Academic literature on the topic 'Quaternions algebra'

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Journal articles on the topic "Quaternions algebra"

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ATA, Erhan, and Ümit Ziya SAVCI. "Generalized Quaternions and Matrix Algebra." Afyon Kocatepe University Journal of Sciences and Engineering 23, no. 3 (2023): 638–47. http://dx.doi.org/10.35414/akufemubid.1182145.

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In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
 Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.
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ABRATE, MARCO. "QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS." Journal of Algebra and Its Applications 08, no. 03 (2009): 289–306. http://dx.doi.org/10.1142/s0219498809003308.

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In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.
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Jiang, Tongsong, Dong Zhang, Zhenwei Guo, Gang Wang, and V. I. Vasil’ev. "Algebraic Techniques for Canonical Forms and Applications in Split Quaternionic Mechanics." Journal of Mathematics 2023 (November 7, 2023): 1–13. http://dx.doi.org/10.1155/2023/4599585.

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The algebra of split quaternions is a recently increasing topic in the study of theory and numerical computation in split quaternionic mechanics. This paper, by means of a real representation of a split quaternion matrix, studies the problem of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization in split quaternionic mechanics.
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Colombaro, Ivano. "Exterior-algebraic formulation of quaternions with applications." Journal of Physics: Conference Series 3027, no. 1 (2025): 012020. https://doi.org/10.1088/1742-6596/3027/1/012020.

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Abstract The purpose of this paper is to describe the formulation of quaternion algebra by means of exterior algebra and calculus, in a three dimensional time-like spacetime. A formal structure is provided, corroborating the equivalence with existing concepts and formulas known in literature. A first application is thus presented by depicting the description of rotations expressed with exterior-algebraic quaternionic notation. Secondly, a formal equivalence between exterior-algebraic quaternions and the classical theory of electromagnetism is recovered, too.
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Dargys, Adolfas, and Artūras Acus. "Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras." Nonlinear Analysis: Modelling and Control 27, no. 6 (2022): 1129–49. http://dx.doi.org/10.15388/namc.2022.27.29528.

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The aim of the paper is to give a uniform picture of complex, hyperbolic, and quaternion algebras from a perspective of the applied Clifford geometric algebra. Closed form expressions for a multivector exponential and logarithm are presented in real geometric algebras Clp;q when n = p + q = 1 (complex and hyperbolic numbers) and n = 2 (Hamilton, split, and conectorine quaternions). Starting from Cl0;1 and Cl1;0 algebras wherein square of a basis vector is either –1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras Cl0;2, Cl1;1, and Cl2;0. The sectors in the multivector coefficient space, where 2D logarithm exists are found. They are related with a square root of the multivector.
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Dai, Songsong. "Quaternionic Fuzzy Sets." Axioms 12, no. 5 (2023): 490. http://dx.doi.org/10.3390/axioms12050490.

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A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership.
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Cao, Wensheng. "Quadratic Equation in Split Quaternions." Axioms 11, no. 5 (2022): 188. http://dx.doi.org/10.3390/axioms11050188.

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Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.
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Eri̇şi̇r, Tülay, Gökhan Mumcu, Sezai̇ Kiziltuğ, and Funda Akar. "A New Construction of Rectifying Direction Curves for Quaternionic Space Q." WSEAS TRANSACTIONS ON MATHEMATICS 24 (March 14, 2025): 114–25. https://doi.org/10.37394/23206.2025.24.13.

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Our article focuses on the study of quaternions topic introduced by Hamilton. Quaternions are a generalization of complex numbers and have multiple applications in mathematical physics. Another application of quaternions is robotics because what generalizes the imaginary axis is the family i, j, k modeling Euler angles and rotations in space. The first part of the article we recall the different definitions of how the algebra of quaternions is well constructed. The main results are given in the third part and concern: spatial quaternionics rectifying-direction (sqRD) curves and and spatial quaternionic rectifying-donor (sqRDnr) curves. We study a new tip of unit speed associated curves in E 3 , which is also used in robotic systems and kinematics, like a spatial quaternionic rectifying-direction curve and spatial quaternionic rectifying-donor curve. Then, we achieve qualification for the curves. Moreover, we present applications of spatial quaternionic rectifying-direction to some specific curves like helix, slant helix, Salkowski and anti-Salkowski curves or rectifying curves. In addition, we establish different theorems which generalize the results obtained on the quaternionic curves in Q. Then, we give some examples are finally discussed. Consequently, Our paper is centered around theoretical analysis in geometry rather than experimental investigations.
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Eri̇şi̇r, Tülay, and Mehmet Ali̇ Güngör. "On Fibonacci spinors." International Journal of Geometric Methods in Modern Physics 17, no. 04 (2020): 2050065. http://dx.doi.org/10.1142/s0219887820500656.

