Les actions de groupes en géométrie symplectique et l'application moment

Ce mémoire porte sur quelques notions appropriées d'actions de groupe sur les variétés symplectiques, à savoir en ordre décroissant de généralité : les actions symplectiques, les actions faiblement hamiltoniennes et les actions hamiltoniennes. Une connaissance des actions de groupes et de la géométrie symplectique étant prérequise, deux chapitres sont consacrés à des présentations élémentaires de ces sujets. Le cas des actions hamiltoniennes est étudié en détail au quatrième chapitre : l'importante application moment y est définie et plusieurs résultats concernant les orbites de la représentation coadjointe, tels que les théorèmes de Kirillov et de Kostant-Souriau, y sont démontrés. Le dernier chapitre se concentre sur les actions hamiltoniennes des tores, l'objectif étant de démontrer le théorème de convexité d'Atiyha-Guillemin-Sternberg. Une discussion d'un théorème de classification de Delzant-Laudenbach est aussi donnée. La présentation se voulant une introduction assez exhaustive à la théorie des actions hamiltoniennes, presque tous les résultats énoncés sont accompagnés de preuves complètes. Divers exemples sont étudiés afin d'aider à bien comprendre les aspects plus subtils qui sont considérés. Plusieurs sujets connexes sont abordés, dont la préquantification géométrique et la réduction de Marsden-Weinstein.

La croissance maxillaire et mandibulaire et l’apnée du sommeil chez les enfants présentant une séquence de Pierre Robin

Introduction : La croissance maxillo-mandibulaire des enfants avec une séquence de Pierre Robin (SPR) est controversée dans la littérature. Certains auteurs croient que la croissance mandibulaire est accélérée après la naissance, mais peu se sont penchés sur la croissance du maxillaire supérieur. Cette étude rétrospective sur dossier vise à analyser la croissance maxillo-mandibulaire des enfants atteints de la SPR. Dans un deuxième temps, nous aurions aimé évaluer la sévérité et l’évolution de l’apnée du sommeil en lien avec la croissance des maxillaires, mais un manque de données a empêché l’atteinte de cet objectif. Matériel et méthode : Les dossiers médicaux et orthodontiques de 93 patients (82 volet apnée et 40 volet croissance) du CHU Ste-Justine avec une SPR isolée ont été révisés puis comparés au groupe contrôle composé d’enfants normaux de l’Université du Michigan. L’analyse statistique de modèle mixte pour mesures répétées de même que celle de Brunner-Langer furent effectuées. Résultats : L’évaluation orthodontique a montré un changement statistiquement significatif pour la relation molaire droite, la présence de chevauchement et de diastème au maxillaire et le surplomb vertical. L’analyse des données céphalométriques nous montre que le maxillaire supérieur, la branche montante et le corps de la mandibule sont tous réduits par rapport à la normale. Ce dernier montre une diminution significative avec l’âge (p = 0,03). L’angle gonial, le SNA, SNB, ANB, l’angle de convexité faciale et l’inclinaison de l’incisive supérieure par rapport à FH sont tous normaux. Par contre, on remarque une augmentation statistiquement significative de cette dernière avec l’âge (p = 0,04). L’angle Y est augmenté tandis que les hauteurs faciales supérieure (HFS) et inférieure (HFI) sont diminuées bien que cette dernière montre une tendance à s’approcher de la normale avec l’âge (p ≤ 0,001). Discussion : Les dimensions des maxillaires sont similaires à plusieurs études. En ce qui concerne la mandibule, la croissance est soit plus lente, soit diminuée. Cette observation est plus marquée lorsque l’on s’approche du pic de croissance puisque l’écart par rapport à la normale s’agrandit. On voit une tendance à la croissance hyperdivergente qui pourrait expliquer l’augmentation de la HFI avec l’âge. Le fait que SNA et SNB soient dans la normale pourrait s’expliquer par une diminution de la longueur de la base crânienne. Conclusion : Il n’y a pas de rattrapage de croissance maxillaire et mandibulaire. Les maxillaires restent micrognathes quoique proportionnels l’un envers l’autre et le profil est convexe tout au long de la croissance. La comparaison des données céphalométriques et des traitements orthodontiques avec ceux des patients présentant une fente palatine isolée devrait se faire sous peu. Nous n’avons pas été en mesure d’atteindre nos objectifs concernant l’apnée du sommeil. Une étude prospective serait à prévoir pour y arriver.

