Ce mémoire porte sur l'étude des surfaces minimales et de courbure moyenne constante dans les espaces homogènes de dimension 3. Nous établissons les formules de Sym-Bobenko pour les surfaces de courbure moyenne constante 1/2 de H^2xR et minimales du groupe de Heisenberg, et donnons des exemples de construction de telles immersions par la méthode DPW. Nous montrons également que des propriétés de symétrie passent aux correspondances de type surfaces sœurs et cousines, ce qui entraîne l'existence de graphes entiers de courbure moyenne constante 1/2 à bout vertical dans H^2xR qui ne sont pas de révolution. Nous reprenons ensuite l'étude des bouts verticaux d'immersions de courbure moyenne constante 1/2 dans H^2xR. Nous munissons une famille de graphes entiers d'une structure de variété lisse et en déduisons un analogue pour H^2xR d'un théorème de A. E. Treibergs pour l'espace de Minkowski. Nous nous intéressons également aux déformations des anneaux de révolution. Une conséquence directe est l'existence d'anneaux immergés qui ne sont pas de révolution. Nous construisons notamment des anneaux dont les bouts n'ont pas le même axe. Enfin, nous décrivons les invariants de Nœther correspondant aux isométries des espaces homogènes pour les surfaces minimales et de courbure moyenne constante. Nous utilisons le formalisme de la géométrie de contact qui permet l'écriture de formules explicites en toute généralité, et nous étudions l'évolution des formes de Nœther sous l'action des isométries des espaces homogènes. Nous calculons ces invariants dans le cas des anneaux déformés de H^2xR, et dans celui des anneaux horizontaux du groupe de Heisenberg / The present dissertation deals with the study of minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces. In a first part, we establish Sym-Bobenko formulæ for constant mean curvature 1/2 surfaces in H^2xR and minimal surfaces in the Heisenberg group, and give examples of construction of such immersions using the DPW method. We also show that certain symmetry properties are shared by sister or cousin surfaces, which implies the existence non rotational entire graphs of constant mean curvature 1/2 in H^2xR with a vertical end.In a second part, we treat in more details the study of vertical ends of constant mean curvature 1/2 immersions in H^2xR. We endow a particular family entire graphs with a structure of smooth manifold and deduce an analogue in H^2xR to a theorem by A. E. Treibergs in the Minkowski space. We are also interested in deforming rotational annuli. A direct consequence is the existence of immersed non rotational annuli, and in particular we construct annuli with ends that do not have the same axis. Finally, we describe the Nœther invariants corresponding to isometries of the ambient homogeneous space for minimal and constant mean curvature surfaces. To do so, we use the formalism of contact geometry which allows general and explicit formulæ. We then study the evolution of Nœther form under the action of isometries in homogeneous spaces. We compute these invariants in the case of deformed annuli in H^2xR, and in the case of horizontal annuli in Heisenberg group

Drawing sketches is certainly one of the most important but at the same time elaborate parts of archaeological work. Currently, 3D scanning technology is affording a number of new applications, and only one of them is using virtual copies instead of the originals as the basis for documentation. Our major contribution are methods for automatically generating stylized images from 3D models. These are not only intuitive and easy to read but also more objective and accurate than traditional drawings. Besides some other useful tools we show several examples from our daily work proving that the system accelerates the whole documentation process considerably.

