The inhibiting effect of a sharp edge on liquid spreading is well observed during drop interaction with textured surfaces. On groove-textured solid surfaces comprising unidirectional parallel grooves, the edge effect of posts results in the squeezing of drop liquid in the direction perpendicular to the grooves and the stretching of drop liquid along the grooves leading to anisotropy in drop flow, popularly known as wetting anisotropy which has been employed in several engineering applications. A recent study observed that the energy loss incurring at the edges of posts via contact angle hysteresis is primarily responsible for the anisotropic spreading of impacting drops on groove-textured surfaces. The present study aims to elucidate the role of edges on the spreading and receding dynamics of water drops. The experiments of drop impact are carried out on semi-infinite rectangular post comprising a pair of parallel 90-deg edges separated by a distance (post width) comparable to the diameter of impacting drop. The equilibrium shape of drops on the semi-infinite rectangular post is analyzed using open source computational tool Surface Evolver to optimize the ratio of initial droplet diameter to post width. Quantitative measurements of drop impact dynamics on semi-infinite rectangular posts are deduced by analysing high speed videos of impact process captured under three different camera views during experiments. Based on the role of post edges on impacting drops, different regimes of the impacting drops are characterized in terms of drop Weber number and the ratio of diameter of impacting drop to post width. Characteristic features of impact dynamics in each of the regimes are identified and discussed. It is seen that edges play a pivotal role on all stages of impact dynamics regardless of Weber number. Impacts in the regime of completely pinned drops on narrow posts are further analyzed to reveal characteristics of post-spreading oscillations.

The thesis is concerned with the contact mechanics behaviour of non-conforming solids. The geometry of the solids considered gives rise to various contact configurations, from concentrated contacts with circular and elliptical configuration to those of finite line nature, as well as those of less concentrated form such as circular flat punches. The radii of curvature of mating bodies in contact or impact give rise to these various nonconforming contact configurations and affect their contact characteristics, from those considered as semi-infinite solids in accord with the classical Hertzian theory to those that deviate from it. Furthermore, layered solids have been considered, some with higher elastic modulus than that of the substrate material (such as hard protective coatings) and some with low elastic moduli, often employed as tribological coatings (such as solid lubricants). Other bonded layered solids behave in viscoelastic manner, with creep relaxation behaviour under load, and are often used to dampen structural vibration upon impact. Analytic models have been developed for all these solids to predict their contact and impact behaviour and obtain pressure distribution, footprint shape and deformation under both elastostatic and transient dynamic conditions. Only few solutions for thin bonded layered elastic solids have been reported for elastostatic analysis. The analytical model developed in this thesis is in accord with those reported in the literature and is extended to the case of impact of balls, and employed for a number of practical applications. The elastostatic impact of a roller against a semi-infinite elastic half-space is also treated by analytic means, which has not been reported in literature. Two and three-dimensional finite element models have been developed and compared with all the derived analytic methods, and good agreement found in all cases. The finite element approach used has been made into a generic tool for all the contact configurations, elastic and viscoelastic. The physics of the contact mechanical problems is fully explained by analytic, numerical and supporting experimentation and agreement found between all these approaches to a high level of conformance. This level of agreement, the development of various analytical impact models for layered solids and finite line configuration, and the development of a multi-layered viscoelastic transducer with agreed numerical predictions account for the main contributions to knowledge. There are a significant number of findings within the thesis, but the major findings relate to the protective nature of hard coatings and high modulus bonded layered solids, and the verified viscoelastic behaviour of low elastic modulus compressible thin bonded layers. Most importantly, the thesis has created a rational framework for contact/impact of solids of low contact contiguity.

This thesis is a study of the detection of separation among the sample points in binary logistic regression models. We propose a new algorithm for detecting separation and demonstrate empirically that it can be computed fast enough to be used routinely as part of the fitting process for logistic regression models. The parameter estimates of a binary logistic regression model fit using the method of maximum likelihood sometimes do not converge to finite values. This phenomenon (also known as monotone likelihood or infinite parameters) occurs because of a condition among the sample points known as separation. There are two classes of separation. When complete separation is present among the sample points, iterative procedures for maximizing the likelihood tend to break down, when it would be clear that there is a problem with the model. However, when quasicomplete separation is present among the sample points, the iterative procedures for maximizing the likelihood tend to satisfy their convergence criterion before revealing any indication of separation. The new algorithm is based on a linear program with a nonnegative objective function that has a positive optimal value when separation is present among the sample points. We compare several approaches for solving this linear program and find that a method based on determining the feasibility of the dual to this linear program provides a numerically reliable test for separation among the sample points. A simulation study shows that this test can be computed in a similar amount of time as fitting the binary logistic regression model using the method of iteratively reweighted least squares: hence the test is fast enough to be used routinely as part of the fitting procedure. An implementation of our algorithm (as well as the other methods described in this thesis) is available in the R package safeBinaryRegression.

