, M. Obata shown in Certain Conditions for a Riemannian Manifold to be isometric With the Sphere, published by the Journal of Mathematical Society of Japan, volume 14, pages 333 to 340, that a Riemannian Manifold M, compact, connected and without boundary, is isometric to a sphere, since the Ricci curvature of M satisfies certain lower bound. This theorem solves the problem of finding manifolds that reach equality in the estimate of Lichnerowicz for the first eigenvalue. In 1977, R. Reilly, in the article Applications of the Hessian operator in a Riemannian Manifold, published in Indianna University Mathematical Journal, volume 23, pages 459 to 452, showed a generalization of the Obata theorem for compact manifolds with boundary. As an example of the technique developed in this demonstration, he presents a new demonstration of the theorem of Alexandrov. This demonstration, as well as the techniques involved are the object of study of this work. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / O objetivo desta dissertação é apresentar uma demonstração de R. Reilly para o Teorema de Alexandrov. O teorema estabelece que As únicas hipersuperfícies compactas, conexas, de curvatura média constante, mergulhadas no espaço Euclidiano são as esferas. O teorema de Alexandrov foi provado por A. D. Alexandrov no artigo Uniqueness Theorems for Surfaces in the Large V, publicado em 1958 pela Vestnik Leningrad University, volume 13, número 19, páginas 5 a 8. Em sua demonstração, Alexandrov usou o famoso Princípio de Tangência, introduzido por ele no citado artigo. No ano de 1962, M. Obata demonstrou em Certain Conditions for a Riemannian Manifold to be Isometric With a Sphere, publicado pelo Journal of Mathematical Society of Japan, volume 14, páginas 333 a 340, que uma variedade Riemanniana M, compacta, conexa e sem bordo, é isométrica a uma esfera, desde que a curvatura de Ricci de M satisfaça determinada limitação inferior. Este teorema resolve o problema de encontrar as variedades que atingem a igualdade na estimativa de Lichnerowicz para o primeiro autovalor. Em 1977, R. Reilly, no artigo Applications of the Hessian Operator in a Riemannian Manifold, publicado no Indianna University Mathematical Journal, volume 23, páginas 459 a 452, demonstrou uma generalização do Teorema de Obata para variedades compactas com bordo. Como exemplo da técnica desenvolvida nesta demonstração, ele apresenta uma nova demonstração do Teorema de Alexandrov. Esta demonstração, bem como as técnicas envolvidas, são o objeto de estudo deste trabalho.

Provamos resultados sobre a geometria de hipersuperfícies em diferentes espaços ambiente. Primeiro, definimos uma aplicação de Gauss generalizada para uma hipersuperfície Mn-1 c/ Nn, onde N é um espaço simétrico de dimensão n ≥ 3. Em particular, generalizamos um resultado de Ruh-Vilms e apresentamos aplicações. Em seguida, estudamos superfícies em espaços de dimensão 3: estudamos a equação da curvatura média em um produto semidireto R2oAR e obtemos estimativas da altura e a existência de gráficos mínimos do tipo Scherk. Finalmente, no espaço ambiente de uma variedade hiperbólica de dimensão 3: nós apresentamos condições suficientes para que um mergulho completo de uma superfície ∑ de topologia finita em N com curvatura média |H∑| ≤ 1 seja próprio. / We prove results concerning the geometry of hypersurfaces on di erent ambient spaces. First, we de ne a generalized Gauss map for a hypersurface Mn-1 c/ Nn, where N is a symmetric space of dimension n ≥ 3. In particular, we generalize a result due to Ruh-Vilms and make some applications. Then, we focus on surfaces on spaces of dimension 3: we study the mean curvature equation of a semidirect product R2 oA R to obtain height estimates and the existence of a Scherk-like minimal graph. Finally, on the ambient space of a hyperbolic manifold N of dimension 3 we give su cient conditions for a complete embedding of a nite topology surface ∑ on N with mean curvature |H∑| ≤ 1 to be proper.

