Ce mémoire porte sur l'étude des antiautomorphismes d'algèbres et en particulier sur les antiautomorphismes linéaires d'algèbres centrales simples (sur un corps commutatif). Si l'algèbre est une algèbre de matrices, alors un tel antiautomorphisme est l'adjonction pour une forme bilinéaire. Ainsi la classification des antiautomorphismes linéaires (resp. de type II) à isomorphisme près est une généralisation de celle des formes bilinéaires (resp. sesquilinéaires) à similitude près. Dans la première partie, on définit la notion d'asymétrie d'une forme sesquilinéaire, et on étudie les éléments d'une algèbre d'endomorphismes qui sont une asymétrie. La notion de produit de formes sesquilinéaires conduit à une théorie de Morita pour les algèbres à antiautomorphismes, qui permet de généraliser la notion de somme orthogonale connue pour les involutions d'algèbres centrales simples aux algèbres à antiautomorphisme Morita équivalentes avec asymétrie. Dans la deuxième partie, après avoir rappelé comment l'asymétrie permet d'obtenir une classification des formes bilinéaires, on généralise au cas non déployé linéaire la notion d'asymétrie et on explique comment on peut espérer obtenir de bons résultats en étudiant l'involution induite sur le centralisateur de l'asymétrie et la pseudo-involution linéaire associée à cette asymètrie. L'étude du principe de Hasse pour les similitudes de formes bilinéaires conduit natu- rellement au calcul de certains groupes de Tate-Schafarevich de tores algébriques de type normique. Ceci permet, dans une troisième partie, de donner des contre-exemples à ce principe sur des corps de nombres, ainsi qu'une interprétation de type corps de classe à l'obstruction à ce principe. Ce type de calculs pour d'autres tores normiques permet de démontrer qu'ils ne sont pas stablement rationnels. Ce résultat permet alors de déterminer les groupes algébriques simples dont le tore générique est rationnel, et délimite donc les cas pour lesquels l'étude du tore générique donne la rationalité du groupe. La quatrième partie est dédiée à la définition et à l'étude d'invariants des algèbres centrales simples à antiautomorphismes qui généralisent ceux donnant de bons résultats de classification pour les involutions : le discriminant, l'algèbre de Clifford et la forme trace. On y développe alors les résultats espérés en petite dimension cohomologique ou en petit degré.

Nous étudions l'euclidianité des corps de nombres pour la norme et quelques unes de ses généralisations. Nous donnons en particulier un algorithme qui calcule le minimum euclidien d'un corps de nombres de signature quelconque. Cela nous permet de prouver que de nombreux corps sont euclidiens ou non pour la norme. Ensuite, nous appliquons cet algorithme à l'étude des classes euclidiennes pour la norme, ce qui permet d'obtenir de nouveaux exemples de corps de nombres avec une classe euclidienne non principale. Par ailleurs, nous déterminons tous les corps cubiques purs avec une classe euclidienne pour la norme. Enfin, nous nous intéressons aux corps de quaternions euclidiens. Après avoir énoncé les propriétés de base, nous étudions quelques cas particuliers. Nous donnons notamment la liste complète des corps de quaternions euclidiens et totalement définis sur un corps de nombres de degré au plus deux.