[pt] Neste trabalho estudamos a teoria clássica das superfícies mínimas em R3, focando na representação de Enneper-Weierstrass e suas consequências. São exibidos vários exemplos, incluindo as superfícies de Jorge-Meeks e de Jorge-Xavier. Também mostramos princípios do máximo para superfícies mínimas e várias aplicações como, por exemplo, o teorema do semi-espaço. Em seguida, nos concentramos na teoria das superfícies mínimas completas de curvatura total finita e, com esta, podemos analisar o desenvolvimento assintótico de fins mínimos completos mergulhados de curvatura total finita. Por fim, a dissertação culmina com o teorema de Schoen, que afirma que as únicas superfícies mínimas completas, conexas, de curvatura total finita e apenas dois fins - ambos mergulhados - são um par de planos e o catenoide. / [en] In this work we study the classical theory of minimal surfaces in R3, with special focus on the Enneper-Weierstrass representation and its consequences. We exhibit many examples, including the Jorge-Meeks and Jorge-Xavier surfaces. We also show maximum principles for minimal surfaces and many applications as, for instance, the half-space theorem. Afterwards, we focus on the theory of complete minimal surfaces with finite total curvature, with which we can analyse the asymptotic development of complete minimal embedded ends with finite total curvature. This dissertation culminates with the Schoen s theorem, which states that the only complete, connected minimal surfaces with finite total curvature and exactly two ends - both embedded - are a pair of planes or a catenoid.

Cette thèse à pour objet l’étude des résonances du laplacien sur les variétés à pointes. Ce sont des variétés dont les bouts sont des pointes hyperboliques réelles. Ces objets ont été introduits par Selberg pour les surfaces à pointes de courbure constante dans les années 50. Leur définition a ensuite été étendue en courbure variable par Lax et Phillips. Les résonances sont les poles d’une famille méromorphe de fonctions propres généralisées du laplacien. Elles sont associées au spectre continu du laplacien. Pour analyser ce spectre continu, plusieurs directions de recherche sont explorées ici. D’une part, on obtient des résultats sur la localisation de ces résonances. En particulier, si la courbure est négative, on montre que pour un ensemble générique de métriques, les résonances se séparent en deux ensembles. Le premier est contenu dans une bande près du spectre continu. L’autre partie est composé de résonances qui s’éloignent du spectre. Ceci laisse une zone de taille log sans résonance.D’autre part, on étudie les mesures microlocales associées à certaines suites de paramètre spectraux. En particulier, on montre que pour des suites de paramètres spectraux qui s’approche du spectre, mais pas trop vite, la mesure microlocale associée est nécessairement la mesure de Liouville. Cette propriété est valable quand la courbure de la variété est négative. / In this thesis, we study the resonances of the Laplace operator on cusp manifolds. They are manifolds whose ends are real hyperbolic cusps. The resonances were introduced by Selberg in the 50's for the constant curvature cusp surfaces. Their definition was later extended to the case of variable curvature by Lax and Phillips. The resonances are the poles of a meromorphic family of generalized eigenfunctions of the Laplace operator. They are associated to the continuous spectrum of the Laplace operator. To analyze this continuous spectrum, different directions of research are investigated.On the one hand, we obtain results on the localization of resonances. In particular, if the curvature is negative, for a generic set of metrics, they split into two sets. The first one is included in a band near the spectrum. The other is composed of resonances that are far from the spectrum. This leaves a log zone without resonances. On the other hand, we study the microlocal measures associated to certain sequences of spectral parameters. In particular we show that for some sequences of parameters that converge to the spectrum, but not too fast, the associated microlocal measure has to be the Liouville measure. This property holds when the curvature is negative.

