Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.

Cette thèse s'intéresse à la factorisation des fonctions zêta des hypersurfaces de Dwork. Candelas, de la Ossa et Rodriguez-Villegas ont mis en évidence, dans le cas de la quintique, un facteur provenant de la symétrie miroir et deux facteurs provenant de courbes de type hypergéométrique. Wan a établit le lien avec la symétrie miroir dans le cas général, mais les facteurs complémentaires n'ont pas été étudiés avec le même niveau de détail que dans le cas de la quintique, et c'est sur eux que se concentre cette thèse. Après un premier chapitre de rappels sur les hypersurfaces de Dwork, on détermine, dans le chapitre 2, une factorisation explicite des fonctions zêta en terme de facteurs provenant d'hypersurfaces de type hypergéométrique. Dans le chapitre 3, on déduit une factorisation à partir d'une décomposition isotypique de la cohomologie des hypersurfaces de Dwork. Finalement, dans le chapitre 4, on relie les deux factorisations précédentes.

Proposition d'une formulation combinatoire pour la conception d'algorithmes de produit matriciel sur les réseaux systoliques linéaires. Étude de la validation des algorithmes systoliques. Deux logiciels sont proposes: le premier, Sisyc, est un simulateur numérique d'algorithmes systoliques; le second, sisyc2, calcule la trace symbolique des algorithmes systoliques et permet lorsqu'il est couple avec un système de calcul formel, de réaliser une simulation formelle

En 2001, Barannikov a montré que la variété de Frobenius provenant de la cohomologie quantique de l'espace projectif complexe est isomorphe à la variété le Frobenius associée à un polynôme de Laurent. <br /> <br /> L'objectif de cette thèse est de généraliser ce résultat. Plus précisément, nous montrons, modulo une conjecture sur la valeur de certains invariants de Gromov-Witten orbifold, que la structure de Frobenius obtenue sur la cohomologie quantique orbifolde de l'espace projectif à poids est isomorphe à celle obtenue à partir d'un certain polynôme de Laurent.

Les trous noirs primordiaux sont une sonde exceptionnelle pour rechercher des effets de nouvelle physique, à l'intersection de la relativité générale, de la mécanique quantique, de la physique des particules et de la cosmologie. Ce mémoire présente quelques pistes d'études relatives à ces objets astrophysiques fascinants. D'abord, autour de leur recherche via l'étude des rayons cosmiques qui seraient émis par évaporation de Hawking. Des liens entre les limites obtenues et les modèles d'inflation sont ensuite proposés afin d'obtenir une borne supérieure très contraignante - et totalement inaccessible aux observables usuelles que sont le fond diffus et les grandes structures - sur la puissance aux petites échelles dans l'Univers primordial. La fin de l'évaporation des trous noirs est etudiée en gravité de corde et leur statut de candidat à la matière noire froide revisité dans le cadre des modèles à brisure d'invariance d'échelle. Enfin, dans le cadre des modèles à basse échelle de Planck (c'est-à-dire présentant de larges dimensions supplémentaires), la formation de trous noirs auprès des collisionneurs est envisagée. Nous montrons que des effets de gravité quantique (couplage de Gauss-Bonnet) pourraient être sondés au LHC. Quelques voies d'investigations futures, liées à la présence d'une constante cosmologique ou au rayonnement cosmique d'énergie extrême sont esquissées.

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi-Monte Carlo applications. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