La philosophie de Spinoza cherche à concilier et réunir trois horizons philosophiques fondamentaux : l’émanation néo-platonicienne (l’expression), le mécanisme cartésien (cause efficiente), et les catégories aristotéliciennes (Substance, attribut, mode). Ce premier point est pris pour acquis. Nous expliquerons que cette tentative sera rendue possible grâce à la conception nouvelle, au 17e siècle, de l’actualité de l’infini. Nous examinerons ensuite les conséquences de cette nouvelle interprétation, qui permet de rendre l’individu transparent à lui-même sur un plan d’immanence, expressif par rapport à une éminence qui le diffuse, mais déterminé dans une substantialité fictive entre objets finis. En proposant le pouvoir de l’imagination et des prophètes comme point de départ et principe actif du conatus, nous montrerons que la distinction, chez Spinoza, demeure toujours une fiction. Pour conclure, nous serons en mesure de signaler en quoi le Zarathoustra de Nietzsche relève d’une volonté de poursuivre le travail entrepris par Spinoza. / Spinoza’s philosophy is an attempt to reconcile three fundamental perspectives : neo-platonician emanation (expression), Cartesian mechanism (efficient cause), and Aristotelian categories (Substance, attribute, mode). This first point is taken for granted. We will argue that this attempt is made possible by the new modernist conception of an actual infinite. We will then examine the consequences of this new interpretation, viz. to allow self-transparency by internal immanence of an individual who is expressive of an eminence that diffuses him, while at the sime time being substantially determined in the knot of finite objects. We will then propose imagination and prophets as a starting point and active principle of the conatus’ effort, in order to show that all distinctions, in Spinoza’s philosophy, rely on a power to forge fictions. In conclusion, we will argue that Nietzsche’s Zarathoustra is an attempt to catch and pursue Spinoza’s intuitions.

Cette thèse étudie les structures dont la théorie au second ordremonadique est décidable, et en particulier la hiérarchie à pile. Onpeut définir celle-ci comme la hiérarchie pour $n$ des graphesd'automates à piles imbriquées $n$ fois ; une définition externe, partransformations de graphes, est également disponible. Nous nousintéressons à l'exemple des ordinaux. Nous montrons que les ordinauxplus petits que $epsilon_0$ sont dans la hiérarchie, ainsi que des graphesporteurs de plus d'information, que l'on appelle "graphecouvrants''. Nous montrons ensuite l'inverse : tous les ordinaux de lahiérarchie sont plus petits que $epsilon_0$. Ce résultat utilise le fait queles ordres d'un niveau sont en fait isomorphes aux structures desfeuilles des arbres déterministes dans l'ordre lexicographique, aumême niveau. Plus généralement, nous obtenons une caractérisation desordres linéaires dispersés dans la hiérarchie. Dans un troisièmetemps, nous resserons l'intérêt aux ordres de type $omega$ --- les mots infinis --- pour montrer que les mots du niveau 2 sont les motsmorphiques, ce qui nous amène à une nouvelle extension au niveau 3 / This thesis studies the structures with decidable monadic second-ordertheory, and in particular the pushdown hierarchy. The latter can bedefined as the family for $n$ of pushdown graphs with $n$ timesimbricated stacks ; another definition is by graph transformations. Westudy the example of ordinals. We show that ordinals smaller that $epsilon_0$are in the hierarchy, along with graphs called "covering graphs'', which carry more data than ordinals. We show then the converse : allordinals of the hierarchy are smaller than $epsilon_0$. This result uses thefact that linear orders of a level are actually isomorphic to thestructure of leaves of deterministic trees by lexicographic ordering, at the same level. More generally, we obtain a characterisation ofscattered linear orders in the hierarchy. We finally focus on the caseof orders of type $omega$ --- infinite words --- and show that morphicwords are exactly words of the second level of the hierarchy. Thisleads us to a new definition of words for level 3

Cette thèse porte sur les modèles génératifs en apprentissage automatique. Deux nouveaux modèles basés sur les réseaux de neurones y sont proposés. Le premier modèle possède une représentation interne où une certaine structure a été imposée afin d’ordonner les caractéristiques apprises. Le deuxième modèle parvient à exploiter la structure topologique des données observées, et d’en tenir compte lors de la phase générative. Cette thèse présente également une des premières applications de l’apprentissage automatique au problème de la tractographie du cerveau. Pour ce faire, un réseau de neurones récurrent est appliqué à des données de diffusion afin d’obtenir une représentation des fibres de la matière blanche sous forme de séquences de points en trois dimensions.

The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively. For theq-state Potts model on an infinite lattice, there areq+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed. The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study aq-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition. Keywords and phrases:Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent.

In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.

Un origami, ou encore une surface à petits carreaux, est l'exemple le plus simple d'une surface de translation. Il s'obtient en collant entre eux un nombre fini de carreaux identiques. Le point le plus intéressant est l'étude du flot linéaire sur un origami, qui est un système dynamique continu lié à la dynamique des billards ou encore celle des échanges d'intervalles. Nous pouvons aussi nous intéresser au stabilisateur de l'action naturelle du groupe spécial linéaire sur les origamis, que nous appelons groupe de Veech de l'origami. Le but de cette thèse est l'étude de ces deux notions sur des exemples d'origamis infinis, obtenus en collant une infinité dénombrable de carreaux entre eux. Ces exemples sont obtenus comme revêtement galoisiens d'origamis finis, avec comme groupe de Galois des groupes abéliens, nilpotents ou plus compliqués. / An origami, or a square-tiled surface, is the simplest example of translation surface. An origami can be viewed as a finite collection of identical squares, glued together along their edges. We can study the linear flow on this origami, which is the geodesic flow for this kind of surfaces. This dynamical system is related to the dynamical system of billiard, or interval exchange transformations. We can also study the Veech group of an origami. The special linear group acts on the space of translation surface, and the Veech group of an origami is the stabilizer of this origami under this action. We know in particular that the Veech group is a fuchsian group. In this thesis, we work on some example of infinite origamis. These origamis are constructed as Galois covering of finite origamis. In these examples, the deck group will be an abelian group, a niltpotent group or something more difficult.

The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.