As pontes de concreto armado são obras de arte de extrema importância para a infraestrutura de transportes do Brasil. Portanto sua inspeção e manutenção são atividades estratégicas. A inspeção visual, ensaios destrutivos e não destrutivos fornecem informações sobre a sua integridade estrutural e auxiliam na tomada de decisões relativas à necessidade de reparos e reforços. Entre os ensaios não destrutivos, avalia-se neste trabalho a aplicação da identificação modal estocástica na detecção de danos em pontes. A técnica baseia-se na medição das vibrações ambientais da estrutura, aquelas que ocorrem durante seu uso, identificação de suas propriedades modais, comparação com as propriedades modais da estrutura íntegra e consequente detecção de danos. Diferentemente da análise dinâmica experimental clássica, na identificação modal estocástica as ações dinâmicas não são medidas e nem controladas durante o ensaio. Por este motivo foram adotadas técnicas de identificação modal baseadas apenas nas vibrações medidas em alguns pontos da estrutura, funções de densidade espectral de potência e transmissibilidades de vibrações entre os pontos. Desta forma as frequências naturais e modos de vibração experimentais puderam ser precisamente identificados em modelos íntegros e danificados de pontes em escala reduzida. Em cada modelo, uma danificação foi imposta em uma de suas longarinas no meio do vão ou no segundo quarto de vão. Após a realização dos ensaios dinâmicos nas condições íntegra e danificada, duas técnicas de identificação de danos foram utilizadas: Diferença de Curvatura Modal (DCM) e Índice de Dano (ID). Ambas as técnicas tiveram sucesso na detecção de danos nos modelos de pontes avaliados. / Reinforced concrete bridges are extremely important elements of Brazilian transportation infrastructure. Consequently their inspection and maintenance are strategic activities. Visual inspection, destructive or nondestructive tests offer relevant information on their structural integrity and support the decision on the need of retrofitting or strengthening. Among existing types of nondestructive tests, this work focuses on the application of stochastic modal identification in damage detection of bridges. This technique is based on the measurement of environmental vibrations that occur during normal operation of the structure, modal identification, comparison of modal properties between damaged and undamaged bridge and finally damage detection. Opposed to classical dynamic experimental analysis, in stochastic modal identification the loads are not measured or known during the test. For this reason modal identification was only based in vibrations measured in selected points of the structure, power spectral density functions and vibration transmissibilities between these points. With this method natural frequencies and experimental modal shapes could be precisely identified in damaged and undamaged small scale models of bridges. The damage was induced in the middle of the span or in the second quarter of the span in one of the girders. After dynamic testing in undamaged and damaged conditions two damage identification techniques were used: Modal Curvature Difference (MCD) and Damage Index (ID). Both techniques detected successfully the damages imposed to the bridge models.

Nos últimos anos têm sido desenvolvidos modelos de regressão beta, que têm uma variedade de aplicações práticas como, por exemplo, a modelagem de taxas, razões ou proporções. No entanto, é comum que dados na forma de proporções apresentem zeros e/ou uns, o que não permite admitir que os dados provêm de uma distribuição contínua. Nesta tese, são propostas, distribuições de mistura entre uma distribuição beta e uma distribuição de Bernoulli, degenerada em zero e degenerada em um para modelar dados observados nos intervalos [0, 1], [0, 1) e (0, 1], respectivamente. As distribuições propostas são inflacionadas no sentido de que a massa de probabilidade em zero e/ou um excede o que é permitido pela distribuição beta. Propriedades dessas distribuições são estudadas, métodos de estimação por máxima verossimilhança e momentos condicionais são comparados. Aplicações a vários conjuntos de dados reais são examinadas. Desenvolvemos também modelos de regressão beta inflacionados assumindo que a distribuição da variável resposta é beta inflacionada. Estudamos estimação por máxima verossimilhança. Derivamos expressões em forma fechada para o vetor escore, a matriz de informação de Fisher e sua inversa. Discutimos estimação intervalar para diferentes quantidades populacionais (parâmetros de regressão, parâmetro de precisão) e testes de hipóteses assintóticos. Derivamos expressões para o viés de segunda ordem dos estimadores de máxima verossimilhança dos parâmetros, possibilitando a obtenção de estimadores corrigidos que são mais precisos que os não corrigidos em amostras finitas. Finalmente, desenvolvemos técnicas de diagnóstico para os modelos de regressão beta inflacionados, sendo adotado o método de influência local baseado na curvatura normal conforme. Ilustramos a teoria desenvolvida em um conjuntos de dados reais. / The last years have seen new developments in the theory of beta regression models, which are useful for modelling random variables that assume values in the standard unit interval such as proportions, rates and fractions. In many situations, the dependent variable contains zeros and/or ones. In such cases, continuous distributions are not suitable for modeling this kind of data. In this thesis we propose mixed continuous-discrete distributions to model data observed on the intervals [0, 1],[0, 1) and (0, 1]. The proposed distributions are inflated beta distributions in the sense that the probability mass at 0 and/or 1 exceeds what is expected for the beta distribution. Properties of the inflated beta distributions are given. Estimation based on maximum likelihood and conditional moments is discussed and compared. Empirical applications using real data set are provided. Further, we develop inflated beta regression models in which the underlying assumption is that the response follows an inflated beta law. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, Fishers information matrix and its inverse. Interval estimation for different population quantities (such as regression parameters, precision parameter, mean response) is discussed and tests of hypotheses on the regression parameters can be performed using asymptotic tests. We also derive the second order biases of the maximum likelihood estimators and use them to define bias-adjusted estimators. The numerical results show that bias reduction can be effective in finite samples. We also develop a set of diagnostic techniques that can be employed to identify departures from the postulated model and influential observations. To that end, we adopt the local influence approach based in the conformal normal curvature. Finally, we consider empirical examples to illustrate the theory developed.