Ce travail de recherche est consacré à l'élaboration des méthodes de traitement d'antenne multicapteur, multicomposante. Le traitement des signaux enregistrés par ce type d'antenne permet l'estimation de la direction d'arrivée et des paramètres de polarisation des ondes arrivant sur l'antenne. Nous montrons comment l'incorporation (d'une manière judicieuse) de l'information multicomposante permet d'améliorer les performances des algorithmes de traitement. L'originalité des méthodes proposées tient à l'utilisation des modèles mathématiques sortant du cadre de l'algèbre vectorielle classique, et qui se trouvent particulièrement bien adaptés à la nature des signaux multicomposantes. <br /><br />Une première approche est fondée sur un modèle tensoriel, permettant de conserver la structure multimodale des signaux. Le tenseur interspectral est introduit pour représenter la covariance des données. Nous proposons deux algorithmes (Vector-MUSIC et Higher-Order MUSIC) basés sur des décompositions orthogonales du tenseur interspectral. Nous montrons, sur des simulations, que l'utilisation du modèle tensoriel et des décompositions multilinéaires associées améliorent les performances des méthodes proposées par rapport à celles atteignables avec les techniques classiques.<br /><br />Nous proposons également une approche en traitement d'antenne multicomposante fondée sur l'utilisation des algèbres hypercomplexes. Les vecteurs de quaternions et biquaternions sont utilisés pour modéliser les signaux polarisés enregistrés par une antenne à deux, trois ou quatre composantes. Deux algorithmes (Quaternion-MUSIC et Biquaternion-MUSIC), basés sur la diagonalisation des matrices de quaternions et de biquaternions, sont introduits. Nous montrons que l'utilisation des nombres hypercomplexes réduit le temps de calcul et améliore la résolution des méthodes.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T14:20:06Z No. of bitstreams: 2 arquivototal.pdf: 9893623 bytes, checksum: 211bb6b2f2a20721abf5b2948352c518 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Approved for entry into archive by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-05-19T14:27:36Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 9893623 bytes, checksum: 211bb6b2f2a20721abf5b2948352c518 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Made available in DSpace on 2015-05-19T14:27:36Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 9893623 bytes, checksum: 211bb6b2f2a20721abf5b2948352c518 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) Previous issue date: 2013-08-14 / Abstract: In this work we study some geometric transformations in the plane and the space. Initially, we present some special types of transformations in the plane and find the matrix of each of these transformations. In the second part we discourse the transformations in the space, emphasizing the rotations. We will use the angles of Euler to determine a rotation in the space around the Cartesian axes and define an equation which allows to rotate a vector around any axis. We also discuss the homogeneous spaces aiming the matrix representation of transformations of translation. Finally, we use the structure of the quaternions group to present a second form to rotation vectors and composition of rotations in the space. We emphasize that this study is essential to describe the motion of objects in the plane and in the space. / Neste trabalho estudamos algumas das transformações geométricas no Plano e no Espaço. Inicialmente, apresentamos alguns tipos de transformações especiais no Plano e encontramos a matriz de cada uma destas transformações. Na segunda parte abordamos as transformações no Espaço, dando ênfase as rotações. Utilizamos os ângulos de Euler para determinar uma rotação no espaço em torno dos eixos cartesianos e definimos uma equação que permite rotacionar um vetores em torno de um eixo qualquer. Também abordamos os espaços homogêneos objetivando a representa ção matricial da transformação de translação. Por último, usamos a estrutura do grupo dos Quatérnios para apresentar uma segunda forma de fazer rotações de vetores e composição de rotações no espaço. Ressaltamos que este estudo é fundamental para descrever o movimento de objetos no plano e no espaço.

abstract: Continuous monitoring of sensor data from smart phones to identify human activities and gestures, puts a heavy load on the smart phone's power consumption. In this research study, the non-Euclidean geometry of the rich sensor data obtained from the user's smart phone is utilized to perform compressive analysis and efficient classification of human activities by employing machine learning techniques. We are interested in the generalization of classical tools for signal approximation to newer spaces, such as rotation data, which is best studied in a non-Euclidean setting, and its application to activity analysis. Attributing to the non-linear nature of the rotation data space, which involve a heavy overload on the smart phone's processor and memory as opposed to feature extraction on the Euclidean space, indexing and compaction of the acquired sensor data is performed prior to feature extraction, to reduce CPU overhead and thereby increase the lifetime of the battery with a little loss in recognition accuracy of the activities. The sensor data represented as unit quaternions, is a more intrinsic representation of the orientation of smart phone compared to Euler angles (which suffers from Gimbal lock problem) or the computationally intensive rotation matrices. Classification algorithms are employed to classify these manifold sequences in the non-Euclidean space. By performing customized indexing (using K-means algorithm) of the evolved manifold sequences before feature extraction, considerable energy savings is achieved in terms of smart phone's battery life. / Dissertation/Thesis / M.S. Electrical Engineering 2014

