Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.

The goal of this thesis is the development of a fast and robust algorithm that is able to detect patterns in flow fields independent from their orientation and adequately visualize the results for a human user. This thesis is an interdisciplinary work in the field of vector field visualization and the field of pattern recognition. A vector field can be best imagined as an area or a volume containing a lot of arrows. The direction of the arrow describes the direction of a flow or force at the point where it starts and the length its velocity or strength. This builds a bridge to vector field visualization, because drawing these arrows is one of the fundamental techniques to illustrate a vector field. The main challenge of vector field visualization is to decide which of them should be drawn. If you do not draw enough arrows, you may miss the feature you are interested in. If you draw too many arrows, your image will be black all over. We assume that the user is interested in a certain feature of the vector field: a certain pattern. To prevent clutter and occlusion of the interesting parts, we first look for this pattern and then apply a visualization that emphasizes its occurrences. In general, the user wants to find all instances of the interesting pattern, no matter if they are smaller or bigger, weaker or stronger or oriented in some other direction than his reference input pattern. But looking for all these transformed versions would take far too long. That is why, we look for an algorithm that detects the occurrences of the pattern independent from these transformations. In the second part of this thesis, we work with moment invariants. Moments are the projections of a function to a function space basis. In order to compare the functions, it is sufficient to compare their moments. Normalization is the act of transforming a function into a predefined standard position. Moment invariants are characteristic numbers like fingerprints that are constructed from moments and do not change under certain transformations. They can be produced by normalization, because if all the functions are in one standard position, their prior position has no influence on their normalized moments. With this technique, we were able to solve the pattern detection task for 2D and 3D flow fields by mathematically proving the invariance of the moments with respect to translation, rotation, and scaling. In practical applications, this invariance is disturbed by the discretization. We applied our method to several analytic and real world data sets and showed that it works on discrete fields in a robust way.

We begin by giving some theoretical background to the underlying concepts of spin representations and spinors. This is done from the perspective of Lie groups and Lie algebras. In particular, we discuss the functionality of Clifford algebras in the determination of the double-covering spin groups. An introduction to K-algebras and Clifford algebras is then given, focusing on the properties of pseudo-Euclidean spaces <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BR%7D%5E%7Bp,q%7D" data-classname="equation" data-title="" />. Some low-dimensional examples are also included, culminating with a characterisation of some Clifford algebras as matrix algebras. Elementary representation theory is then introduced and quickly followed by the definition of the Clifford-Lipschitz and spin groups. The work of Lundholm and Svensson (2016), Vaz and da Rocha (2016), and Schwichtenberg (2018) is then united to construct a definition of the spin representations. An attempt at formulating a definition of spinors from a mathematical perspective is then given; formed by combining multiple approaches and definitions of the above-mentioned authors, as well as drawing inspiration from important cases in theoretical physics, in particular that of SO(3) and the Lorentz group SO(1,3).

The Clifford A. Phaneuf Environmental Center is home to ECOS, The Environmental Center for Our Schools, in Springfield Massachusetts. The ECOS program provides a chance for elementary and middle school students in Springfield public schools to experience and learn about the natural world. The built environment provides opportunities to teach about ecosystems and human connections to nature. A new design for the Clifford A. Phaneuf Environmental Center will teach students about the natural world experientially through the building’s own connections to the environment.

This thesis examines the relationship between Cold War science fiction fix-up novels and social cycle theory. The study engages with textual, cultural, and comparative analysis to elucidate and analyze links between the fix-up novel format, a cyclical conception of human history, and the Cold War setting of the construction and publication of three SF novels. The objects of this study are three Cold War era fix-up novels with origins in World War II pulp science fiction magazine short stories: The Martian Chronicles by Ray Bradbury, City by Clifford D. Simak, and A Canticle for Leibowitz by Walter M. Miller Jr. The project examines these three novels alongside the reflective nature of the fix-up novel format, the authors' interactions with social cycle theory, and the Cold War cultural considerations of ideological instability and the threat of annihilation. By examining these works through the lens of retroactive continuity, social cycle theory, and the Cold War cultural imaginary, this thesis demonstrates the complex interplay between literature, culture, and history, and the ways in which SF authors have used their works to engage with the pressing concerns of their time. / Master of Arts / The Cold War era novels The Martian Chronicles by Ray Bradbury, City by Clifford D. Simak, and A Canticle for Leibowitz by Walter M. Miller Jr. share origins in pulp science fiction magazine short stories. These authors consolidated and revised previously published short stories to produce these works, known as fix-up novels. These three fix-up novels interact with representations of human progress as cyclical or non-linear. This project examines how the Cold War setting of the authors may have influenced science fiction authors' conceptions of human progress as cyclical. This thesis studies how the revision process of creating fix-up novels combined with the transition from World War II to the societal anxieties of the Cold War may have impacted the cultural messages of these novels.

