Ce mémoire organisé en deux parties présente deux méthodes de représentation de la solution transitoire d'un problème de tenue à la mer basées sur l'utilisation de solutions harmoniques. La première partie est consacrée à l'étude une méthode baptisée méthode de décomposition en modes résonants appliquée au problème de tenue à la mer d'une plaque élastique mince. Cette méthode qui peut être vue comme un prolongement analytique de la transformation de Laplace consiste à représenter la réponse transitoire à l'aide d'une superposition discrète de modes résonants exponentiellement amortis. La question à laquelle nous tentons de donner une

Effects of drugs are in clinical trials often measured on categorical scales. These measurements are increasingly being analyzed using mixed-effects logistic regression. However, the experience with such analyzes is limited and only a few models are used. The aim of this thesis was to investigate the performance and improve the use of models and methods for mixed-effects categorical data analysis. The Laplacian method was shown to produce biased parameter estimates if (i) the data variability is large or (ii) the distribution of the responses is skewed. Two solutions are suggested; the Gaussian quadrature method and the back-step method. Two assumptions made with the proportional odds model have also been investigated. The assumption with proportional odds for all categories was shown to be unsuitable for analysis of data arising from a ranking scale of effects with several underlying causes. An alternative model, the differential odds model, was developed and shown to be an improvement, in regard to statistical significance as well as predictive performance, over the proportional odds model for such data. The appropriateness of the likelihood ratio test was investigated for an analysis where dependence between observations is ignored, i.e. performing the analysis using the proportional odds model. The type I error was found to be affected; thus assessing the actual critical value is prudent in order to verify the statistical significance level. An alternative approach is to use a Markov model, in which dependence between observations is incorporated. In the case of polychotomous data such model may involve considerable complexity and thus, a strategy for the reduction of the time-consuming model building with the Markov model and sleep data is presented. This thesis will hopefully contribute to a more confident use of models for categorical data analysis within the area of pharmacokinetic and pharmacodynamic modelling in the future.

This thesis presents the numerical algorithms to carry out the face transformation. The main efforts are denoted to the finite volume method (FVM) with the Delaunay triangulation to solve the Laplace equations in the harmonic transformation undergone in face images. The advantages of the FVM with the Delaunay triangulation are: (1) Easy to formulate the linear algebraic equations, (2) Good to retain the geometric and physical properties, (3) less CPU time needed. The numerical and graphical experiments are reported for the face transformations from a female to a male, and vice versa. The computed sequential and absolute errors are and , where N is division number of a pixel into subpixels. Such computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in [Li and Bui, 1998c].

A Constant Proportion Debt Obligation is a credit derivative which has been introduced to generate a surplus return over a riskless market return. The surplus payments should be obtained by synthetically investing in a risky asset (such as a credit index) and using a linear leverage strategy which is capped for bounding the risk. In this thesis, we investigate two approaches for investigation of constant proportion debt obligations. First, we search for an optimal leverage strategy which minimises the mean-square distance between the final payment and the final wealth of constant proportion debt obligation by the use of optimal control methods. We show that the optimal leverage function for constant proportion debt obligations in a mean-square sense coincides with the one used in practice for geometric type diffusion processes. However, the optimal strategy will lead to a shortfall for some cases. The second approach of this thesis is to develop a pricing formula for constant proportion debt obligations. To do so, we consider both the early defaults and the default on the final payoff features of constant proportion debt obligations. We observe that a constant proportion debt obligation can be modelled as a barrier option with rebate. In this respect, given the knowledge on barrier options, the pricing equation is derived for a particular leverage strategy.

We study the properties of thinning and Markov chain thinning of renewal processes. Among others, we investigate whether some special renewal processes can be obtained through Markov chain thinning.

Die Temperaturverhältnisse beim Ziehen und Wärmebehandeln von Draht werden mit mathematisch-analytischen Methoden auf der Grundlage der FOURIERschen Wärmeleitungsgleichung eingehend untersucht. Insbesondere wird unter den spezifischen Wärmeübergangsbedingungen zwischen Draht und Ziehdüse sowie zwischen Draht und Ziehtrommel deren thermische Wechselwirkung analysiert. Ein Näherungsverfahren zur Berechnung der Drahttemperaturen in Zugfolgen unter Berücksichtigung des Ziehdüseneinflusses wird angegeben und mit einem Beispiel zum Nassziehen stark verzinkten Stahldrahts illustriert. Aus geschwindigkeitsabhängig gemessenen Änderungen des Drahtdurchmessers werden unter thermoelastischer Ziehringdurchmesserkorrektur Schmierfilmdicken bestimmt. Diffusionsgleichungen werden analysiert und ein Zusammenhang zur Reaktionskinetik wird hergestellt. Ein neues reaktionskinetisches Werkstoffmodell, das insbesondere auch im Falle stärker anisothermer Verhältnisse, also bei Kurzzeitwärmebehandlung anwendbar ist, wird vorgestellt.

Les voies colloïdales de préparation de catalyseurs hétérogènes offrent des perspectives prometteuses dans le domaine du raffinage et de la pétrochimie.<br />Leur caractérisation dynamique reste pourtant insuffisante pour pouvoir décrire complètement les phénomènes physico-chimiques contrôlant l'état final du catalyseur.<br /><br />La relaxométrie RMN des systèmes hétérogènes possède une résolution temporelle suffisante pour le suivi de ce type de synthèse. Elle permet<br />d'accéder à l'avancement de la<br />réaction et de caractériser la formation des agrégats, leur synérèse et leur structure fractale.<br /><br />Avant l'application de cette technique au suivi de synthèses de nanoparticules d'oxyde de palladium,<br />des aspects fondamentaux nécessaires à son développement ont été abordés, comme l'étude de<br />la couche de solvatation liée en surface.<br />Ceci permet d'élargir le champ d'application de cette technique à tout système colloïdal d'oxyde :<br />allant des poudres aux sols en passant par les agrégats.

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.

In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.

Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases. In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method. An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T. In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.