Cette thèse est consacrée à l'étude d'une conjecture de Lang et Silverman de minoration de la hauteur de Néron-Tate sur les variétés abéliennes sur les corps de nombres. Le premier chapitre décrit le matériel nécessaire à l'étude des chapitres suivants et fixe les notations et normalisations. On montre dans le second chapitre que la conjecture est vraie pour certaines classes de variétés abéliennes de dimension 2, en particulier pour les jacobiennes ayant potentiellement bonne réduction et restant loin des produits de courbes elliptiques dans l'espace de modules. Le second chapitre renferme aussi des corollaires allant dans la direction de la conjecture de borne uniforme sur la torsion et de majoration uniforme du nombre de points rationnels d'une courbe de genre 2. Le troisième chapitre généralise les résultats de minoration du second chapitre aux jacobiennes de courbes hyperelliptiques de genre g supérieur ou égal à 2. Le quatrième chapitre contient une étude de la restriction des scalaires à la Weil et une étude asymptotique de la hauteur des points de Heegner sur les jacobiennes de courbes modulaires. Le cinquième chapitre est une annexe contenant des formules explicites utiles pour la dimension 2 et un paragraphe sur un raisonnement par isogénies.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

The main purpose of this study is to develop a flow solver that employs triangular grids to solve two-dimensional, viscous, laminar, steady, incompressible flows. The flow solver is based on Least Squares Spectral Element Method (LSSEM). It has p-type adaptive mesh refinement/coarsening capability and supports p-type nonconforming element interfaces. To validate the developed flow solver several benchmark problems are studied and successful results are obtained. The performances of two different triangular nodal distributions, namely Lobatto distribution and Fekete distribution, are compared in terms of accuracy and implementation complexity. Accuracies provided by triangular and quadrilateral grids of equal computational size are compared. Adaptive mesh refinement studies are conducted using three different error indicators, including a novel one based on elemental mass loss. Effect of modifying the least-squares functional by multiplying the continuity equation by a weight factor is investigated in regards to mass conservation.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

Ce manuscrit est dédié à la détection et la caractérisation de points d'intérêt dans les maillages. Nous montrons tout d'abord les limitations de la mesure de courbure sur des contours francs, mesure habituellement utilisée dans le domaine de l'analyse de maillages. Nous présentons ensuite une généralisation de l'opérateur SUSAN pour les maillages, nommé SUSAN-3D. La mesure de saillance proposée quantifie les variations locales de la surface et classe directement les points analysés en cinq catégories : saillant, crête, plat, vallée et creux. Les maillages considérés sont à variété uniforme avec ou sans bords et peuvent être réguliers ou irréguliers, denses ou non et bruités ou non. Nous étudions ensuite les performances de SUSAN-3D en les comparant à celles de deux opérateurs de courbure : l'opérateur de Meyer et l'opérateur de Stokely. Deux méthodes de comparaison des mesures de saillance et courbure sont proposées et utilisées sur deux types d'objets : des sphères et des cubes. Les sphères permettent l'étude de la précision sur des surfaces différentiables et les cubes sur deux types de contours non-différentiables : les arêtes et les coins. Nous montrons au travers de ces études les avantages de notre méthode qui sont une forte répétabilité de la mesure, une faible sensibilité au bruit et la capacité d'analyser les surfaces peu denses. Enfin, nous présentons une extension multi-échelle et une automatisation de la détermination des échelles d'analyse qui font de SUSAN-3D un opérateur générique et autonome d'analyse et de caractérisation pour les maillages

This thesis work deals with the most actual topic in Computer Vision field which is scene understanding and this using matching of 3D feature point images. The objective is to make use of Saab’s latest breakthrough in extraction of 3D feature points, to identify the best alignment of at least two 3D feature point images. The thesis gives a theoretical overview of the latest algorithms used for feature detection, description and matching. The work continues with a brief description of the simultaneous localization and mapping (SLAM) technique, ending with a case study on evaluation of the newly developed software solution for SLAM, called slam6d. Slam6d is a tool that registers point clouds into a common coordinate system. It does an automatic high-accurate registration of the laser scans. In the case study the use of slam6d is extended in registering 3D feature point images extracted from a stereo camera and the results of registration are analyzed. In the case study we start with registration of one single 3D feature point image captured from stationary image sensor continuing with registration of multiple images following a trail. Finally the conclusion from the case study results is that slam6d can register non-laser scan extracted feature point images with high-accuracy in case of single image but it introduces some overlapping results in the case of multiple images following a trail.