Dans ce Manuscrit, nous étudions l'influence de la géométrie sur les équations de Hardy-Sobolev perturbées ou non sur toute variété Riemannienne compacte sans bord de dimension supérieure ou égale à 3. Plus précisément, dans le cas non perturbé nous démontrons que pour toute dimension de la variété strictement supérieure à, l'existence d'une solution (ou plutôt une condition suffisante d'existence) dépendra de la géométrie locale autour de la singularité. En revanche, dans le cas où la dimension est égale à 3, c'est la géométrie globale (particulièrement, la masse de la fonction de Green) de la variété qui comptera. Dans le cas d'une équation à terme perturbatif sous-critique, nous démontrons que l'existence d'une solution dépendra uniquement de la perturbation pour les grandes dimensions et qu'une interaction entre la géométrie globale de la variété et la perturbation apparaîtra en dimension 3. Enfin, nous établissons une inégalité optimale de Hardy-Sobolev Riemannienne, la variété étant avec ou sans bord, où nous démontrons que la première meilleure constante est celle des inégalités Euclidiennes et est atteinte / In this Manuscript, we investigate the influence of geometry on the Hardy-Sobolev equations on the compact Riemannian manifolds without boundary of dimension greateror equal to 3. More precisely, we prove in the non perturbative case that the existence of solutions depends only on the local geometry around the singularity when the dimension is greater or equal to 4 while it is the global geometry of the manifold when the dimension is equal to 3 that matters. In the presence of a perturbative subcritical term, we prove that the existence of solutions depends only on the perturbation when the dimension is greater or equal to 4 while an interaction between the perturbation and the global geometry appears in dimension 3. Finally, we establish an Optimal Hardy-Sobolev inequality for all compact Riemannian manifolds, with or without boundary, where we prove that the Riemannian sharp constant is the one for the Euclidean inequality and is achieved

Les travaux présentés dans cette thèse portent sur le rôle que peuvent jouer les différentes courbures extrinsèques d’une hypersurface dans l’étude de sa géométrie, en particulier dans le cas des variétés spinorielles. Dans un premier temps, nous nous intéressons au cas de la courbure moyenne et construisons une nouvelle famille de surfaces minimales non simplement connexes dans le groupe de Lie Sol3, en adaptant une méthode déjà utilisée par Daniel et Hauswirth dans Nil3 et utilisant les propriétés de l’application de Gauss d’une surface. Ensuite, nous démontrons le Théorème d’Alexandrov généralisé aux Hr-courbures dans l’espace euclidien Rn+1 et dans l’espace hyperbolique Hn+1 en testant un spineur adéquat dans des inégalités de type holographiques établies récemment par Hijazi, Montiel et Raulot. Grâce à ces inégalités, nous démontrons également l'Inégalité de Heintze-Karcher dans l'espace euclidien. Enfin, nous donnons des majorations extrinsèques de la première valeur propre de l’opérateur de Dirac des surfaces de S2 x S1(r) et des sphères de Berger Sb3 (τ) grâce aux restrictions de spineurs ambiants construits par Roth, et nous en caractérisons les cas d’égalité. / In this thesis we are interested in the role played by the extrinsic curvatures of a hypersurface in the study