The work is situated within the theory of slice analysis, a generalization of complex analysis for hypercomplex numbers, considering function of both quaternionic and Clifford variables, in both one and several variables. %We first characterize some partial slice sets of The primary focus of the thesis is on the harmonic and polyharmonic properties of slice regular functions. We derive explicit formulas for the iteration of the Laplacian on slice regular functions, proving that their degree of harmonicity increases with the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables. Additionally, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of the Fueter and Fueter-Sce theorems in the several variables context. Finally, we establish that regular polynomials of sufficiently low degree are the unique slice regular functions in the kernel of the iteration of the Laplacian, whose power is less than Sce index.

Les signaux analytiques multidimensionnels nous permettent d'avoir des possibilités de calculer les phases et modules. Cependant, peu de travaux se trouvent sur les signaux analytiques multidimensionnels qui effectuent une extensibilité appropriée pour les applications à la fois sur du traitement des données médicales 2D et 3D. Cette thèse a pour objectif de proposer des nouvelles méthodes pour le traitement des images médicales 2D/3D pour les applications de détection d'enveloppe et d'estimation du mouvement. Premièrement, une représentation générale du signal quaternionique 2D est proposée dans le cadre de l'algèbre de Clifford et cette idée est étendue pour modéliser un signal analytique hypercomplexe 3D. La méthode proposée décrit que le signal analytique complexe 2D, est égal aux combinaisons du signal original et de ses transformées de Hilbert partielles et totale. Cette écriture est étendue au cas du signal analytique hypercomplexe 3D. Le résultat obtenu est que le signal analytique hypercomplexe de Clifford peut être calculé par la transformée de Fourier complexe classique. Basé sur ce signal analytique de Clifford 3D, une application de détection d'enveloppe en imagerie ultrasonore 3D est présentée. Les résultats montrent une amélioration du contraste de 7% par rapport aux méthodes de détection d'enveloppe 1D et 2D. Deuxièmement, cette thèse propose une approche basée sur deux phases spatiales du signal analytique 2D appliqué aux séquences cardiaques. En combinant l'information de ces phases des signaux analytiques de deux images successives, nous proposons un estimateur analytique pour les déplacements locaux 2D. Pour améliorer la précision de l'estimation du mouvement, un modèle bilinéaire local de déformation est utilisé dans un algorithme itératif. Cette méthode basée sur la phase permet au déplacement d'être estimé avec une précision inférieure au pixel et est robuste à la variation d'intensité des images dans le temps. Les résultats de sept séquences simulées d'imagerie par résonance magnétique (IRM) marquées montrent que notre méthode est plus précise comparée à des méthodes récentes utilisant la phase du signal monogène ou des méthodes classiques basées sur l'équation du flot optique. Les erreurs d'estimation de mouvement de la méthode proposée sont réduites d'environ 33% par rapport aux méthodes testées. En outre, les déplacements entre deux images sont cumulés en temps, pour obtenir la trajectoire d'un point du myocarde. En effet, des trajectoires ont été calculées sur deux patients présentant des infarctus. Les amplitudes des trajectoires des points du myocarde appartenant aux régions pathologiques sont clairement réduites par rapport à celles des régions normales. Les trajectoires des points du myocarde, estimées par notre approche basée sur la phase de signal analytique, sont donc un bon indicateur de la dynamique cardiaque locale. D'ailleurs, elles s'avèrent cohérentes à la déformation estimée du myocarde. / Different mathematical tools, such as multidimensional analytic signals, provide possibilities to calculate multidimensional phases and modules. However, little work can be found on multidimensional analytic signals that perform appropriate extensibility for the applications on both of the 2D and 3D medical data processing. In this thesis, based on the Hahn 1D complex analytic, we aim to proposed a multidimensional extension approach from the 2D to a new 3D hypercomplex analytic signal in the framework of Clifford algebra. With the complex/hypercomplex analytic signals, we propose new 2D/3D medical image processing methods for the application of ultrasound envelope detection and cardiac motion estimation. Firstly, a general representation of 2D quaternion signal is proposed in the framework of Clifford algebra and this idea is extended to generate 3D hypercomplex analytic signal. The proposed method describes that the complex/hypercomplex 2D analytic signals, together with 3D hypercomplex analytic signal, are equal to different combinations of the original signal and its partial and total Hilbert transforms, which means that the hypercomplex Clifford analytic signal can be calculated by the classical Fourier transform. Based on the proposed 3D Clifford analytic signal, an application of 3D ultrasound envelope detection is presented. The results show a contrast optimization of about 7% comparing with 1D and 2D envelope detection methods. Secondly, this thesis proposes an approach based on two spatial phases of the 2D analytic signal applied to cardiac sequences. By combining the information of these phases issued from analytic signals of two successive frames, we propose an analytical estimator for 2D local displacements. To improve the accuracy of the motion estimation, a local bilinear deformation model is used within an iterative estimation scheme. This phase-based method allows the displacement to be estimated with subpixel accuracy and is robust to image intensity variation in time. Results from seven realistic simulated tagged magnetic resonance imaging (MRI) sequences show that our method is more accurate compared with the state-of-the-art method. The motion estimation errors (end point error) of the proposed method are reduced by about 33% compared with that of the tested methods. In addition, the frame-to-frame displacements are further accumulated in time, to allow for the calculation of myocardial point trajectories. Indeed, from the estimated trajectories in time on two patients with infarcts, the shape of the trajectories of myocardial points belonging to pathological regions are clearly reduced in magnitude compared with the ones from normal regions. Myocardial point trajectories, estimated from our phase-based analytic signal approach, are therefore a good indicator of the local cardiac dynamics. Moreover, they are shown to be coherent with the estimated deformation of the myocardium.