Historiquement et socialement, le visage est chez l'humain une modalité de prédilection pour déterminer l'identité et l'état émotionnel d'une personne. Il est naturellement exploité en vision par ordinateur pour les problèmes de reconnaissance de personnes et d'émotions. Les algorithmes d'analyse faciale automatique doivent relever de nombreux défis : ils doivent être robustes aux conditions d'acquisition ainsi qu'aux expressions du visage, à l'identité, au vieillissement ou aux occultations selon le scénario. La modalité 3D a ainsi été récemment investiguée. Elle a l'avantage de permettre aux algorithmes d'être, en principe, robustes aux conditions d'éclairage ainsi qu'à la pose. Cette thèse est consacrée à l'analyse de visages en 3D, et plus précisément la reconnaissance faciale ainsi que la reconnaissance d'expressions faciales en 3D sans texture. Nous avons dans un premier temps axé notre travail sur l'apport que pouvait constituer une approche régions aux problèmes d'analyse faciale en 3D. L'idée générale est que le visage, pour réaliser les expressions faciales, est déformé localement par l'activation de muscles ou de groupes musculaires. Il est alors concevable de décomposer le visage en régions mimiques et statiques, et d'en tirer ainsi profit en analyse faciale. Nous avons proposé une paramétrisation spécifique, basée sur les distances géodésiques, pour rendre la localisation des régions mimiques et statiques le plus robustes possible aux expressions. Nous avons également proposé une approche régions pour la reconnaissance d'expressions du visage, qui permet de compenser les erreurs liées à la localisation automatique de points d'intérêt. Les deux approches proposées dans ce chapitre ont été évaluées sur des bases standards de l'état de l'art. Nous avons également souhaité aborder le problème de l'analyse faciale en 3D sous un autre angle, en adoptant un système de cartes de représentation de la surface 3D. Nous avons ainsi proposé de projeter sur le plan 2D des informations liées à la topologie de la surface 3D, à l'aide d'un descripteur géométrique inspiré d'une mesure de courbure moyenne. Les problèmes de reconnaissance faciale et de reconnaissance d'expressions 3D sont alors ramenés à ceux de l'analyse faciale en 2D. Nous avons par exemple utilisé SIFT pour l'extraction puis l'appariement de points d'intérêt en reconnaissance faciale. En reconnaissance d'expressions, nous avons utilisé une méthode de description des visages basée sur les histogrammes de gradients orientés, puis classé les expressions à l'aide de SVM multi-classes. Dans les deux cas, une méthode de fusion simple permet l'agrégation des résultats obtenus à différentes échelles. Ces deux propositions ont été évaluées sur la base BU-3DFE, montrant de bonnes performances tout en étant complètement automatiques. Enfin, nous nous sommes intéressés à l'impact des dégradations des modèles 3D sur les performances des algorithmes d'analyse faciale. Ces dégradations peuvent avoir plusieurs origines, de la capture physique du visage humain au traitement des données en vue de leur interprétation par l'algorithme. Après une étude des origines et une théorisation des types de dégradations potentielles, nous avons défini une méthodologie permettant de chiffrer leur impact sur des algorithmes d'analyse faciale en 3D. Le principe est d'exploiter une base de données considérée sans défauts, puis de lui appliquer des dégradations canoniques et quantifiables. Les algorithmes d'analyse sont alors testés en comparaison sur les bases dégradées et originales. Nous avons ainsi comparé le comportement de 4 algorithmes de reconnaissance faciale en 3D, ainsi que leur fusion, en présence de dégradations, validant par la diversité des résultats obtenus la pertinence de ce type d'évaluation. / Historically and socially, the human face is one of the most natural modalities for determining the identity and the emotional state of a person. It has been exploited by computer vision scientists within the automatic facial analysis domain. Still, proposed algorithms classically encounter a number of shortcomings. They must be robust to varied acquisition conditions. Depending on the scenario, they must take into account intra-class variations such as expression, identity (for facial expression recognition), aging, occlusions. Thus, the 3D modality has been suggested as a counterpoint for a number of those issues. In principle, 3D views of an object are insensitive to lightning conditions. They are, theoretically, pose-independant as well. The present thesis work is dedicated to 3D Face Analysis. More precisely, it is focused on non-textured 3D Face Recognition and 3D Facial Expression Recognition. In the first instance, we have studied the benefits of a region-based approach to 3D Face Analysis problems. The general concept is