A demanda crescente por petróleo impulsiona a exploração marítima desta riqueza para águas cada vez mais profundas. O aumento da lâmina dágua exige novas soluções de engenharia principalmente no que se refere à operação de unidades flutuantes de produção. Dentre os desafios impostos pelos novos ambientes de prospecção, destaca-se o processo de ancoragem, neste texto explorado sob a ótica da chamada estaca-torpedo. Embora já tenha sido empregada com sucesso na ancoragem de risers e FPSOs, esta solução encontra-se em constante desenvolvimento, sendo a principal fonte de informação acerca dos lançamentos da estaca-torpedo proveniente de uma unidade de medição inercial (UMI). A presente pesquisa baseou-se no estudo desta UMI e teve como objetivos principais verificar seu desempenho e compreender a empregabilidade deste tipo de monitoração em operações submarinas de maneira mais ampla. Além do estudo detalhado dos sensores, foi dada especial atenção aos algoritmos empregados no tratamento dos sinais provenientes da UMI. Foram estudadas técnicas de correção do sinal, quantificação de ruído, desafios inerentes aos sistemas do tipo strapdown e o processo de integração. Como resultado final foi desenvolvido um algoritmo baseado em quatérnios, alternativo ao atualmente empregado para o processamento dos sinais provenientes da UMI que equipa a estaca-torpedo. / The increasing demand on crude oils constantly pushes the offshore exploitation to deeper waters. As the water depth grows, new engineering challenges arise, especially concerning to the operation of floating production units. Among all the technical issues inherent to the new prospection environment, the mooring system is a significant topic and the development of the torpedo-pile takes place at this scenario. This mooring system has already been successfully applied in anchoring risers and FPSOs; nevertheless, it is in constant study and improvement. The major source of information about the torpedo-pile deployment comes from an inertial measurement unit (IMU). The research presented here is based on this IMU and had as main objective verify its performance and also comprehend the applicability of such kind of unit in other subsea processes. Along with the detailed sensors study, it was given special attention to the algorithms used to process the signals from the IMU. Signal correction techniques and noise quantification were investigated as long as challenges intrinsically related to strapdown navigation systems and the integration process. In the end, an alternative data processing algorithm based on quaternions was produced, to be employed in torpedo-pile launching together with its IMU.

It is commonly accepted that a starting point of the science of diffusion is the phenomenological diffusion equation postulated by German physiologist Adolf Fick inspired by experiments on diffusion by Thomas Graham and Robert Brown. Fick’s diffusion equation has been interpreted decades later by Albert Einstein and Marian Smoluchowski. Here we will show that the theory of diffusion has its elegant mathematical foundations formulated three decades before Fick by French mathematician Augustin Cauchy (~1822). The diffusion equation is straightforward consequence of his model of the elastic solid - the classical balance equations for isotropic, elastic crystal. Basing on the Cauchy model and using the quaternion algebra we present a rigorous derivation of the quaternion form of the diffusion equation. The fundamental consequences of all derived equations and relations for physics, chemistry and the future prospects are presented.

<p>Unexplained signals emanating from the pervasive Venusian clouds have intrigued scien- tists for more than half a century. Efforts to account for a myriad of perplexing measurements have motivated the development of new atmospheric missions to Venus. Fulfillment of science objectives in the inhospitable Venusian environment necessitates a range of mission archi- tectures, each of which poses significant design challenges which guide the development of new technologies. Missions of greater scope require increasingly intricate sample collection and measurement techniques, leading inevitably to the demand for long-duration science operations within the clouds. Sustained flight in the Venusian atmosphere may be afforded by high-altitude balloons. As such platforms have been studied extensively for terrestrial applications, it is prudent to consider which designs are most suitable for a given set of science goals. We present four balloon design options to accommodate Venus atmospheric science missions. We discuss system masses, envelope volumes, material characteristics, and mechanisms for altitude control as determined by the science objectives and cloud condi- tions. We consider three variations of a baseline mission architecture and corresponding gondola designs accommodating their distinct science instrument suites. The scientific value of long-duration in situ atmospheric sampling is surpassed only by the return of Venusian cloud material to Earth for investigation in a laboratory setting. Sample capture, orbit ac- quisition, and subsequent return to Earth may be accommodated by a high-altitude balloon, rocket, and rendezvous vehicle. Precise understanding of the balloon-rocket dynamics inside the clouds is required for launch attitude acquisition and may inform sample capture strat- egy. We present a generalized dynamical model for a balloon-gondola system. We model the system as a triple-spherical-floating-compound pendulum (TSFCP). Dynamical analyses of pendular motion frequently rely on the use of Eulerian angles as generalized coordinates, in- evitably resulting in the residence of trigonometric functions in the denominator. We present nonsingular equations of motion in terms of Euler parameters. We characterize a notional Venus sample return platform and simulate its motion in the Venusian atmosphere. We discuss the behavior of the system in response to wind gusts and initial perturbation. We consider limitations of the model as well as opportunities for its extension into future work.</p>

