Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Einstein's equations in vacuum spacetimes with two spacelike killing vectors are explored using affine projection tensor geometry. By doing a semi-conformal transformation on the metric, a new "fiducial" geometry is constructed using a projection tensor fields. This fiducial geometry provides coordinate independent information about the underlying structure of the spacetime without the use of an explicit form of the metric tensor.

The purpose of this thesis is to further develop numerical methods for solving surface problems by utilizing tangential calculus and the trace finite element method. Direct computation on the surface is possible by the use of tangential calculus, in contrast to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. Using tangential calculus, the problem formulation is only dependent on the position and normal vectors of the 3D surface. Tangential calculus thus enables a clean, simple and inexpensive formulation and implementation of finite element methods for surface problems. Meshing techniques are greatly simplified from the end-user perspective by utilizing an unfitted finite element method called the Trace Finite Element Method, in which the basic idea is to embed the surface in a higher dimensional mesh and use the shape functions of this background mesh for the discretization of the partial differential equation. This method makes it possible to model surfaces implicitly and solve surface problems without the need for expensive meshing/re-meshing techniques especially for moving surfaces or surfaces embedded in 3D solids, so called embedded interface problems. Using these two approaches, numerical methods for solving three surface problems are proposed: 1) minimal surface problems, in which the form that minimizes the mean curvature was computed by iterative update of a level-set function discretized using TraceFEM and driven by advection, for which the velocity field was given by the mean curvature flow, 2) elastic membrane problems discretized using linear and higher order TraceFEM, which makes it straightforward to embed complex geometries of membrane models into an elastic bulk for reinforcement and 3) stabilized, accurate vertex normal and mean curvature estimation with local refinement on triangulated surfaces. In this thesis the basics of the two main approaches are presented, some aspects such as stabilization and surface reconstruction are further developed, evaluated and numerically analyzed, details on implementations are provided and the current state of work is presented.

This diploma thesis provides an evaluation of Exponential families of distributions which has a special position in mathematical statistics. Diploma will learn the basic concepts and facts associated with the distribution of exponential type. Especially with focusing on the advantages of exponential families in classical parametric statistics, thus in theory of estimation and hypothesis testing. Emphasis will be placed on one-parameter and multi-parameters systems.

Title: Exponential families in statistical inference Author: Sally Abdel-Maksoud Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D. Supervisor's e-mail address: Daniel.Hlubinka@mff.cuni.cz Abstract: This diploma thesis provides an evaluation of Exponential families of distributions which has a special position in mathematical statistic including appropriate properties for estimation of population parameters, hypothesis testing and other inference problems. Diploma will introduce the basic concepts and facts associated with the distribution of exponential type especially with focusing on the advantages of exponential families in classical parametric statistics, thus in theory of estimation and hypothesis testing. Emphasis will be placed on one-parameter and multi- parameters systems. It also exposes an important concepts about the curvature of a statistical problem including the curvature in exponential families. We will define a quantity that measure how nearly "exponential" the families are. This quantity is said to be the statistical curvature of the family. We will show that the family with a small curvature enjoy the good properties of exponential families Moreover, the properties of the curvature, hypotheses testing and some...

