This is an overview of geometric algebras in dimensions 0-4 from the perspective of the concept of number itself. It is developed from a historic viewpoint and investigates and develops a pedagogic approach emphasizing the geometric aspects of the subject. There is a focus broadly on three main but interconnected areas: the relation between the discrete and the continuous, the centrality of complex numbers, and the hypothesis that the octonions may be expressed in the even subalgebra of four dimensions. This will not be proved, and the focus is on the overall perspective and presentation. One central result is a proposal for the identification of the Cayley-Dickson process with the gluing together of spheres, extending it to start from ℕ. This leads to a projective representation of a unified elliptic/parabolic/hyperbolic geometry of 4-dimensional space. Coupled with this is a discrete representation of it that can be classified with Geometric Algebra.

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'PALM' is an acronym of Pathway Language of Mapping, which is inspired by the movement of the northern lights (aurora borealis). The display dynamic patterns of brilliant lights that appear as curtains, rays, spiral, or dynamic flickers covering the entire sky. These moving visual images are resembled in the terms of sound spectra, which allow actualizing a temporal sonic world in panoramic sound landscape is heard as moving texture and timbre in twenty one groups under ten large shifts. To succeed in an approach of the listening perception, the groups of movements by frequency modulations in interlacing texture are shaped selectively through the observation of the 24-hour cycle live cam of the northern lights.

Orientador: Manoel Ferreira Borges Neto / Banca: Gilberto Aparecido Pratavieira / Banca: José Márcio Machado / Resumo: Resumo O objetivo deste trabalho é estabelecer similaridades entre os complexos e os hipercomplexos, motivados em explorar idéias de Murnaghan, que introduziu, pela primeira vez, em uma apresentação elementar, a teoria dos quatérnios baseados no teorema de Moivre. É mostrada em detalhes uma analogia da relação complexa clássica de Moivre para quatérnios, e em brevidade para octônios generalizados, e apresenta-se as conexões com os operadores da teoria de Fueter e as funções transcendentais. A extensão do teorema de Moivre é estudada para quatérnios em definindo-se uma função exponencial quaterniônica. / Abstract: Abstract In this work we establish similarities between the complex and the hipercomplex numbers, motivated in exploring ideas of Murnaghan, that introduced, for the first time, in an elementary presentation, the theory of the quaternions based on the theorem of Moivre. We show an analogy of the classic complex relation of Moivre for quaternions, and briefly discuss generalized octonions, as well as to present connections to operators of the theory of Fueter and transcendental functions. We consider them to study the extension of the theorem of Moivre for quaternions, in defining a exponential function on the quaternions. / Mestre

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02-20Bitstream added on 2014-06-13T18:47:50Z : No. of bitstreams: 1 oliveira_ac_me_sjrp.pdf: 346640 bytes, checksum: dfe236f71e5ef2c050420d64d2c48a70 (MD5) / O objetivo deste trabalho é estabelecer similaridades entre os complexos e os hipercomplexos, motivados em explorar idéias de Murnaghan, que introduziu, pela primeira vez, em uma apresentação elementar, a teoria dos quatérnios baseados no teorema de Moivre. É mostrada em detalhes uma analogia da relação complexa clássica de Moivre para quatérnios, e em brevidade para octônios generalizados, e apresenta-se as conexões com os operadores da teoria de Fueter e as funções transcendentais. A extensão do teorema de Moivre é estudada para quatérnios em definindo-se uma função exponencial quaterniônica. / In this work we establish similarities between the complex and the hipercomplex numbers, motivated in exploring ideas of Murnaghan, that introduced, for the first time, in an elementary presentation, the theory of the quaternions based on the theorem of Moivre. We show an analogy of the classic complex relation of Moivre for quaternions, and briefly discuss generalized octonions, as well as to present connections to operators of the theory of Fueter and transcendental functions. We consider them to study the extension of the theorem of Moivre for quaternions, in defining a exponential function on the quaternions.

