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Ce mémoire a pour but de présenter quelques résultats classiques de théorie des noeuds et de faire un parallèle entre cette théorie et les espaces de représentations associés au groupe d'un noeud. Le premier chapitre est consacré à une introduction de la théorie des noeuds dans lequel nous allons définir les surfaces de Seifert, le polynôme d'Alexander, le nombre d'entrelacement, les matrices de Seifert, le groupe d'un noeud. Quelques notions plus complexes vont être présentées comme le revêtement double ramifié le long d'un noeud, qui nous permettra d'établir une relation entre l'ordre du groupe d'homologie de ce revètement double et le polynôme d'Alexander évalué en -1. Le second chapitre présente le goupe SU(2) et le lien existant entre la conjugaison par un de ces élements et les rotations dans l'espace. Sont ensuite introduites les notions d'espaces de représentations illlustrées par le calcul explicite de celui du cercle, du noeud de trèfle, du double de Whitehead et du tore. Dans ce même chapitre nous présentons les twists de Dehn, chirurgies de Dehn et le résultat de Lickorish concernant l'obtention à partir de S³, de toute 3-variété fermée orientable, par chirurgie entière le long d'un entrelacs. Le troisième chapitre se concentre sur les espaces de représentations dans le groupe binaire dihédrale. Ces quelques pages proposent une ébauche de construction par chirurgie du polynôme d'Alexander évalué en -1. Nous présentons une approche géométrique de la construction de cet invariant en utilisant les relations skeins et la formule de Conway. Les dernières pages de ce mémoire sont consacrées à une brève introduction à l'invariant de Casson. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Mathématiques, Théorie des noeuds, Espaces de représentations, Invariant de Casson, Quaternions.

In the thesis, we present the structures of interacting multiple model (IMM) algorithm-based filter design for real-time motion orientation estimation and tracking by using inertial sensor measurements in three-dimensional space. The major sensor such as gyroscope, though has high-sensitivity characteristics, suffers from bias build-up and error drift over time. The complementary sensors such as accelerometer and magnetometer, on the other hand, have low sensitivity, but do not suffer from bias problems. By using individual inertial and magnetic sensors, measurements of multiple modes can be interactively computed. The IMM based designs show the advantages of weighting individual sensors in different motion states. We propose a signal processing architecture based on the IMM algorithm. It is composed of three parallel Kalman filters (KFs), each deals with measured signals from accelerometer, magnetometer and gyroscope, respectively. The accelerometer cannot effectively sense the rotation around the vertical axis; while the magnetometer can only sense the rotation around vertical axis. Therefore, estimation accuracy with the parallel filtering arrangement of the IMM algorithm-based structure may be affected. A scheme using the residual signal, which is computed in the IMM, provides the information of gyroscope-based KF to the other two filters for feasible calculation of update weights. Related research also usually combined the information of major and complementary sensors in estimator designs. In the literature, existing ¡§Triad¡¨ methods with quaternion-based extended Kalman filter (EKF), process the measurements from major and complementary sensors. To compensate the functions, we propose to use a gyroscope-based EKF and a Triad EKF in forming a parallel multiple model-based structure. The analysis and performance evaluation shows advantages and disadvantages of using EKFs and KFs in IMM-based filtering approachs. Simulation results validate the proposed estimator design concept, and show that the scheme is capable of reducing the overall estimation errors by flexible computation of model weights.

Techn. University, Diss., 2004--Berlin.

Davis and Januszkiewicz introduced in 1981 a family of compact real manifolds, the Quasi-Toric Manifolds, with a group action by a torus, a direct product of circle (T) groups. Their manifolds have an orbit space which is a simple polytope with a distinct isotropy subgroup associated to each face of the polytope, subject to some consistency conditions. They defined a characteristic function which captured the properties of the isotropy subgroups, and showed that their manifolds can be classified by the polytope and characteristic function. They further showed that the cohomology ring of the manifold can be written down directly from properties derived from the polytope and the characteristic function. This work considers the question of how far the circle group T can be replaced by the group of unit quaternions Q in the construction and description of quasi-toric manifolds. Unlike T, the group Q is not commutative, so the actions of Q n on the product H n of the set of quaternions using quaternionic multiplication are studied in detail. Then, in direct analogy to the quasi-toric manifolds, a family of compact real manifolds, the Quoric Manifolds, is introduced which have an action by Q n, and whose orbit space is a polytope. A characteristic functor is defined on the faces of the polytope which captures the properties of the isotropy classes of the orbits of the action. It is shown that quoric manifolds can be classified in a manner similar to the quasi-toric manifolds, by the polytope and characteristic functor. A restricted family, the global quoric manifolds, which satisfy an additional condition are defined. It is shown that an infinite number of polytopes exist in any dimension over which a global quoric manifold can be defined. It is shown that any global quoric manifold can be described as a quotient space of a moment angle complex over the polytope, and that its integral cohomology ring can be calculated, taking a form analagous to that in the quasi-toric case.

This investigation is designed to find quaternion operators which will generate selected space groups and which are more convenient to manipulate in some important types of problems.

Let ΓM be the fundamental group of a knot or link complement M. The discrete faithful representation of ΓM into PSL2(C) has an associated quaternion algebra. We can extend this notation to other representations, which are encoded by the character variety X(ΓM). The generalization is the canonical quaternion algebra and can be used to find unifying features of irreducible representations, such as the splitting behavior of their associated quaternion algebras. Within this dissertation, we will determine properties of the canonical quaternion algebra for the Whitehead link complement and explore how the algebra can descend to quaternion algebras of the Dehn (d, m)-surgeries thereon. / Mathematics

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In this dissertation an overview on the real representation of quaternions in distribution theory is given. The density functions of the p-variate and matrix-variate quaternion normal distributions are derived from first principles, while that of the quaternion Wishart distribution is derived from the real associated Wishart distribution via the characteristic function. Applications of this theory in hypothesis testing is presented, and the density function of Wilks's statistic is derived for quaternion Wishart matrices. / Dissertation (MSc)--University of Pretoria, 2010. / Statistics / unrestricted

A Wearable Motion Analysis System (WMAS) was developed to evaluate gait, particularly parameters that are indicative of mild traumatic brain injury. The WMAS consisted on six Opal IMUs attached on the sternum, waist, left and right thigh and left and right shank. Algorithms were developed to calculate the knee flexion angle, stride length and cadence parameters during slow, normal and fast gait speeds. The WMAS was validated for repeatability using a robotic arm and accuracy using the Vicon motion capture system, the gold standard for gait analysis. The WMAS calculated the gait parameters to within a clinically acceptable range and is a powerful tool for gait analysis and potential concussion diagnosis outside of a laboratory setting.