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Spinors are used in physics quite extensively. Basically, the forms of use include Dirac four-spinors, Pauli three-spinors and quaternions. Quaternions in mathematics are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Fibonacci spinors using the Fibonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini formulas which are given for some series of numbers in mathematics for Fibonacci spinors.
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Pogorui, Anatoliy, and Tamila Kolomiiets. "Some algebraic properties of complex Segre quaternoins." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 158–59. http://dx.doi.org/10.37069/1683-4720-2019-33-13.

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This paper deals with the basic properties the algebra of Segre quaternions over the field of complex numbers. We study idempotents, ideals, matrix representation and the Peirce decomposition of this algebra. We also investigate the structure of zeros of a polynomial in Segre complex quaternions by reducing it to the system of four polynomial equations in the complex field. In addition, Cauchy-Riemann type conditions are obtained for the differentiability of a function on the complex Segre quaternionic algebra.
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Dissertations / Theses on the topic "Quaternions algebra"

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GÜNAŞTI, Gökmen. "Quaternions Algebra, Their applications in Rotations and Beyond Quaternions." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-20267.

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The theory of quaternions was discovered in the middle of nineteenth century and they were commonly used to represent rotations.This thesis is written to review the basic properties of quaternions algebra and their applications in representing rotation of a body in 3-dimensional Euclidean space. Also, last sections in this thesis explore why the use of quaternions are more advantages than Euler angle sequences and can quaternions themselves be further generalized to another number systems?
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Lopes, Wilder Bezerra. "Geometric-algebra adaptive filters." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/3/3142/tde-22092016-143525/.

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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.
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Rodríguez, Ordóñez Hugo. "Topological study of nonsingular bilinear maps /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1251841791&sid=5&Fmt=2&clientId=11238&RQT=309&VName=PQD.

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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.
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Silva, Julio César Conegundes da 1986. "G2 e as álgebras normadas." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305803.

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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
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Floderová, Hana. "Geometrické struktury založené na kvaternionech." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2010. http://www.nusl.cz/ntk/nusl-229021.

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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.
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Freitas, José Roberto. "Equações algébricas nos quatérnios de Hamilton." Universidade Tecnológica Federal do Paraná, 2013. http://repositorio.utfpr.edu.br/jspui/handle/1/594.

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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.
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Resende, Adriana Souza. "Introdução elementar às álgebras Clifford 'CL IND.2' 'CL IND. 3'." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306698.

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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
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Vieira, Vandenberg Lopes. "Grupos fuchsianos aritmeticos identificados em ordens dos quaternios para construção de constelações de sinais." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261079.

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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
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Parcollet, Titouan. "Quaternion neural networks A survey of quaternion neural networks - Chapter 2 Real to H-space Autoencoders for Theme Identification in Telephone Conversations - Chapter 7." Thesis, Avignon, 2019. http://www.theses.fr/2019AVIG0233.

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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
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Lesesvre, Didier. "Arithmetic Statistics for Quaternion Algebras." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCD040.

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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
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Books on the topic "Quaternions algebra"

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P, Ward J. Quaternions and Cayley numbers: Algebra and applications. Kluwer Academic Publishers, 1997.

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Ward, J. P. Quaternions and Cayley Numbers: Algebra and Applications. Springer Netherlands, 1997.

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Conway, John Horton. On quaternions and octonions: Their geometry, arithmetic, and symmetry. AK Peters, 2003.

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Chuang, Chih-Yun. Brandt matrices and theta series over global function fields. American Mathematical Society, 2015.

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Behrns, Vernon N. An introduction to the algebra of hypernumbers. Dorrance Publishing Co., 2007.

