Academic literature on the topic 'Quaternions algebra'

This document introduces a new class of adaptive filters, namely Geometric- Algebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics, among others, a general leastsquares cost function is posed, from which two types of GAAFs are designed. The first one, called standard, provides a generalization of regular adaptive filters for any subalgebra of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing almost perfect agreement for different levels of measurement noise. The second type, called pose estimation, is designed to estimate rigid transformations { rotation and translation - in n-dimensional spaces. The GA-LMS performance is assessed in a 3-dimensional registration problem, in which it is able to estimate the rigid transformation that aligns two point clouds that share common parts.<br>Este documento introduz uma nova classe de filtros adaptativos, entitulados Geometric-Algebra Adaptive Filters (GAAFs). Eles s~ao projetados via formulação do problema de minimização (uma função custo de mínimos quadrados) do ponto de vista de álgebra geométrica (GA), uma abrangente linguagem matemática apropriada para a descrição de transformações geométricas. Adicionalmente, diferente do que ocorre na formulação com álgebra linear, cálculo geométrico (a extensão de álgebra geométrica que possibilita o uso de cálculo diferencial) permite aplicar as mesmas técnicas de derivação independentemente do tipo de dados (subálgebra), isto é, números reais, números complexos, quaternions etc. Usando essas e outras características, uma função custo geral de mínimos quadrados é proposta, da qual dois tipos de GAAFs são gerados. O primeiro, chamado standard, generaliza filtros adaptativos da literatura concebidos sob a perspectiva de subálgebras de GA. As seguintes variantes do filtro least-mean squares (LMS) s~ao obtidas como casos particulares: LMS real, LMS complexo e LMS quaternions. Uma análise mean-square é desenvolvida e corroborada por simulações para diferentes níveis de ruído de medição em um cenário de identificação de sistemas. O segundo tipo, chamado pose estimation, é projetado para estimar transformações rígidas - rotação e translação { em espaços n-dimensionais. A performance do filtro GA-LMS é avaliada em uma aplicação de alinhamento tridimensional na qual ele estima a tranformação rígida que alinha duas nuvens de pontos com partes em comum.

Thesis (Ph. D.)--University of Oregon, 2006.<br>Typescript. Includes vita and abstract. Includes bibliographical references (leaves - ). Also available for download via the World Wide Web; free to University of Oregon users.

Orientador: Luiz Antonio Barrera San Martin<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-21T14:02:28Z (GMT). No. of bitstreams: 1 Silva_JulioCesarConegundesda_M.pdf: 1171316 bytes, checksum: 4e6e6eb2a3f1c066ac73e86495c06428 (MD5) Previous issue date: 2012<br>Resumo: ...Observações: Por apresentar basicamente fórmulas, o resumo, na íntegra, poderá ser visualizado no texto completo da tese digital<br>Abstract: ...Note: The complete abstract is available with the full electronic document<br>Mestrado<br>Matematica<br>Mestre em Matemática

A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.

Capes<br>A descoberta dos quatérnios pelo matemático britânico William Rowan Hamilton (1805-1865) permitiu uma nova abordagem na resolução de equações algébricas, fornecendo uma estrutura algébrica mais geral onde buscar soluções. Generalizando o caso clássico (sobre os complexos) apresentamos neste trabalho um tratamento da equação algébrica geral com coeficientes quatérnios. Verificamos que o número de raízes pode ser maior que o grau, e muitas vezes, pode mesmo ser infinito. Damos ênfase ao caso da equação quadrática, obtendo fórmulas para as raízes. Também nos detemos na obtenção de uma raiz enésima quatérnia de um quatérnio e de um número real.<br>The discovery of quartenions by the mathematician Willian Rowan Hamilton (1805-1865) allowed a new approach regards solving algebraic equations, providing a broad algebraic structure to seek solutions. As a generalization of the classical case (about the complexes), here we present a treatment of the general algebraic equation with quaternions coefficients. We found that the number of roots can be greater than the degree and often can be infinite. We give emphasis to the case of the quadratic equation, obtaining its solution formulas. We also dealt with obtaining a quaternary root of a quaternion and a real number.

Orientador: Waldyr Alves Rodrigues Junior<br>Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-15T23:09:32Z (GMT). No. of bitstreams: 1 Resende_AdrianaSouza_M.pdf: 17553204 bytes, checksum: a66cefe30e9957cc4351e03d3aec35b2 (MD5) Previous issue date: 2010<br>Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto<br>Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book<br>Mestrado<br>Ágebra<br>Mestre em Matemática