I detta kandidatarbete undersöker och jämför vi två olika metoder för att approximera gauss- och medelkrökningen hos en yta i rummet som är given som en mängd av punkter. Det är viktigt att försöka få en bra analogi mellan diskret krökning och analytisk krökning då man ofta startar med en mängd punkter i de praktiska fallen, som t ex i tillverkningsindustrin, igenkänning av objekt (inscannade bilder) och datorgrafik. Givet dessa punkter och en bra approximation av gauss- och medelkrökningen kan man få mer information om ytans geometri och beteende. För att kunna förstå dessa begrepp och metoder/algoritmer så behandlas först den bakomliggande teorin och sedan metoderna. Den första metoden är att återge ytan med hjälp av Bézierytor, vilka vi kan utföra geometriska operationer på utan problem och även få fram gauss- och medelkrökningen. Den andra metoden kommer från artikeln ``Discrete Differential-Geometry Operators for Triangulated 2-Manifolds'' av Mark Meyer, Mathieu Desbrun, Peter Schröder och Alan H. Barr. Deras approximationer av krökningarna kräver en triangulering av ytan, vilket de inte ger någon algoritm för. De tittar på ett område runt varje punkt och approximerar krökningarna genom detta område, även Gauss-Bonnets sats används för approximering av gausskrökningen. Mina simuleringar visar att Bézierytornas approximationer av gauss- och medelkrökningar är konvergenta och att alla värden ligger relativt nära varandra. Artikelns algoritm fungerar bra för gauss- och medelkrökning men deras algoritm beror väldigt mycket på trianguleringen vilket gör att man behöver ha krav på den triangulerade ytan, vilket i sig är ett svårt problem att lösa. / In this thesis we analyze and compare two different methods for approximating the Gauss and mean curvature on a surface, which is given as a set of points. It is important to find a method that agrees well with the analytic Gauss and mean curvatures and guarantees robust estimations. There is a great interest in Gauss and mean curvature since these two curvatures give information about the local geometry of the surface around the point at which these curvatures are calculated. The thesis begins with a short overview of differential theory and then the methods are explained and described. The reason for this is to give the reader an understanding of the theory before explaining the methods. The first method is called Bézier surfaces, which interpolates the given points. These surfaces are differentiable which makes it possible to approximate the Gauss and mean curvature, and are therefore very well suited for our problem. The second method comes from the research article ``Discrete Differential-Geometry Operators for Triangulated 2-Manifolds'' by Mark Meyer, Mathieu Desbrun, Peter Schröder and Alan H. Barr. Their algorithm requires a triangulated surface, which itself is a hard problem to solve (at least if one has requirements on the triangulation). Their approximations of the Gauss and mean curvatures use a well chosen area around the point, and the Gauss curvature also makes use of the Gauss-Bonnet theorem. My simulations show that Bézier surfaces approximate both Gauss and mean curvature well, and the approximations seem to converge to the analytic value when the information gets better. The articles algorithm also works well for approximating both curvatures, though this method seems to depend somewhat on the triangulation. This gives some requirements on the triangulation and will therefore be a harder problem to solve. The approximations do not converge when given a triangulation with obtuse triangles, though it shows signs to do so.

In this thesis, sensitivity and error analysis of a differential rectification method were performed by using digital images taken by a frame camera onboard BILSAT and pushbroom scanner on ASTER. Three methods were implemented for Sensitivity and Uncertainty analysis: Monte Carlo, covariance analysis and FAST (Fourier Amplitude Sensitivity Test). A parameter estimation procedure was carried out on the basis of so called Mixed Model extended by some suitable additional regularization parameters to stabilize the solution for improper geometrical conditions of the imaging system. The effectiveness and accuracy of the differential rectification method were compared with other rectification methods and the results were analyzed. Furthermore the differential method is adapted to the pushbroom scanners and software which provides rectified images from raw satellite images was developed.

Numerical solutions of mathematical modelizations of heat and mass transfer in cubical and cylindrical reactors of solar adsorption refrigeration systems are studied. For the resolution of the equations describing the coupling between heat and mass transfer, Bubnov-Galerkin method is used. An exact solution for time dependent heat transfer in cylindrical multilayered annulus is presented. Separation of variables method has been used to investigate the temperature behavior. An analytical double series relation is proposed as a solution for the temperature distribution, and Fourier coefficients in each layer are obtained by solving some set of equations related to thermal boundary conditions at inside and outside of the cylinder.

In 1975, H. Fujimoto generalized Nevanlinna's known results for meromorphic fonctions to the case of meromorphic mappings of Cn into PN(C). He proved that for two linearly nondegenerate meromorphic mappings f and g of C into PN(C). if they have the saine inverse images counted with multiplicities for 3N + 2 hyperplanes in general position in PN(C) then f = g. After that, this problem has been studied intensively by a number of mathematicans as H. Fujimoto, W. Stoll, L. Smiley, M. Ru, G. Dethloff - T. V. Tan, D. D. Thai - S. D. Quang, Chen-Yan and so on. Parallel to the development of Nevanlinna theory, the value distribution theory of the Gauss map of minimal surfaces immersed in Rm vas studied by many mathematicans as R. Osserman, S.S. Chern, F. Xavier, H. Fujimoto, S. J. Kao, M. Ru and many other mathematicans. In this thesis, we continuous studing some problems on these directions. The main goals of the thesis are followings. * Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) sharing 2N + 2 fixed hyperplanes.* Unicity theorems with truncated multiplicities of meromorphic mappings of Cn into PN(C) for moving targets, and a small set of identity.