La déformation de matériaux élastique dont l'une au moins des dimensions est petite apparaît dans un grand nombre de structures naturelles ou artificielles pour lesquelles une courbure spontanée est présente. Dans ces travaux de thèse, nous couplons plusieurs approches théoriques à des expériences macroscopiques sur des rubans élastiques afin de comprendre la dynamique d'enroulement de biomembranes ouvertes d'un trou. La motivation est issue d'observations récentes d'enroulements obtenues au cours de la sortie de parasites de la Malaria de globules rouges infectés, et de l'explosion de vésicules polymère. Dans une première partie, nous étudions théoriquement la stabilité d'un pore et la propagation de l'enroulement sur une biomembrane sphérique ouverte. Nous modélisons de façon géométrique l'enroulement toroïdal de la membrane par une spirale d'Archimède de révolution et décentrée. Avec cette hypothèse, nous montrons que la stabilité du pore vis-à-vis de l'enroulement dépend fortement de la tension de ligne et du cisaillement et nous discutons ces résultats dans le cadre de l'enroulement de membranes MIRBCs. De plus, en prenant en compte les différentes sources de dissipation, nous obtenons un très bon accord entre les données expérimentales obtenues pour les MIRBCs et la dynamique d'enroulement obtenue par le calcul. Notre approche montre en particulier que la dissipation dans la membrane due à la redistribution de la matière durant l'enroulement domine sur la dissipation visqueuse dans le milieu.Cependant, la complexité de la géométrie sphérique, ainsi que le nombre limité d'observations microscopiques à l'échelle de la membrane sont une entrave au développement de modèles plus détaillés qui permettraient de décrire complètement le couplage entre écoulement et déformation. Nous avons donc étudié dans une seconde partie la déformation d'enroulement dans le cas de rubans élastiques ayant une courbure spontanée dans différents milieux visqueux et pour différentes conditions élastiques. A grands nombres de Reynolds, en raison de la localisation de la courbure pour les rubans au cours de la propagation du front d'enroulement le long du matériau, nous montrons que l'enroulement atteint rapidement une vitesse de propagation constante. Dans ce régime, le ruban s'enroule sur lui-même de façon compacte, sur un cylindre dont la taille est prévue à partir de la solution de l'onde stationnaire pour l'Elastica. A faible nombre de Reynolds, cependant, se rapprochant des conditions d'enroulement d'une membrane microscopique, nous mettons en évidence l'influence des forces de lubrification sur la nature non-compacte de l'enroulement. La taille globale de la spirale de ruban augmente dans le temps conduisant à une diminution de la puissance élastique libérée et donc à une diminution de la vitesse. Nous discutons dans quelle mesure ces résultats peuvent faire avancer la modélisation de l'enroulement dans les MIRBCs et les vésicules polymère. / Curling deformation of thin elastic surfaces appears in numerous natural and man-made structures where a spontaneous curvature is present. In this thesis, we couple theoretical approaches and macroscopic experiments on elastic ribbons to understand the dynamics of curling of opened bio-membranes, motivated by the need to better understand recent microscopic observations during egress of Malaria infected red blood cells (MIRBC) and bursting of artificial polymersomes.In a first part, we study theoretically pore stability and curling propagation of an initially opened spherical bio-membrane. We model geometrically curling deformation as the revolution of a decentered Archimedean spiral, leading to a prescribed toroidal wrapping of the membrane. In this configuration, we show how the stability of a pore to curling depends strongly on both line-tension and shear elasticity and we discuss these results in relation to the curling of MIRBCs membranes. Moreover, taking into account viscous dissipations, the consequent dynamics we calculate agrees quantitatively well with experimental data obtained during opening of MIRBCs. Our approach shows in particular how the membrane dissipation resulting from the surface redistribution dominates curling dynamics over outer viscous dissipation.However, the complexity of the spherical geometry and the lack of detailed images in microscopic observations hamper the development of more accurate models where the coupling between flow and deformation is fully understood. Subsequently, we study in a second part the curling deformation of macroscopic naturally curved elastic ribbons in different viscous media and elastic conditions. At high Reynolds numbers, due to the tendency of ribbons to localize bending deformations when a curling front travels down the material, we show that curling reaches rapidly a constant propagating velocity. In this regime, the ribbon wraps itself into a compact roll whose size is predicted through the solitary wave solution of the associated Elastica. At low Reynolds numbers, however, closer to the hydrodynamic conditions of curling in microscopic membranes, we show that the strong lubrication forces induce a non-compact curling. The overall size of the spiraling ribbon increases in time leading to a temporal decrease of the released elastic power and therefore a consequent decrease in velocity. We discuss how such discovery sheds a new light on the modeling of curling in MIRBCs and polymersomes.

Wavy type Single Chamber Solid Oxide Fuel Cells (SC-SOFCs) have been shown to be conducive to improving the effective electrochemical reaction area contributing to higher performance, compared with planar type SC-SOFCs of the same diameter. This study presents a fabrication process for wavy type SC-SOFCs with a single fabrication step via co-sintering of a triple-layer structure. The monitoring and observation of the curvature evolution of bi- and triple-layer structures during co-sintering has resulted in an improved process with reduced manufacturing time and effort, as regards the co-sintering process for multi-layer structures. Investigation using in-situ monitoring helps different shrinkage behaviours of each porous layer to minimise mismatched stresses along with avoidance of severe warping and cracking. In the co-sintering of the multi-layer structures, the induced in-plane stresses contribute to curvature evolution in the structure, which can be utilised in the design of a curved multi-layer structure via the co-sintering process. For intermediate temperature SOFCs, the materials used are NiO/CGO for anode; CGO for electrolyte; and LSCF for cathode. These materials are tape-casted with 20μm thickness and assembled for bi- and triple-layer structures by hot pressing. Sintering mismatch stresses have been analysed in bi-layer structures, consisting of NiO/CGO-CGO and CGO-LSCF. The maximum sintering mismatch stress was calculated at interface of bi-layer structure in the top layer. In order to achieve the desired wavy type triple-layer structure, flexible green layers of each component were stacked up and laid on alumina rods to support the curvature during the process. In-situ observation, to monitor the shrinkage of each material and the curvature evolution of the structures, was performed using a long focus microscope (Infinity K-2). With these values, the main factors such as viscosity, shrinkage rate of each material, and curvature rate are investigated to determine the sintering mismatch stresses. This enables the prediction of curvature for triple-layer structure and the prediction is validated by in-situ monitoring of the triple-layer structure co-sintering process. Zero-deflection condition is confirmed to maintain initial shape during co-sintering and helps to minimise the development of undesired curvature in the triple-layer structure. Performance testing of the wavy cell was carried out in a methane-air mixture (CH4:O2 =1:1). In comparison with a planar SC-SOFC, it showed higher OCV which might be attributed to not only macro deformation, but also microstructural distribution affecting the effective gas diffusion paths and electrochemical active sites.

