In this thesis, a bundle F →(M,ω) → B is said to be Lagrangian if (M,ω) is a 2n- dimensional symplectic manifold and the fibres are compact and connected Lagrangian submanifolds of (M,ω), i.e. ω |F = 0 for all F. This condition implies that the fibres and the base space are n-dimensional. Such bundles arise naturally in the study of a special class of dynamical systems in Hamiltonian mechanics, namely those called completely integrable Hamiltonian systems. A celebrated theorem due to Liouville [39], Mineur [46] and Arnol`d [2] provides a semi-global (i.e. in the neighbourhood of a fibre) symplectic classification of Lagrangian bundles, given by the existence of local action-angle coordinates. A proof of this theorem, due to Markus and Meyer [41] and Duistermaat [20], shows that the fibres and base space of a Lagrangian bundle are naturally integral affine manifolds, i.e. they admit atlases whose changes of coordinates can be extended to affine transformations of Rn which preserve the standard cocompact lattice Zn Rn. This thesis studies the problem of constructing Lagrangian bundles from the point of view of affinely at geometry. The first step to study this question is to construct topological universal Lagrangian bundles using the affine structure on the fibres. These bundles classify Lagrangian bundles topologically in the sense that every such bundle arises as the pullback of one universal bundle. However, not all bundles which are isomorphic to the pullback of a topological universal Lagrangian bundle are Lagrangian, as there exist further smooth and symplectic invariants. Even for bundles which admit local action-angle coordinates (these are classified up to isomorphism by topological universal Lagrangian bundles), there is a cohomological obstruction to the existence of an appropriate symplectic form on the total space, which has been studied by Dazord and Delzant in [18]. Such bundles are called almost Lagrangian. The second half of this thesis constructs the obstruction of Dazord and Delzant using the spectral sequence of a topological universal Lagrangian bundle. Moreover, this obstruction is shown to be related to a cohomological invariant associated to the integral affine geometry of the base space, called the radiance obstruction. In particular, it is shown that the integral a ne geometry of the base space of an almost Lagrangian bundle determines whether the bundle is, in fact, Lagrangian. New examples of (almost) Lagrangian bundles are provided to illustrate the theory developed.

The purpose of this thesis is the study of the Lie algebras of affine vector fields and curvature collineations of space-time, the aim being, in the first case, to obtain upper bounds on the dimension of the Lie algebra of affine vector fields (under the assumption that the space-time is non-flat) as well as to obtain a characterization of such vector fields in terms of other types of symmetries. In the case of curvature collineations the aim was that of characterizing space-times which may admit an infinite-dimensional Lie algebra of curvature collineations as well as to find local characterizations of such vector fields. Chapters 1 and 2 consist of introductory material, in Differential Geometry (Ch.l) and General Relativity (Ch.2). In Chapter 3 we study homothetic vector fields which admit fixed points. The general results of Alekseevsky (a) and Hall (b) are presented, some being deduced by different methods. Some further details and results are also given. Chapter 4 is concerned with space-times that can admit proper affine vector fields. Using the holonomy classification obtained by Hall (c) it is shown that there are essentially two classes to consider. These classes are analysed in detail and upper bounds on the dimension of the Lie algebra of affine vector fields of such space-times are obtained. In both cases local characterizations of affine vector fields are obtained. Chapter 5 is concerned with space-times which may admit proper curvature collineations. Using the results of Halford and McIntosh (d) , Hall and McIntosh (e) and Hall (f) we were able to divide our study into several classes The last two of these classes are formed by those space-times which admit a (1 or 2-dimensional) non-null distribution spanned by vector fields which contract the Riemann tensor to zero. A complete analysis of each class is made and some general results concerning the infinite-dimensionality problem are proved. The chapter ends with some comments in the cases when the distribution mentioned above is null.