Orientadores: Reginaldo Palazzo Jr., Mercio Botelho Faria<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação<br>Made available in DSpace on 2018-08-08T06:25:10Z (GMT). No. of bitstreams: 1 Vieira_VandenbergLopes_D.pdf: 990187 bytes, checksum: 2212b8074f5503f78aa813ce4422cc4b (MD5) Previous issue date: 2007<br>Resumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como elemento base para construção de constelações de sinais geometricamente uniformes. E através desse procedimento que identificamos as estruturas algébrica e geométrica além de construir códigos geometricamente uniformes em espaços homogêneos. Propomos, a partir desses reticulados, a construção de grupos fuchsianos aritméticos Tp obtidos de tesselações hiperbólicas {p; q}, derivados de álgebras de divisão dos quaternios A sobre corpos de números K. Generalizamos o processo de identificação desses grupos em ordens dos quatérnios (reticulados hiperbólicos), associadas às constelações de sinais geometricamente uniformes, provenientes de grupos discretos. Esse procedimento permite rotular os sinais das constelações construídas por elementos de uma estrutura algébrica<br>Abstract: Within the context of digital communications system in homogeneous space in particular, in hyperbolic spaces, it is necessary to establish systematic procedure for the construction of lattices O ; as the basic entity for construction of eometrically uniforms signal constellations. By this procedure we identify the algebraic and geometric structures to construct geometrically uniforms codes in homogeneous spaces. We propose, from lattices, the construction of arithmetic fuchsian groups ¡p obtained by hyperbolic tessellations {p; q}, derived from division quaternion algebras A over numbers fields K. We generalize the process of identification of these groups in quaternion orders (hyperbolic lattices), which are associated with geometrically uniforms signal constellations, proceeding from discrete groups. This procedure allows us to realize the labelling of the signals belonging to such constellations by elements of an algebraic structure<br>Doutorado<br>Telecomunicações e Telemática<br>Doutor em Engenharia Elétrica

Au cours des dernières années, l’apprentissage profond est devenu l’approche privilégiée pour le développement d’une intelligence artificielle moderne (IA). L’augmentation importante de la puissance de calcul, ainsi que la quantité sans cesse croissante de données disponibles ont fait des réseaux de neurones profonds la solution la plus performante pour la resolution de problèmes complexes. Cependant, la capacité à parfaitement représenter la multidimensionalité des données réelles reste un défi majeur pour les architectures neuronales artificielles.Pour résoudre ce problème, les réseaux de neurones basés sur les algèbres des nombres complexes et hypercomplexes ont été développés. En particulier, les réseaux de neurones de quaternions (QNN) ont été proposés pour traiter les données tridi- mensionnelles et quadridimensionnelles, sur la base des quaternions représentant des rotations dans notre espace tridimensionnel. Malheureusement, et contrairement aux réseaux de neurones à valeurs complexes qui sont de nos jours acceptés comme une alternative aux réseaux de neurones réels, les QNNs souffrent de nombreuses lacunes qui sont en partie comblées par les différents travaux détaillés par ce manuscrit.Ainsi, la thèse se compose de trois parties qui introduisent progressivement les concepts manquants, afin de faire des QNNs une alternative aux réseaux neuronaux à valeurs réelles. La premiere partie présente et répertorie les précédentes découvertes relatives aux quaternions et aux réseaux de neurones de quaternions, afin de définir une base pour la construction des QNNs modernes.La deuxième partie introduit des réseaux neuronaux de quaternions état de l’art, afin de permettre une comparaison dans des contextes identiques avec les architectures modernes traditionnelles. Plus précisément, les QNNs étaient majoritairement limités par leurs architectures trop simples, souvent composées d’une seule couche cachée comportant peu de neurones. Premièrement, les paradigmes fondamentaux, tels que les autoencodeurs et les réseaux de neurones profonds sont présentés. Ensuite, les très répandus et étudiés réseaux de neurones convolutionnels et récurrents sont étendus à l’espace des quaternions. De nombreuses experiences sur différentes applications réelles, telles que la vision par ordinateur, la compréhension du langage parlé ainsi que la reconnaissance automatique de la parole sont menées pour comparer les modèles de quaternions introduits aux réseaux neuronaux conventionnels. Dans ces contextes bien spécifiques, les QNNs ont obtenus de meilleures performances ainsi qu’une réduction importante du nombre de paramètres neuronaux nécessaires à la phase d’apprentissage.Les QNNs sont ensuite étendus à des conditions d’entrainement permettant de traiter toutes les représentations en entrée des modèles de quaternions. Dans un scénario traditionnel impliquant des QNNs, les caractéristiques d’entrée sont manuellement segmentées en quatre composants, afin de correspondre à la representation induite par les quaternions. Malheureusement, il est difficile d’assurer qu’une telle segmentation est optimale pour résoudre le problème considéré. De plus, une segmentation manuelle réduit fondamentalement l’application des QNNs à des tâches naturellement définies dans un espace à au plus quatre dimensions. De ce fait, la troisième partie de cette thèse introduit un modèle supervisé et un modèle non supervisé permettant l’extraction de caractéristiques d’entrée désentrelacées et significatives dans l’espace des quaternions, à partir de n’importe quel type de signal réel uni-dimentionnel, permettant l’utilisation des QNNs indépendamment de la dimensionnalité des vecteurs d’entrée et de la tâche considérée. Les expériences menées sur la reconnaissance de la parole et la classification de documents parlés montrent que les approches proposées sont plus performantes que les représentations traditionnelles de quaternions<br>In the recent years, deep learning has become the leading approach to modern artificial intelligence (AI). The important improvement in terms of processing time required for learning AI based models alongside with the growing amount of available data made of deep neural networks (DNN) the strongest solution to solve complex real-world problems. However, a major challenge of artificial neural architectures lies on better considering the high-dimensionality of the data.To alleviate this issue, neural networks (NN) based on complex and hypercomplex algebras have been developped. The natural multidimensionality of the data is elegantly embedded within complex and hypercomplex neurons composing the model. In particular, quaternion neural networks (QNN) have been proposed to deal with up to four dimensional features, based on the quaternion representation of rotations and orientations. Unfortunately, and conversely to complex-valued neural networks that are nowadays known as a strong alternative to real-valued neural networks, QNNs suffer from numerous limitations that are carrefuly addressed in the different parts detailled in this thesis.The thesis consists in three parts that gradually introduce the missing concepts of QNNs, to make them a strong alternative to real-valued NNs. The first part introduces and list previous findings on quaternion numbers and quaternion neural networks to define the context and strong basics for building elaborated QNNs.The second part introduces state-of-the-art quaternion neural networks for a fair comparison with real-valued neural architectures. More precisely, QNNs were limited by their simple architectures that were mostly composed of a single and shallow hidden layer. In this part, we propose to bridge the gap between quaternion and real-valued models by presenting different quaternion architectures. First, basic paradigms such as autoencoders and deep fully-connected neural networks are introduced. Then, more elaborated convolutional and recurrent neural networks are extended to the quaternion domain. Experiments to compare QNNs over equivalents NNs have been conducted on real-world tasks across various domains, including computer vision, spoken language understanding and speech recognition. QNNs increase performances while reducing the needed number of neural parameters compared to real-valued neural networks.Then, QNNs are extended to unconventional settings. In a conventional QNN scenario, input features are manually segmented into three or four components, enabling further quaternion processing. Unfortunately, there is no evidence that such manual segmentation is the representation that suits the most to solve the considered task. Morevover, a manual segmentation drastically reduces the field of application of QNNs to four dimensional use-cases. Therefore the third part introduces a supervised and an unsupervised model to extract meaningful and disantengled quaternion input features, from any real-valued input signal, enabling the use of QNNs regardless of the dimensionality of the considered task. Conducted experiments on speech recognition and document classification show that the proposed approaches outperform traditional quaternion features