L’amélioration de la sécurité routière ainsi que le développement des systèmes de transports intelligents sont des enjeux d’avenir dans le secteur automobile avec un essor considérable du véhicule semi autonome et autonome. Les systèmes de sécurité active qui équipent de plus en plus les véhicules commercialisés utilisent des capteurs radar (longue et courte portée) fonctionnant dans les bandes 24 GHz ou 77 GHz. L’étude et la mise au point de tels capteurs peuvent être facilitées via l’utilisation d’une plate-forme de simulation générique permettant de simuler un système radar couplé à son environnement selon des scénarios types prédéfinis. Il est alors nécessaire de disposer d’une représentation fiable et réaliste de l’environnement et des objets présents.Cette thèse aborde la caractérisation et la modélisation du canal de propagation et plus largement de l’environnement radioélectrique en ondes millimétriques pour des applications radar, en termes de phénomènes de propagation (trajets multiples, réflexion, diffraction …) et de cibles électriquement larges. Une combinaison de méthodes asymptotiques a été mise en œuvre afin de permettre l'analyse de problèmes électriquement larges en bande W, tout en réduisant les exigences en temps de calcul et en capacité de mémoire. La précision du simulateur a été évaluée à l’aide d’une campagne de mesures de SER de cibles canoniques et complexes de petite taille (inférieure 6cm) dans une chambre anéchoïque. Le banc de mesure mis en œuvre a permis également de valider une procédure expérimentale de détermination de la signature radar. En effet, la procédure expérimentale a été généralisée à la mesure de la signature radar d’objets de taille réelle, dans un milieu « indoor ». Les mesures effectuées ont montré une bonne adéquation avec les résultats présentés dans la littérature. En outre, ces données expérimentales permettent d’extraire une description de la cible par des points brillants qui modélisent les phénomènes de diffusion et de réflexion spéculaire. La réponse à haute fréquence d’une cible peut être approchée par la somme de réponses de ses points brillants. On propose ainsi de simplifier les signatures mesurées pour maximiser l'efficacité de calcul. Comparé aux modèles géométriques détaillés d’une cible complexe, le modèle de points brillants conduit à une meilleure efficacité des simulations de propagation basées sur des rayons dans des scénarios routiers. Le modèle tient également compte de l’anisotropie des diffuseurs (dans le plan azimutal) en modélisant leurs amplitudes par des gaussiennes. / Improving road safety as well as the development of intelligent transport systems are issues of the future in the automotive sector with a considerable rise of the semi-autonomous and autonomous vehicle. The active safety systems that increasingly equip commercial vehicles use radar sensors (long and short range) operating in the 24 GHz or 77GHz bands. The study and development of such sensors can be facilitated through the use of a generic simulation platform to simulate a radar system coupled to its environment according to predefined standard scenarios. It is then necessary to have a reliable and realistic representation of the environment as well as targets. This thesis deals with the characterization and modelling of the propagation channel for radar applications, in terms of propagation phenomena (multipath, reflection, diffraction …) and electrically large targets. A combination of asymptotic methods was developed for the analysis of electrically large problems in W band, while reducing the requirements in CPU time and memory. The accuracy of the simulator was evaluated with radar cross section measurement of canonical and complex small targets (not exceeding 6 cm) in an anechoic chamber. The developed bench measurement also made it possible to validate an experimental procedure for determining the radar signature. Indeed, the experimental characterization was generalized to characterize various automotive related targets in an “indoor” environment. Measurement results matched well with the results presented in the literature. Moreover, the experimental data allows the extraction of a simple target description in terms of scattering points which model the diffusion and specular reflection phenomena. The high frequency response of a target can be approached by the sum of the responses of its scattering centres. It is thus proposed to simplify the measured signatures in order to increase the computation efficiency. Compared to detailed geometrical representation of a complex target, scattering centre model leads to better efficiency of ray-based propagation simulations of road scenarios. The model also takes into account the scattering centre anisotropy (in the azimuth plan) by modelling their amplitudes by Gaussian ones.