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Paşa, Hüseyin Tevfik. Hüseyin Tevfik Paşa ve "Linear algebra". İstanbul Teknik Üniversitesi Bilim ve Teknoloji Tarihi Araştırma Merkezi, 1988.

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Dixon, Geoffrey M. Division algebras: Octonions, quaternions, complex numbers, and the algebraic design of physics. Kluwer Academic Publishers, 1994.

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Voight, John. Quaternion Algebras. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56694-4.

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Wolfgang, Sprössig, Gürlebeck Klaus, and Volkswagenstiftung, eds. Proceedings of the symposium: Analytical and numerical methods in quaternionic and Clifford analysis : Seiffen 1996. s.n., 1996.

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Fragoulopoulou, Maria. Tensor products of enveloping locally C*-algebras. Mathematisches Institut der Universität Münster, 1997.

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Book chapters on the topic "Quaternions algebra"

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Gorodentsev, Alexey L. "Quaternions and Spinors." In Algebra I. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-45285-2_20.

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Goldman, Ron. "The Algebra of Quaternion Multiplication." In Rethinking Quaternions. Springer International Publishing, 2010. http://dx.doi.org/10.1007/978-3-031-79549-7_4.

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Goldman, Ron. "Summary—Formulas From Quaternion Algebra." In Rethinking Quaternions. Springer International Publishing, 2010. http://dx.doi.org/10.1007/978-3-031-79549-7_10.

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Vince, John. "Quaternion Algebra." In Quaternions for Computer Graphics. Springer London, 2011. http://dx.doi.org/10.1007/978-0-85729-760-0_5.

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Vince, John. "Quaternion Algebra." In Quaternions for Computer Graphics. Springer London, 2021. http://dx.doi.org/10.1007/978-1-4471-7509-4_6.

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Kantor, I. L., and A. S. Solodovnikov. "Quaternions and Vector Algebra." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_4.

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Goldman, Ronald. "Algebra." In Dual Quaternions and Their Associated Clifford Algebras. CRC Press, 2023. http://dx.doi.org/10.1201/9781003398141-3.

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Vince, John. "Number Sets and Algebra." In Quaternions for Computer Graphics. Springer London, 2011. http://dx.doi.org/10.1007/978-0-85729-760-0_2.

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Ward, J. P. "Fundamentals of Linear Algebra." In Quaternions and Cayley Numbers. Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5768-1_1.

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Kamberov, George, Peter Norman, Franz Pedit, and Ulrich Pinkall. "Chapter 3. Spinor Algebra." In Quaternions, Spinors, and Surfaces. American Mathematical Society, 2002. http://dx.doi.org/10.1090/conm/299/03.

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Conference papers on the topic "Quaternions algebra"

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Purwar, Anurag, and Q. J. Ge. "Polar Decomposition of Unit Dual Quaternions." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70882.

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This paper seeks to extend the notion of polar decomposition from matrix algebra to dual quaternion algebra. The goal is to obtain a simple, efficient and explicit method for determining the polar decompositions (PD) of spatial displacements in Euclidean three-space that belong to a special Euclidean Group known as SE(3). It has been known that such a decomposition is equivalent to the projection of an element of SE(3) onto SO(4) that yields hyper spherical displacements that best approximate rigid-body displacements. It is shown in this paper that a dual quaternion representing an element of SE(3) can be decomposed into a pair of unit quaternions, called double quaternion, that represents an element of SO(4). Furthermore, this decomposition process may be interpreted as the projection of a point in four-dimensional space onto a unit hypersphere. Examples are provided to illustrate that the results obtained from this dual-quaternion based polar decomposition are same as those obtained from the matrix based polar decomposition.
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Terze, Zdravko, Andreas Mueller, and Dario Zlatar. "Redundancy-Free Integration of Rotational Quaternions in Minimal Form." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-35118.