On Some Infinite Convex Invariants

The present study on some infinite convex invariants. The origin of convexity can be traced back to the period of Archimedes and Euclid. At the turn of the nineteenth centaury , convexicity became an independent branch of mathematics with its own problems, methods and theories. The convexity can be sorted out into two kinds, the first type deals with generalization of particular problems such as separation of convex sets[EL], extremality[FA], [DAV] or continuous selection Michael[M1] and the second type involved with a multi- purpose system of axioms. The theory of convex invariants has grown out of the classical results of Helly, Radon and Caratheodory in Euclidean spaces. Levi gave the first general definition of the invariants Helly number and Radon number. The notation of a convex structure was introduced by Jamison[JA4] and that of generating degree was introduced by Van de Vel[VAD8]. We also prove that for a non-coarse convex structure, rank is less than or equal to the generating degree, and also generalize Tverberg’s theorem using infinite partition numbers. Compare the transfinite topological and transfinite convex dimensions

Convex extendable trees

The concept of convex extendability is introduced to answer the problem of finding the smallest distance convex simple graph containing a given tree. A problem of similar type with respect to minimal path convexity is also discussed.

Studies on Fuzzy Matroitls and Related Topics

The doctoral thesis focuses on the Studies on fuzzy Matroids and related topics.Since the publication of the classical paper on fuzzy sets by L. A. Zadeh in 1965.the theory of fuzzy mathematics has gained more and more recognition from many researchers in a wide range of scientific fields. Among various branches of pure and applied mathematics, convexity was one of the areas where the notion of fuzzy set was applied. Many researchers have been involved in extending the notion of abstract convexity to the broader framework of fuzzy setting. As a result, a number of concepts have been formulated and explored. However. many concepts are yet to be fuzzified. The main objective of this thesis was to extend some basic concepts and results in convexity theory to the fuzzy setting. The concept like matroids, independent structures. classical convex invariants like Helly number, Caratheodoty number, Radon number and Exchange number form an important area of study in crisp convexity theory. In this thesis, we try to generalize some of these concepts to the fuzzy setting. Finally, we have defined different types of fuzzy matroids derived from vector spaces and discussed some of their properties.

Consejos superiores de la administración, neocorporativismo y participación orgánica

La Constitución Política de 1991 introdujo el concepto de participación como dato fundamental de su estructura. Un estudio ligero permite evidenciar una cantidad de artículos que regulan la materia participativa sin embargo, dichos preceptos se encuentran sin una relación de convexidad precisa. En cuanto al tema de la participación del administrado en la administración pública, como género de la especie que engloba las normas precisas sobre el ejercicio de funciones administrativas por particulares, el problema es de mayor complejidad; ello debido a que el problema participativo supone el estudio de teorías sociológicas que explican el fenómeno desde perspectivas que superan el ámbito normativo de los preceptos que consagran dicho fenómeno. El presente artículo tiene como propósito estudiar desde un punto de vista teórico el problema de la participación como fundamento del ejercicio de funciones administrativas por particulares con el objeto de explicar su fundamento último. Para este efecto, el artículo propone un panorama general de los fundamentos tradicionales del derecho administrativo y su eficacia actual, dentro del marco de una sociedad compleja en permanente conexión con el Estado Social de Derecho. A su vez expone teorías explicativas de la relación mencionada y sus efectos en la producción de normas jurídicas. Por último, expone un estudio de caso para la aplicación de los fundamentos teóricos expuestos en este artículo.

Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Shape preserving approximation using least squares splines

Least squares polynomial splines are an effective tool for data fitting, but they may fail to preserve essential properties of the underlying function, such as monotonicity or convexity. The shape restrictions are translated into linear inequality conditions on spline coefficients. The basis functions are selected in such a way that these conditions take a simple form, and the problem becomes non-negative least squares problem, for which effecitive and robust methods of solution exist. Multidimensional monotone approximation is achieved by using tensor-product splines with the appropriate restrictions. Additional inter polation conditions can also be introduced. The conversion formulas to traditional B-spline representation are provided.

Review of applications of the Cutting Angle methods

The theory of abstract convexity provides us with the necessary tools for building accurate one-sided approximations of functions. Cutting angle methods have recently emerged as a tool for global optimization of families of abstract convex functions. Their applicability have been subsequently extended to other problems, such as scattered data interpolation. This paper reviews three different applications of cutting angle methods, namely global optimization, generation of nonuniform random variates and multivatiate interpolation.

Extended cutting angle method of global optimization

Methods of Lipschitz optimization allow one to find and confirm the global minimum of multivariate Lipschitz functions using a finite number of function evaluations. This paper extends the Cutting Angle method, in which the optimization problem is solved by building a sequence of piecewise linear underestimates of the objective function. We use a more flexible set of support functions, which yields a better underestimate of a Lipschitz objective function. An efficient algorithm for enumeration of all local minima of the underestimate is presented, along with the results of numerical experiments. One dimensional Pijavski-Shubert method arises as a special case of the proposed approach.

Partners in crime : collusive corruption and search

This paper analyzes corruption as a collusive act which requires the participation of two willing partners. An agent intending to engage in a corrupt act must search for a like-minded partner. When many people in the economy are corrupt, such a search is more likely to be fruitful. Thus when an agent engages in a search, he raises the net benefit of searching for other similar agents in the economy, creating an externality. This introduces a non-convexity in the model, which consequently has multiple equilibria. The economy can be in stable equilibrium with a high or low level of corruption.

Starting from the high-corruption equilibrium, a sufficient increase in vigilance triggers a negative cascade, leading the economy to a new equilibrium in which no agent finds it profitable to search for corrupt partners. The no-corruption equilibrium continues to be stable if vigilance is then relaxed. This suggests that the correct way to deal with corruption is to launch a ``big push'' with large amounts of resources. Once the level of corruption declines, these resources can be withdrawn.

Construction of k-Lipschitz triangular norms and conorms from empirical data

This paper examines the practical construction of k-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on k-convex additive generators and translate k-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees k-Lipschitz property of the resulting triangular norms and conorms.

Bounded lower subdifferentiability optimization techniques : applications

In this article we develop a global optimization algorithm for quasiconvex programming where the objective function is a Lipschitz function which may have "flat parts". We adapt the Extended Cutting Angle method to quasiconvex functions, which reduces significantly the number of iterations and objective function evaluations, and consequently the total computing time. Applications of such an algorithm to mathematical programming problems inwhich the objective function is derived from economic systems and location problems are described. Computational results are presented.

Vehicle detection and classification by measuring and processing magnetic signal

This paper presents novel vehicle detection and classification method by measuring and processing magnetic signal based on single micro-electro- mechanical system (MEMS) magnetic sensor. When a vehicle moves over the ground, it generates a succession of impacts on the earth's magnetic field, which can be detected by single magnetic sensor. The magnetic signal measured by the magnetic sensor is related to the moving direction and the type of the vehicle. Generally, the recognition rate using single sensor detector is not high. In order to improve the recognition rate, a novel feature extraction algorithm and a novel vehicle classification and recognition algorithm are presented. The concavity and convexity areas, and the angles of concave and convex parts of the waveform are extracted. An improved support vector machine (ISVM) classifier is developed to perform vehicle classification and recognition. The effectiveness of the proposed approach is verified by outdoor experiments.

On the use of restricted dissimilarity and dissimilarity-like functions for defining penalty functions

In this work we study the relation between restricted dissimilarity functions-and, more generally, dissimilarity-like functions- and penalty functions and the possibility of building the latter using the former. Several results on convexity and quasiconvexity are also considered.