(in English) The main topic of this thesis is both analysis and interpretation of the relationship between funeral rites, political power and society. Funeral rites are understood as a complex of performative, symbolic social acts, which are themselves comprised of various distinctive phases and signs. The meaning of these acts and signs is important because it can change the features and practice of the collective it is assigned to. The main purpose of this thesis is not only to perform a thick description of different kinds of communist funeral rite in the given time period but to interprate it in the wider social and cultural context as well. The concept of political religion as a heuristic tool was used in the thesis in order to distinguish an ambivalent nature of communist ideology which manifested itself in funeral rites. Funeral rites are analyzed as models for human behavior, which represent key values and norms of communist ideology. At the same time, funeral rites are analyzed as forms of cultural management. That is the reason, why are put under scrutiny not only the rites which are significant for society as a whole but also the funeral rites of individual actors.

Le sujet de cette thèse est l'apport d'applications du formalisme des algèbres de Clifford au traitement d'images multicanaux. Nous y introduisons également l'utilisation du cadre des fibrés vectoriels en traitement d'image. La Partie 1 est consacrée à la segmentation d'images multicanaux. Nous généralisons l'approche de Di Zenzo pour la détection de contours en construisant des tenseurs métriques adaptés au choix de la segmentation. En utilisant le cadre des fibrés en algèbres de Clifford, nous montrons que le choix d'une segmentation d'une image est directement lié au choix d'une métrique, d'une connexion et d'une section sur un tel fibré. La Partie 2 est consacrée à la régularisation. Nous utilisons le cadre des équations de la chaleur associées à des Laplaciens généralisés sur des fibrés vectoriels. Le résultat principal que nous obtenons est qu'en considérant l'équation de la chaleur associée à l'opérateur de Hodge sur le fibré de Clifford d'une variété Riemannienne bien choisie, nous obtenons un cadre global pour régulariser de manière anisotrope des images (vidéos) multicanaux, et des champs s'y rapportant tels des champs de vecteurs ou des champs de repères orthonormés. Enfin, dans la Partie 3, nous nous intéressons à l'analyse spectrale via la définition d'une transformée de Fourier d'une image multicanaux. Cette définition repose sur une théorie abstraite de la transformée de Fourier basée sur la notion de représentation de groupe. De ce point de vue, la transformée de Fourier usuelle pour les images en niveau de gris est basée sur les représentations irréductibles du groupe des translations du plan. Nous l'étendons aux images multicanaux en lui associant les représentations réductibles de ce groupe.

The work of this Ph.D. thesis in mathematics concerns the problem of determining existence, uniqueness, and structure of zero-energy states in supersymmetric matrix models, which arise from a quantum mechanical description of the physics of relativistic membranes, reduced Yang-Mills gauge theory, and of nonperturbative features of string theory, respectively M-theory. Several new approaches to this problem are introduced and considered in the course of seven scientific papers, including: construction by recursive methods (Papers A and D), deformations and alternative models (Papers B and C), averaging with respect to symmetries (Paper E), and weighted supersymmetry and index theory (Papers F and G). The mathematical tools used and developed for these approaches include Clifford algebras and associated representation theory, structure of supersymmetric quantum mechanics, as well as spectral theory of (matrix-) Schrödinger operators. / QC20100629