Dans cette thèse nous étudions deux problèmes dans le domaine de la géométrie arithmétique, concernant respectivement les points de torsion des variétés abéliennes et le polylogarithme motivique sur les schémas abéliens. La conjecture de Manin-Mumford (démontrée par Raynaud en 1983) affirme que si A est une variété abélienne et X est une sous-variété de A ne contenant aucune translatée d'une sous-variété abélienne de A, alors X ne contient qu'un nombre fini de points de torsion de A. En 1996, Buium présenta une forme effective de la conjecture dans le cas des courbes. Dans cette thèse, nous montrons que l'argument de Buium peut être utilisé aussi en dimension supérieure pour prouver une version quantitative de la conjecture pour une classe de sous-variétés avec fibré cotangent ample étudiée par Debarre. Nous généralisons aussi à toute dimension un résultat sur la dispersion des relèvements p-divisibles non ramifiés obtenu par Raynaud dans le cas des courbes. En 2014, Kings and Roessler ont montré que la réalisation en cohomologie de Deligne analytique de la part de degré zéro du polylogarithme motivique sur les schémas abéliens peut être reliée aux formes de torsion analytique de Bismut-Koehler du fibré de Poincaré. Dans cette thèse, nous utilisons la théorie de l'intersection arithmétique dans la version de Burgos pour raffiner ce résultat dans le cas où la base du schéma abélien est propre. / In this thesis we approach two independent problems in the field of arithmetic geometry, one regarding the torsion points of abelian varieties and the other the motivic polylogarithm on abelian schemes. The Manin-Mumford conjecture (proved by Raynaud in 1983) states that if A is an abelian variety and X is a subvariety of A not containing any translate of an abelian subvariety of A, then X can only have a finite number of points that are of finite order in A. In 1996, Buium presented an effective form of the conjecture in the case of curves. In this thesis, we show that Buium's argument can be made applicable in higher dimensions to prove a quantitative version of the conjecture for a class of subvarieties with ample cotangent studied by Debarre. Our proof also generalizes to any dimension a result on the sparsity of p-divisible unramified liftings obtained by Raynaud in the case of curves. In 2014, Kings and Roessler showed that the realisation in analytic Deligne cohomology of the degree zero part of the motivic polylogarithm on abelian schemes can be described in terms of the Bismut-Koehler higher analytic torsion form of the Poincaré bundle. In this thesis, using the arithmetic intersection theory in the sense of Burgos, we give a refinement of Kings and Roessler's result in the case in which the base of the abelian scheme is proper.

Interval graphs are well studied structures. Intervals can represent resources like jobs to be sched-uled. Finding maximum independent set in interval graphs would correspond to scheduling maximum number of non-conflicting jobs on the computer. Most optimization problems on interval graphs like independent set, vertex cover, dominating set, maximum clique, etc can be solved efficiently using combinatorial algorithms in polynomial time. Hitting, Covering and Packing problems have been ex-tensively studied in the last few decades and have applications in diverse areas. While they are NP-hard for most settings, they are polynomial solvable for intervals. In this thesis, we consider the generaliza-tions of hitting, covering and packing problems for intervals. We model these problems as min-cost flow problems using non-trivial reduction and solve it using standard flow algorithms. Demand-hitting problem which is a generalization of hitting problem is defined as follows: Given N intervals, a positive integer demand for every interval, M points, a real weight for every point, select a subset of points H, such that every interval contains at least as many points in H as its demand and sum of weight of the points in H is minimized. Note that if the demand is one for all intervals, we get the standard hitting set problem. In this case, we give a dynamic programming based O(M + N) time algorithm assuming that intervals and points are sorted. A special case of the demand-hitting set is the K-hitting set problem where the demand of all the intervals is K. For the K-hitting set problem, we give a O(M2N) time flow based algorithm. For the demand-hitting problem, we make an assumption that no interval is contained in another interval. Under this assumption, we give a O(M2N) time flow based algorithm. Demand-covering problem which is a generalization of covering problem is defined as follows: Given N intervals, a real weight for every interval, M points, a positive integer demand for every point, select a subset of intervals C, such that every point is contained in at least as many intervals in C as its demand and sum of weight of the intervals in C is minimized. Note that if the demand of points are one, we get the standard covering set problem. In this case, we give a dynamic programming based O(M + N log N) time algorithm assuming that points are sorted. A special case of the demand-covering set is the K-covering set problem where the demand of all the points is K. For the K-covering set problem, we give a O(MN2) time flow based algorithm. For the demand-covering problem, we give a O(MN2) time flow based algorithm. K-pack points problem which is a generalization of packing problem is defined as follows: Given N intervals, an integer K, M points, a real weight for every point, select a subset of points Y , such that every interval contains at most K points from Y and sum of weight of the points in Y is maximized. Note that if K is one, we get the standard pack points problem. In this case, we give a dynamic pro-gramming based O(M + N) time algorithm assuming that points and intervals are sorted. For K-pack points problem, we give O(M2 log M) time flow based algorithm assuming that intervals and points are sorted.

The work is focused on monitoring and detail survey of eventual changes in Ostřice river bed in Horní Planá cadastre area. It was done in two phases in different seasons. The work objective is to outline and secure position of cross sections on the sides of the river polygon, followed by their survey. The work consists of two sections {--} theoretical and practical. The theoretical section first explains establishment of a dot field by geodetic points. This is followed by description of a detail dot field, requirements for the individual points, methods of their survey, detail mapping of these points as well as establishment of water stream lengthwise sections and cross sections. The practical part includes characteristics of the surveyed location. In addition there is description of preparatory work, necessary for land survey and the survey itself, using both existing and newly established points of the dot field. At the end there is explanation of auxiliary and detail points calculation and method of their graphic processing. The conclusion includes assessment of changes between April and August surveys of the subject territory.