The problem addressed in this research is the in-flight generation of trajectories for autonomous unmanned aircraft, which requires a method of generating pseudo-optimal trajectories in near-real-time, on-board the aircraft, and without external intervention. The focus of this research is the enhancement of a particular inverse dynamics direct method that is a candidate solution to the problem. This research introduces the following contributions to the method. A quaternion-based inverse dynamics model is introduced that represents all orientations without singularities, permits smooth interpolation of orientations, and generates more accurate controls than the previous Euler-angle model. Algorithmic modifications are introduced that: overcome singularities arising from parameterization and discretization; combine analytic and finite difference expressions to improve the accuracy of controls and constraints; remove roll ill-conditioning when the normal load factor is near zero, and extend the method to handle negative-g orientations. It is also shown in this research that quadratic interpolation improves the accuracy and speed of constraint evaluation. The method is known to lead to a multimodal constrained nonlinear optimization problem. The performance of the method with four nonlinear programming algorithms was investigated: a differential evolution algorithm was found to be capable of over 99% successful convergence, to generate solutions with better optimality than the quasi- Newton and derivative-free algorithms against which it was tested, but to be up to an order of magnitude slower than those algorithms. The effects of the degree and form of polynomial airspeed parameterization on optimization performance were investigated, and results were obtained that quantify the achievable optimality as a function of the parameterization degree. Overall, it was found that the method is a potentially viable method of on-board near- real-time trajectory generation for unmanned aircraft but for this potential to be realized in practice further improvements in computational speed are desirable. Candidate optimization strategies are identified for future research.

L'objectif de ce travail de thèse est de concevoir des algorithmes de commande et de navigation pour le suivi des cibles dynamiques au sol en utilisant des véhicules aériens. Les quaternions, qui fournissent une alternative aux représentations classiques de la dynamique des véhicules aériens, ont été choisis comme une base pour développer des contrôleurs robustes et des algorithmes de navigation agile, en raison de leurs avantages tels que l'absence de singularités et discontinuités, et leur simplicité mathématique lors de la manipulation des rotations. Les approches de commande explorées à l'aide de quaternions dans cette thèse commencent par le retour d'état, la passivité, et des contrôleurs basés sur l'énergie, jusqu'à des modes glissants, et des approches de saturation en trois dimensions. Ensuite, des stratégies de navigation autonomes et semi-autonomes pour quadrirotors ont été explorées. Un algorithme a été développé pour le pilotage d'un quadrirotor en utilisant des gestes d'un utilisateur portant un bracelet. Afin de faciliter le fonctionnement des multi rotors dans des scénarios défavorables, une stratégie de déploiement agressive a été proposée ou un quadrirotor est lancé à la main avec ses moteurs éteints. Finalement, des techniques de navigation autonomes pour le suivi des cibles dynamiques ont été conçues. Un algorithme de génération de trajectoire basée sur des équations différentielles a été introduit pour le suivi d'un véhicule terrestre tout en décrivant des cercles. Enfin un algorithme de planification de chemin distribué a été développé pour une flottille de drones, afin de suivre de façon autonome des cibles au sol, en résolvant un problème d'optimisation en ligne. / The objective of this thesis work is to design control and navigation algorithms for tracking of dynamic ground targets using aerial vehicles. Quaternions, which provide an alternative to the classical representations of aerial vehicle dynamics, have been chosen as a basement to develop robust controllers and agile navigation algorithm, due to their advantages such as the absence of singularities and discontinuities and their mathematical simplicity when handling rotations. The quaternion-based control approaches explored in this thesis range from state feedback, passivity, and energy-based controllers, up to sliding modes, and three-dimensional saturation approaches. Then, autonomous and semi-autonomous navigation strategies for quadrotors were explored. An algorithm has been developed for controlling a quadrotor using gestures from a user wearing an armband. To facilitate the operation of multirotors in adverse scenarios, an aggressive deployment strategy has been proposed where a quadrotor is launched by hand With its motors turned off. Finally, autonomous navigation techniques for tracking dynamic targets have been designed. A trajectory generation algorithm based on differential equations has been introduced to track a land vehicle while describing circles. Finally a distributed path planning algorithm has been developed for a fleet of drones to autonomously track ground targets by solving an online optimization problem.