that a face, when performing facial expressions, is deformed locally by the activation of muscles or groups of muscles. We then assumed that it was possible to decompose the face into several regions of interest, assumed to be either mimic or static. We have proposed a specific facial surface parametrization, based upon geodesic distance. It is designed to make region localization as robust as possible regarding expression variations. We have also used a region-based approach for 3D facial expression recognition, which allows us to compensate for errors relative to automatic landmark localization. We also wanted to experiment with a Representation Map system. Here, the main idea is to project 3D surface topology data on the 2D plan. This translation to the 2D domain allows us to benefit from the large amount of related works in the litterature. We first represent the face as a set of maps representing different scales, with the help of a geometric operator inspired by the Mean Curvature measure. For Facial Recognition, we perform a SIFT keypoints extraction. Then, we match extracted keypoints between corresponding maps. As for Facial Expression Recognition, we normalize and describe every map thanks to the Histograms of Oriented Gradients algorithm. We further classify expressions using multi-class SVM. In both cases, a simple fusion step allows us to aggregate the results obtained on every single map. Finally, we have studied the impact of 3D models degradations over the performances of 3D facial analysis algorithms. A 3D facial scan may be an altered representation of its real life model, because of several reasons, which range from the physical caption of the human model to data processing. We propose a methodology that allows us to quantify the impact of every single type of degradation over the performances of 3D face analysis algorithms. The principle is to build a database regarded as free of defaults, then to apply measurable degradations to it. Algorithms are further tested on clean and degraded datasets, which allows us to quantify the performance loss caused by degradations. As an experimental proof of concept, we have tested four different algorithms, as well as their fusion, following the aforementioned protocol. With respect to the various types of contemplated degradations, the diversity of observed behaviours shows the relevance of our approach.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T19:25:58Z No. of bitstreams: 1 ERALDO ALMEIDA LIMA JÚNIOR - DISSERTAÇÃO PPGMAT 2011..pdf: 415901 bytes, checksum: 427abfdae7c5a546735d4a6b14f72bfe (MD5) / Made available in DSpace on 2018-07-25T19:25:58Z (GMT). No. of bitstreams: 1 ERALDO ALMEIDA LIMA JÚNIOR - DISSERTAÇÃO PPGMAT 2011..pdf: 415901 bytes, checksum: 427abfdae7c5a546735d4a6b14f72bfe (MD5) Previous issue date: 2011-07 / Neste trabalho, apresentamos um estudo das hipersuperfícies tipo-espaço imersas no ambiente −R × Hn, exibindo condições para que tais hipersuperfícies sejam slices {t0}×Hn. Para uma melhor compreensão das demonstrações e dos resultados, inserimos processos de diferenciação, cálculos de gradientes e Laplacianos que, juntamente com o princípio do máximo de Omori-Yau, foram cruciais no desenvolvimento dos resultados que, em sua maioria são do tipo Bernstein. Também incluímos um resultado do tipo Calabi. / In this work we present a study of the spacelike hypersurfaces immersed in the manifold −R × Hn providing sufficient conditions for such hypersurfaces be slices, {t0}×Hn. For a better understanding of the proofs and results, we have added differentiation processes, gradient computations and Laplacians which jointly with the Omori-Yau Maximum Principle were crucial in the developing of the results whose are mostly Bernstein-type. In the elapsing we also included Calabi-type results.

Made available in DSpace on 2015-05-15T11:46:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 816279 bytes, checksum: 28c5081e37dbd539abb463a0ed89b87c (MD5) Previous issue date: 2012-07-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this paper we studied surfaces with constant mean curvature and surfaces with constant Gaussian curvature in the Sol space which are invariant under the action of two one-parameter subgroups of isometries of the ambient space. Furthermore, we classify the surfaces that satisfy a relationship of type k1 = mk2, where k1 and k2 are the principal curvatures of the surface and m ∈ R. / O presente trabalho aborda um estudo das superfícies com curvatura média constante e das superfícies com curvatura Gaussiana constante no espaço Sol que são invariantes sob a ação de dois grupos a 1-parâmetro de isometrias do espaço ambiente. Além disso, classificamos as superfícies que satisfazem uma relação do tipo k1 = mk2, onde k1 e k2 são as curvaturas principais da superfície e m ∈ R.