This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

Les avancées dans les technologies ont permis le développement de systèmes d’aide à la conduite (ADAS) pour prévenir les accidents routiers causés par les erreurs de conduite au manque d’attention de conducteurs. Plusieurs types sont déjà disponibles sur le marché, comme l’ABS et l’ESC (ou ESP), utilisant uniquement des capteurs proprioceptifs. Les capteurs extéroceptifs sont présents dans les ADAS plus récents, comme LKAS (maintien dans la voie) et LDWS. Cependant, l‘ESC agit dans la dynamique du véhicule en situations d¹urgence alors que les systèmes d’alerte de sortie de voie sont conçus pour les situations de faible sollicitation latérale. Cette thèse traite le développement d’un ADAS, nommé ESC-perceptif, qui intègre les informations des capteurs extéroceptifs (camera vidéo) avec le contrôle de la vitesse de lacet afin d’éviter les sorties de voie, y compris pour des conditions de fortes sollicitations latérales. La prise en compte de la saturation de forces de contact pneumatiques-chaussée est essentielle pour la conception de ce système. La non-linéairité des efforts pneumatiques est traité par l'approche des systèmes affines par morceaux (PWA). Cela permet de mener l'analyse et la synthèse de contrôleurs en combinant les fonctions de Lyapunov avec la résolution de problèmes d’optimisations sous contraintes d¹inégalités matricielles linéaires et bilinéaires. Au long de la thèse, plusieurs contrôleurs PWA pour le développement de ADAS sont présentés. L’ESC-perceptif, basé uniquement sur les capteurs disponibles sur les véhicules commercialisés est validé expérimentalement sur véhicule prototype. / Advances in the technology of sensors and actuators have enabled the development of driver assistance systems (ADAS) to prevent road accidents due to drivers mistakes or inattention. Several types are already deployed in the commercialised vehicles, such as, ABS and ESC by means of proprioceptive sensors. Exteroceptive sensors can be seen in systems such as, LKAS (Lane Keeping Assistance Systems) and LDWS (Lane Departure Warning Systems). While the ESC deals with the vehicle dynamics in emergency situations, the systems to avoid lane departure are currently designed to work in conditions of weak lateral solicitation. This thesis deals with the development of a ADAS, named perceptive-ESC, which integrates the information from the exteroceptive sensors (provided by a video camera) with the yaw rate control in order to avoid unintended lane departure even in situation of strong lateral solicitation or degraded road adhesion. Considering the saturation of the lateral tyre forces is essential for the conception of the perceptive-ESC, therefore the nonlinear behaviour of the lateral tyre forces is taken into account by the use of Piecewise Affine (PWA) Systems which analysis and control synthesis are based on quadratic Lyapunov functions casted as optimisation problems with linear and bilinear matrix inequalities constraints. Throughout the thesis, several PWA controllers for driver assistance systems are presented in which the complexity is gradually increased from simply enhancing the vehicle handling to the perceptive-ESC based only on sensors available in the currently commercialised passenger cars, which has been validated by practical experiments on a prototype vehicle.

Partial constraints are often available in visual processing tasks requiring the matching of contours in two images. We propose a non- iterative scheme to determine contour matches using locally affine transformations. The method assumes that contours are approximated by the orthographic projection of planar patches within oriented neighborhoods of varying size. For degenerate cases, a minimal matching solution is chosen closest to the minimal pure translation. Performance on noisy synthetic and natural contour imagery is reported.

This thesis is a study of circle packings for arbitrary combinatorial tori in the geometric setting of affine tori. Certain new tools needed for this study, such as face labels instead of the usual vertex labels, are described. It is shown that to each combinatorial torus there corresponds a two real parameter family of affine packing labels. A construction of circle packings for combinatorial fundamental domains from affine packing labels is given. It is demonstrated that such circle packings have two affine side-pairing maps, and also that these side-pairing maps depend continuously on the two real parameters.

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AbstractThis thesis has taken an axiomatic system for a finite geometry with a five-point line, introduced definitions of various familiar figures and relationships, and determined properties of the figures in this system. Many of the theorems found herein are true in ordinary Euclidean geometry, but several interesting properties also arise in contrast to the usual.Given consideration in this development are properties of parallelism, perpendicularity, and congruence, in a study of lines, segments, triangles, and quadrilaterals. Also included in the presentation is an introduction to circles and parabolas.Ball State UniversityMuncie, IN 47306

Range arithmetic is a way of calculating with variables that hold ranges of real values. This ability to manage uncertainty during computation has many applications. Examples in graphics include rendering and surface modeling, and there are more general applications like global optimization and solving systems of nonlinear equations. This thesis focuses on affine arithmetic, one kind of range arithmetic. The main drawbacks of affine arithmetic are that it taxes processors with heavy use of floating point arithmetic and uses expensive sparse vectors to represent noise symbols. Stream processors like graphics processing units (GPUs) excel at intense computation, since they were originally designed for high throughput media applications. Heavy control flow and irregular data structures pose problems though, so the conventional implementation of affine arithmetic with dynamically managed sparse vectors runs slowly at best. The goal of this thesis is to map affine arithmetic efficiently onto GPUs by turning sparse vectors into shorter dense vectors at compile time using static analysis. In addition, we look at how to improve efficiency further during the static analysis using unique symbol condensation. We demonstrate our implementation and performance of the condensation on several graphics applications.