Made available in DSpace on 2016-04-28T14:16:20Z (GMT). No. of bitstreams: 1 Donizetti Fermino Louro.pdf: 5486550 bytes, checksum: 06eabea84200c8c5e9e39a82ee79d8f2 (MD5) Previous issue date: 2014-06-15 / The historiography of triplets of Hamilton aims to address the mathematical reasoning that preceded the development of quaternions and their contribution to the study of three-dimensional objects. The study of triplets as geometrical representation of square roots and negative quantities prior to the formalization of hypercomplex numbers, in particular the algebra of quaternions. The origin and the quaternion's fundaments based on triplets, applied in the rotational motion of threedimensional objects in space, evokes further study of their complex structures, a priori. The search of Hamilton for a hypercomplex number able to represent rotations in space presented geometrical concerns, as well as, a vector approach of plane, then their applications determine isometries that still underlie the mathematical morphology of the digital dynamic image / A historiografia dos tripletos de Hamilton pretende abordar o raciocínio matemático que antecedeu o desenvolvimento dos quaternions e sua contribuição ao estudo de objetos tridimensionais. O estudo dos tripletos como representação geométrica das raízes quadradas e quantidades negativas antecedeu a formalização dos números hipercomplexos, em específico da álgebra de quaternions. A origem e os fundamentos dos quaternions baseados nos tripletos, aplicados no movimento rotacional de objetos tridimensionais no espaço suscitam um estudo mais aprofundado de suas estruturas complexas, a priori. A busca de Hamilton por um número hipercomplexo capaz de representar rotações do espaço apresentava preocupações geométricas, assim como uma abordagem vetorial do plano, logo suas aplicações determinariam isometrias que ainda fundamentam a morfologia da imagem digital dinâmica

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.

A presente dissertação volta-se à análise da alteração superveniente das circunstâncias fáticas, especificamente, em contratos interempresariais. Tais contratos, compreendidos como aqueles em que, ao menos, uma das partes é ente empresarial e ambos os contratantes visam auferir lucro, têm características distintas de outros contratos de natureza civil (contratos existenciais), mormente no que se refere à assunção de riscos pelas partes envolvidas. Dado o caráter arriscado da atividade empresarial, a intervenção exógena nos contratos interempresariais visando à tutela dos contratantes deve ser reduzida, de forma a se privilegiar tanto quanto possível a força obrigatória dos contratos (pacta sunt servanda). Nessa medida, as nuances dos contratos interempresariais, em conjunto com a imprescindível atividade estatal de regulação do mercado (fenômeno poliédrico), demandam uma aplicação diferenciada do modelo da onerosidade excessiva previsto nos artigos 478 a 480 do Código Civil, sob pena de desvirtuamento da lógica empresarial e desestímulo ao tráfico mercantil. O primeiro capítulo deste trabalho é destinado à individualização dos principais vetores da atividade empresarial e os respectivos reflexos daqueles na formação dos contratos empresariais. Em seguida, no segundo capítulo da empreitada, são analisadas as teorias revisionistas de maior influência na construção do modelo de onerosidade excessiva brasileiro, bem como as especificidades do próprio modelo em si. Por fim, o terceiro e conclusivo capítulo visa à identificação do suporte fático da onerosidade excessiva em contratos empresariais, bem como à análise dos efeitos decorrentes da aplicação do modelo a referidas formas de contratação. / This dissertation is focused on the analysis of the change of circumstances, regarding specifically entrepreneurial contracts. These contracts, understood as those in which at least one of the parties involved is an entrepreneurial entity and both of the parties seek to obtain profits with the deal, can be defined by distinguished characteristics when compared to civil contracts in general (contratos existenciais), mainly when it comes to the risk assumption by the parties in the contract. Given the risk inherent to business activities, the exogenous intervention in entrepreneurial contracts aiming the protection of one of the parties should be reduced as much as possible, in such a way to respect the binding force of the contracts (pacta sunt servanda). Accordingly, the specificities of the entrepreneurial contracts, together with the indispensable public regulation of the market (polyhedral phenomenon), require a distinguished applicability of the rules provided by the Brazilian Civil Code under the articles 478 to 480, otherwise entrepreneurial rationality of commercial contracts may be ruined and, as a result, the commercial traffic discouraged. The first chapter of this study is dedicated to finding the main features of the business activity and their implications to the formation of entrepreneurial contracts. Subsequently, in the second chapter of the essay, the theories related to change of circumstances that most influenced the construction of a Brazilian theory are analyzed, as well as the peculiarities of the Brazilian theory itself. Finally, the third and concluding chapter seeks to identify the hypothesis provided by articles 478 to 480 for entrepreneurial contracts, as well to analyze the effects arising from the application of this set of rules to referred contracts.