of its geometry, especially in the case of spin manifolds. First, we focus our attention on the mean curvature and construct a new family of non simply connected minimal surfaces in the Lie group Sol3, by adapting a method used by Daniel and Hauswirth in Nil3 based on the properties of the Gauss map of a surface. Then we give a new spinorial proof of the Alexandrov Theorem extended to all Hr-curvatures in the euclidean space Rn+1 and in the hyperbolic space Hn+1, using a well-chosen test-spinor in the holographic inequalities recently obtained by Hijazi, Montiel and Raulot. These inequalities lead to a new proof of the Heintze-Karcher Inequality as well. Finally we use restrictions of particular ambient spinor fields constructed by Roth to give some extrinsic upper bounds for the first nonnegative eigenvalue of the Dirac operator of surfaces immersed into S2 x S1(r) and into the Berger spheres Sb3 (τ), and we describe the equality cases.

This thesis is devoted to the optical properties of low-dimensional structures based on such two-dimensional materials as graphene, silicene and phosphorene. We investigate optical properties of a variety of quasi-one dimensional and quasi-zero-dimensional structures, which are promising for future optoelectronics. Primarily we focus on their low-energy optical properties and how these properties are influenced by the structures’ geometry, external fields, intrinsic strain and edge disorder. As a consequence of this endeavor, we find several interesting effects such as correlation between the optical properties of tubes and ribbons whose periodic and ‘hard wall’ boundary conditions are matched and a universal value of matrix element in narrow-gap tubes and ribbons characterizing probability of transitions across the band gap opened up by intrinsic strain originating from the tube’s surface curvature or ribbon’s edge relaxation. The analytical study of the gapped 2D Dirac materials such as silicene and germanene, which have some similarity to the aforementioned quasi-one-dimensional systems in terms of physical description, reveals a valley- and polarization-dependent selection rules. It was also found that absorption coefficient should change in gapped materials with increasing frequency and become a half of its value for gap edge transitions when the spectrum is linear. Our analysis of the electronic properties of flat clusters of silicene and phosphorene relates the emergence and the number of the peculiar edge states localized at zero energy, so-called zero-energy states, which are know to be of topological origin, to the cluster’s structural characteristics such as shape and size. This allows to predict the presence and the number of such states avoiding complicated topological arguments and provides a recipes for design of metallic and dielectric clusters. We show that zero-energy states are optically active and can be efficiently manipulated by external electric field. However, the edge disorder is important to take into account. We present a new fractal-based methodology to study the effects of the edge disorder which can be applied also to modeling of composite materials. These finding should be useful in design of optoelectronic devices such as tunable emitters and detectors in a wide region of electromagnetic spectrum ranging form the mid-infrared and THz to the optical frequencies.