Les formes automorphes sont des objets centraux en théorie des nombres. En dépit de leur omniprésence, elles demeurent mystérieuses et leur comportement est loin d'être entièrement compris. Considérer ces formes automorphes au sein de familles a un effet régularisant, et ouvre la voie aux résultats en moyenne : voilà l'esprit des statistiques arithmétiques. La famille de toutes les représentations automorphes d'un groupe réductif donné, appelée famille universelle du groupe, est particulièrement importante. Dans le cas des formes intérieures de GL(2), autrement dit les groupes d'unités d'algèbres de quaternions, la formule des traces de Selberg est une puissante méthode d'approche. Il existe une notion de taille sur les formes automorphes, le conducteur analytique, permettant de tronquer la famille universelle en un ensemble fini pour lequel ces problèmes de statistiques arithmétiques ont un sens.Une loi de comptage pour la famille universelle tronquée est établie, avec un terme d'erreur gagnant par une puissance dans le cas totalement défini, et une constante à forte teneur géométrique. Cette loi de Weyl est généralisée en un résultat d'équirépartition par rapport à une mesure explicite, et mène à vérifier les conjectures de Sato-Tate dans ce cadre. Des statistiques sur les petits zéros des fonctions L associées sont établies, menant à dévoiler partiellement le type de symétrie des algèbres de quaternions.Plusieurs indices sont mentionnés laissant à croire que d'autres groupes sont abordables par les mêmes méthodes, et les lois de comptage conjecturales pour certains groupes unitaires et symplectiques de petits rangs sont énoncées<br>Automorphic forms are central objects in modern number theory. Despite their ubiquity, they remain mysterious and their behavior is far from understood. Embedding them in wider families has a smoothing effect, allowing results on average: these are the aims of arithmetic statistics. The whole family of automorphic representations of a given reductive group, referred to as its universal family, is of fundamental importance. In the case of inner forms of GL(2), that is to say groups of units of quaternion algebras, the Selberg trace formula is a powerful method to handle it. There is a way to define a suitable notion of size, the analytic conductor, allowing to truncate the universal family to a finite one amenable to arithmetic statistics methods. A counting law for the truncated universal family is established, with a power savings error term in the totally definite case and a geometrically meaningful constant. This Weyl's law is generalized to an equidistribution result with respect to an explicit measure, and leads to answer the Sato-Tate conjectures in this case. Statistics on low-lying zeros are provided, leading to uncover part of the type of symmetry of quaternion algebras.Strong evidence is provided that further ground groups should be amenable as well to the same methods and conjectural counting laws are given in the case of symplectic and unitary groups of low ranks