Provamos resultados sobre a geometria de hipersuperfícies em diferentes espaços ambiente. Primeiro, definimos uma aplicação de Gauss generalizada para uma hipersuperfície Mn-1 c/ Nn, onde N é um espaço simétrico de dimensão n ≥ 3. Em particular, generalizamos um resultado de Ruh-Vilms e apresentamos aplicações. Em seguida, estudamos superfícies em espaços de dimensão 3: estudamos a equação da curvatura média em um produto semidireto R2oAR e obtemos estimativas da altura e a existência de gráficos mínimos do tipo Scherk. Finalmente, no espaço ambiente de uma variedade hiperbólica de dimensão 3: nós apresentamos condições suficientes para que um mergulho completo de uma superfície ∑ de topologia finita em N com curvatura média |H∑| ≤ 1 seja próprio. / We prove results concerning the geometry of hypersurfaces on di erent ambient spaces. First, we de ne a generalized Gauss map for a hypersurface Mn-1 c/ Nn, where N is a symmetric space of dimension n ≥ 3. In particular, we generalize a result due to Ruh-Vilms and make some applications. Then, we focus on surfaces on spaces of dimension 3: we study the mean curvature equation of a semidirect product R2 oA R to obtain height estimates and the existence of a Scherk-like minimal graph. Finally, on the ambient space of a hyperbolic manifold N of dimension 3 we give su cient conditions for a complete embedding of a nite topology surface ∑ on N with mean curvature |H∑| ≤ 1 to be proper.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / FundaÃÃo de Amparo à Pesquisa do Estado do Amazonas / Esta tese està composta de quatro partes distintas. Na primeira parte, vamos dar uma nova caracterizaÃÃo da esfera euclidiana como a Ãnica variedade Riemanniana compacta com curvatura escalar constante e admitindo um campo de vetores conforme nÃo trivial que à tambÃm Ricci conforme. Na segunda parte, provaremos algumas propriedades dos quase sÃlitons de Ricci, as quais permitem estabelecer condiÃÃes de rigidez desses objetos, bem como caracterizar as estruturas de quase sÃlitons de Ricci gradiente na esfera euclidiana. ImersÃes isomÃtricas tambÃm serÃo consideradas; classificaremos os quase sÃlitons de Ricci imersos em formas espaciais, atravÃs de uma condiÃÃo algÃbrica sobre a funÃÃo sÃliton. AlÃm disso, vamos caracterizar, atravÃs de uma condiÃÃo sobre o operador de umbilicidade, as hipersuperfÃcies n-dimensionais de uma forma espacial, com curvatura mÃdia constante, tendo duas curvaturas principais distintas e com multiplicidades p e n - p. Na terceira parte, provaremos um resultado de rigidez e algumas fÃrmulas integrais para uma mÃtrica m-quasi-Einstein generalizada compacta. Na Ãltima parte, vamos apresentar uma relaÃÃo entre a curvatura gaussiana e o Ãngulo de contato de superfÃcies imersas na esfera euclidiana tridimensional,a qual permite concluir que a superfÃcie à plana, se o Ãngulo de contato for constante. AlÃm disso, deduziremos que o toro de Clifford à a Ãnica superfÃcie compacta com curvatura mÃdia constante tendo tal propriedade. / This thesis is composed of four distinct parts. In the first part, we shall give a new characterization of the Euclidean sphere as the only compact Riemannian manifold with constant scalar curvature carrying a conformal vector eld non-trivial which is also Ricci conformal. In the second part, we shall prove some properties of almost Ricci solitons, which allow us to establish conditions for rigidity of these objects, as well as characterize the structures of gradient almost Ricci soliton in Euclidean sphere. Isometric immersions also will be considered, we shall classify almost Ricci solitons immersed in space forms, through algebraic condition on soliton function. Furthermore, we characterize under a condition of the umbilicity operator, n-dimensional hypersurfaces in a space form with constant mean curvature, admitting two distinct principal curvatures with multiplicities p and n - p. In the third part, we prove a result of rigidity and some integral formulae for a compact generalized m-quasi-Einstein metric. In the last part, we present a relation between the Gaussian curvature and the contact angle of surfaces immersed in Euclidean three-dimensional sphere, which allows us to conclude that such a surface is at provided its contact angle is constant. Moreover, we deduce that Clifford tori are the unique compact surfaces with constant mean curvature having such property.