Les voies d’endocytose qui régissent l’internalisation d’éléments extracellulaires peuvent être classées selon que la protéine de manteau, la clathrine, est impliquée ou non dans le processus. Les voies indépendantes de la clathrine sont utilisées par de nombreuses toxines, des virus et des protéines endogènes. Les mécanismes permettant d’induire le recrutement des protéines cargoes et la déformation de la membrane plasmique dans le contexte de l'endocytose clathrine-indépendant restent encore mal compris. Cette étude montre que la galectine 3, une protéine humaine qui se lie aux glucides, induit la formation d’invaginations de la membrane plasmique de manière indépendante de la clathrine. Les glycosphingolipides (molécules jouant un rôle majeur dans la physiologie de la cellule) sont essentielles pour permettre à la galectine 3 d’induire ces invaginations et d’être internalisée dans la cellule. Les structures tubulaires induites par la galectine 3 présentent une morphologie étonnamment similaire à celle de compartiments intermédiaires de transport décrits dans la littérature pour l’endocytose indépendante de la clathrine. Des cargos utilisant la voie indépendante de la clathrine, tels que CD44 et les intégrines α5 et β1, sont retrouvés dans les tubules induits par la galectine 3. De plus, cette dernière est nécessaire à l’internalisation de CD44. Cela indique donc que la galectine 3 pourrait relier des protéines cargos glycosylés à des glycosphyngolipides de la membrane plasmique et ainsi induire une déformation de la membrane et leur internalisation dans les cellules. Ce mécanisme diffère de celui utilisé par la toxine pentamérique de Shiga et par la toxine cholérique, qui sont leur protéines cargos propores et interagissant directement avec le glycosphyngolipide leur servant de récepteur. Les tubules induits par la galectine 3 sont distincts de ceux induit par les autres lectines. Celles-ci présentent des spécificités différentes de liaison aux glucides, montrant ainsi la l'importance des interactions entre les lectines et les sucres dans ce processus. De plus, nous avons constaté que la galectine 3 module l’équilibre à l’état basal de l’intégrine β1 à la surface de la cellule. Cette protéine étant capitale pour les phénomènes d’adhésion et de migration cellulaires, nous avons donc exploré le rôle conjoint de la galectine 3 et des glycosphingolipides dans la migration cellulaire. La galectine 3 inhibe la migration des cellules humaines de carcinomes mammaires alors qu’elle stimule au contraire celle de cellules de tumeurs mammaires murines. Or, nous avons montré que la régulation par la galactine 3 de la migration de différentes lignées cellulaires est dépendante des glycosphingolipides. Il ressort donc de cette étude que la galectine 3 et les glycosphingolipides contribuent de manière synergique au processus d’induction de déformation de la membrane, à l’endocytose de protéines cargos et à la migration cellulaire. / Endocytic processes which govern the uptake of extracellular material into the cell can be classified based on their dependence on the coat protein, clathrin. Clathrin-independent mechanisms are used by many toxins, viruses and endogenous proteins. How cargo is recruited and membranes are bent is not well understood in these cases. Here, we discovered that galectin 3, a human carbohydrate binding protein induced the clathrin-independent formation of endocytic plasma membrane invaginations. Glycosphingolipids, which have established functions in key physiological processes, were found to be essential for the formation of galectin 3-induced invaginations and for the efficient uptake of the protein into the cell. Galectin 3-induced tubular structures were found to have a strikingly similar morphology to that of the clathrin-independent carriers described in literature. Clathrin-independent endocytic cargoes such as CD44, α5 and β1 integrin were present in galectin 3-induced tubules, and galectin activity and glycosphingolipids were required for the uptake of CD44. This indicated that galectin 3 could link glycosylated cargoes with glycosphingolipids for cargo recruitment and membrane bending. In contrast, the pentameric Shiga and cholera toxins are their own cargoes and drive membrane deformations by directly binding to their respective glycosphingolipid receptors. Galectin 3-induced tubules were distinct from those induced by lectins with different carbohydrate binding specificities, which revealed the importance of lectin-glycan interaction in this process. Further, we observed that galectin 3 modulated the steady state surface dynamics of β1 integrin, a protein which like CD44 is critical for cell adhesion and migration. Subsequently, we explored the interplay of galectins and glycosphingolipids in cell migration. Galectin 3 inhibited cell migration in human breast carcinoma cells, and stimulated migration in a mouse mammary tumor cell line. However, the regulation of migration by galectin 3 was in both cases found to be dependent on glycosphingolipids. In conclusion, galectin 3 and glycosphingolipids synergistically contribute to the clathrin-independent curvature generation process, cargo endocytosis and cell migration.