A presente dissertação volta-se à análise da alteração superveniente das circunstâncias fáticas, especificamente, em contratos interempresariais. Tais contratos, compreendidos como aqueles em que, ao menos, uma das partes é ente empresarial e ambos os contratantes visam auferir lucro, têm características distintas de outros contratos de natureza civil (contratos existenciais), mormente no que se refere à assunção de riscos pelas partes envolvidas. Dado o caráter arriscado da atividade empresarial, a intervenção exógena nos contratos interempresariais visando à tutela dos contratantes deve ser reduzida, de forma a se privilegiar tanto quanto possível a força obrigatória dos contratos (pacta sunt servanda). Nessa medida, as nuances dos contratos interempresariais, em conjunto com a imprescindível atividade estatal de regulação do mercado (fenômeno poliédrico), demandam uma aplicação diferenciada do modelo da onerosidade excessiva previsto nos artigos 478 a 480 do Código Civil, sob pena de desvirtuamento da lógica empresarial e desestímulo ao tráfico mercantil. O primeiro capítulo deste trabalho é destinado à individualização dos principais vetores da atividade empresarial e os respectivos reflexos daqueles na formação dos contratos empresariais. Em seguida, no segundo capítulo da empreitada, são analisadas as teorias revisionistas de maior influência na construção do modelo de onerosidade excessiva brasileiro, bem como as especificidades do próprio modelo em si. Por fim, o terceiro e conclusivo capítulo visa à identificação do suporte fático da onerosidade excessiva em contratos empresariais, bem como à análise dos efeitos decorrentes da aplicação do modelo a referidas formas de contratação. / This dissertation is focused on the analysis of the change of circumstances, regarding specifically entrepreneurial contracts. These contracts, understood as those in which at least one of the parties involved is an entrepreneurial entity and both of the parties seek to obtain profits with the deal, can be defined by distinguished characteristics when compared to civil contracts in general (contratos existenciais), mainly when it comes to the risk assumption by the parties in the contract. Given the risk inherent to business activities, the exogenous intervention in entrepreneurial contracts aiming the protection of one of the parties should be reduced as much as possible, in such a way to respect the binding force of the contracts (pacta sunt servanda). Accordingly, the specificities of the entrepreneurial contracts, together with the indispensable public regulation of the market (polyhedral phenomenon), require a distinguished applicability of the rules provided by the Brazilian Civil Code under the articles 478 to 480, otherwise entrepreneurial rationality of commercial contracts may be ruined and, as a result, the commercial traffic discouraged. The first chapter of this study is dedicated to finding the main features of the business activity and their implications to the formation of entrepreneurial contracts. Subsequently, in the second chapter of the essay, the theories related to change of circumstances that most influenced the construction of a Brazilian theory are analyzed, as well as the peculiarities of the Brazilian theory itself. Finally, the third and concluding chapter seeks to identify the hypothesis provided by articles 478 to 480 for entrepreneurial contracts, as well to analyze the effects arising from the application of this set of rules to referred contracts.