O presente estudo teve como objetivo avaliar a resistência à fadiga cíclica flexural dos instrumentos de níquel- titânio, Hyflex CM (Coltène, EUA) e TF Adaptive (SybronEndo, EUA) em diferentes situações experimentais. Todas as limas que foram selecionadas possuíam conicidade 0,04 e diâmetro de ponta 35. Utilizou-se um dispositivo desenvolvido especificamente para realizar o ensaio flexural dinâmico. Os instrumentos TF Adaptive foram divididos em 3 grupos de acordo com o ângulo de curvatura do ensaio: 45º, 60º e 90º e cada grupo subdividido em 2 subgrupos de acordo com o tipo de movimento: rotação contínua e Adaptive. Cada subgrupo era composto por 15 instrumentos TF Adaptive, totalizando 90 instrumentos. Quinze instrumentos Hyflex CM formavam o grupo 4, no ensaio com ângulo de curvatura 90 graus e rotação contínua. A simulação foi realizada em canais artificiais de aço com ângulo de 45, 60, 90 graus e raio 5m m. O número de ciclos e o tempo em segundos até a fratura foram tabulados e analisados. Entretanto, a fadiga cíclica flexural foi significante maior nos três grupos em movimento Adaptive. E as limas TF Adaptive em seu próprio movimento tiveram maior número de ciclos e tempo até a fratura quando comparadas as Hyflex CM no ensaio de 90 graus. Portanto, conclui-se que o sistema Adaptive (limas TF Adaptive + movimento Adaptive) foi mais seguro à resistência á fadiga flexural, e no ensaio de 90 graus o sistema Adaptive foi mais resistente quando comparado com as limas Hyflex CM no movimento de rotação contínua. / The aim of this work to evaluate the cyclic fatigue resistance flexural the instruments of nickel-titanium, HyFlex CM (Coltène, USA) and TF Adaptive (SybronEndo, USA) in different experimental situations. All files that were selected had 0.04 taper and tip diameter 35. We used a device developed specifically to perform the dynamic flexural test. The TF Adaptive instruments were divided into 3 groups according to the angle of curvature of the test: 45, 60 and 90 and further divided in two subgroups according to the type of movement: continuous rotation and Adaptive. Each group consisted of 15 instruments TF Adaptive totaling 90 instruments. Fifteen HyFlex CM instruments formed the group 4 in the trial of bend angle 90 degrees and continuous rotation. The simulation was performed in artificial steel angled channels 45, 60, 90 degrees and radius 5m m. The number of cycles and the time in seconds until fracture were tabulated and analyzed. However, the cyclical flexural fatigue was significantly greater in the three groups Adaptive motion. And the TF Adaptive files on your own movement had a higher number of cycles and time to fracture when the HyFlex CM compared in 90-degree test. Therefore, it is concluded that the adaptive system (TF Adaptive Motion + Adaptive files) was safer resistance to flexural fatigue, and 90 degrees test Adaptive system is more resistant when compared with the HyFlex files into continuous.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

Orientador: Cláudia Valéria Seullner Brandão / Resumo: Os gatos possuem papel de destaque como animais de estimação, assim como em estudos oftalmológicos comparativos. Pelo conhecimento dos autores, até o momento, a avaliação de bulbo de gatos utilizando tomógrafos e topógrafo associados, por meio dos sistemas de câmeras Scheimpflug duplas e um disco Plácido, não foi descrita na literatura. O objetivo deste estudo foi analisar a córnea em gatos considerando as curvaturas anterior e posterior, o poder total e espessura da córnea pelo sistema de duplo Scheimpflug e disco de plácido. Na ceratometria anterior (SimK) verificou-se valores de 38,37 ± 0,83D, ceratometria posterior -4,89±0,34D e poder corneal total médio 38,04 ± 0,94D. Quanto à espessura corneal central foi obtido valor médio de 617,34 ± 53,38 μm, superior aos encontrados em outros trabalhos utilizando a paquimetria ultrassônica. A utilização do sistema duplo Scheimpflug e disco de plácido foi efetiva em gatos domésticos e permitiu a obtenção de valores ainda não descritos na literatura para a espécie, tais como ceratometria anterior nas zonas da córnea, ceratometria corneal posterior, poder corneal total, volume corneal. / Abstract: Cats have an important role as pets and in ophthalmologic comparative studies. To the autors’ knowledge, this is the first study that evaluates cornea of cats using a device that combine tomography and topography, through dual Scheimpflug cameras and a Placid disc. The aim of this study was to evaluate anterior and posterior keratometry, total corneal power and pachymetry using dual Scheimpflug and Placid disc system. Concerning anterior keratometry (SimK), the value obtained was 38,37 ± 0,83D, posterior keratometry -4,89±0,34D and total corneal power 38,04 ± 0,94D. Regarding the pachymetry, the mean CCT value obtained was 617,34 ± 53,38 μm, superior to those find in other studies using ultrasonic pachymetry. The use of dual Scheimpflug and Placid disc system was effective in domestic cats and enabled the description of values that have not been described in the literature in felines yet, such as anterior keratomemetry in the corneal zones, posterior keratommetry, total corneal power, corneal volume. / Mestre