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Redundancy-free computational procedure for solving dynamics of rigid body by using quaternions as the rotational kinematic parameters will be presented in the paper. On the contrary to the standard algorithm that is based on redundant DAE-formulation of rotational dynamics of rigid body that includes algebraic equation of quaternions’ unit-length that has to be solved during marching-in-time, the proposed method will be based on the integration of a local rotational vector in the minimal form at the Lie-algebra level of the SO(3) rotational group during every integration step. After local rotational vector for the current step is determined by using standard (possibly higher-order) integration ODE routine, the rotational integration point is projected to Sp(1) quaternion-group via pertinent exponential map. The result of the procedure is redundancy-free integration algorithm for rigid body rotational motion based on the rotational quaternions that allows for straightforward minimal-form-ODE integration of the rotational dynamics.
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Condurache, Daniel. "Dual Lie Algebra Representations of Rigid Body Motion With Dual Cayley Maps: An Overview." In ASME 2024 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2024. http://dx.doi.org/10.1115/detc2024-143707.

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Abstract The main objective of this research is to develop a new minimal parameterization technique for the displacement and motion of rigid bodies using hypercomplex dual algebra. Our study is based on the properties of dual tensors and dual quaternions, more precisely, their Lie groups and algebras. Based on the higher-order modified fractional Cayley transforms, for the first time, a complete and unitary parameterization framework, which gives the possibility of developing direct unitary solutions for the calculation of the leading entities of kinematic representation of displacement and motion of rigid body: Euler dual vector, higher-order Rodrigues dual vector, dual quaternions, dual sin family parameters and dual orthographic projection. The representation is coordinate-free and in a closed-form.
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Ge, Q. J. "On the Matrix Algebra Realization of the Theory of Biquaternions." In ASME 1994 Design Technical Conferences collocated with the ASME 1994 International Computers in Engineering Conference and Exhibition and the ASME 1994 8th Annual Database Symposium. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/detc1994-0221.

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Abstract This paper describes a matrix algebra presentation of Clifford’s theory of biquaternions. We examine 4 × 4 skew-symmetric matrices and use their exponentials to relate quaternions to equal-angle double rotations in Euclidean four-space E4. We show how double rotations in E4 expressed in terms of plane coordinates lead to elliptic biquaternions in both Plücker and Study forms and present the fundamental Plücker and Study conditions that govern the biquaternions. Finally, we show that a spatial displacement in E3 in terms of parabolic biquaternions (or dual quaternions) is a limiting case of a double rotation in E4 in terms of elliptic biquaternions.
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Yu, Zihan, Qiaode Jeffrey Ge, and Mark P. Langer. "Construction of Confidence Regions for Uncertain Spatial Displacements With Dual Rodrigues Parameters." In ASME 2024 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2024. http://dx.doi.org/10.1115/detc2024-143410.

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Abstract This paper follows our recent work on the computation of kinematic confidence regions from a given set of uncertain spatial displacements with specified confidence levels. Dual quaternion algebra is used to compute the mean displacement as well as relative displacements from the mean. In constructing a 6D confidence ellipsoid, however, we use dual Rodrigue parameters resulting from dual quaternions. The advantages of using dual quaternions and dual Rodrigues parameters are discussed in comparison with those of three translation parameters and three Euler angles, which were used for the development of the so-called the Rotational and Translational Confidence Limit (RTCL) method. The set of six dual Rodrigue parameters are used to define a parametric space in which a 6 × 6 covariance matrix and a 6D confidence ellipsoid are obtained. An inverse operation is then applied to first obtain dual quaternions and then to recover the rotation matrix and translation vector for each point on the 6D ellipsoid. Through examples, we demonstrate the efficacy of our approach by comparing it with the RTCL method known in literature. Our findings indicate that our method, based on the dual-Rodrigues formulation, yields more compact and effective swept volumes than the RTCL method, particularly in cases involving screw displacements.
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Agrawal, O. P. "Quaternions, Hamilton Operators, and Kinematics of Mechanical Systems." In ASME 1987 Design Technology Conferences. American Society of Mechanical Engineers, 1987. http://dx.doi.org/10.1115/detc1987-0095.

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Abstract In this paper, quaternions are briefly reviewed and their associated matrix algebra is developed. Two Hamilton operators are defined and some of their properties are studied. The properties of these operators are then applied to find kinematic relations of a body undergoing spatial rotation and to find a recursive relation for intermediate-axes. The formulation presented provides an easy approach to kinematic analysis of spatial mechanical systems.
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Benger, Werner. "Illustrating Geometric Algebra and Differential Geometry in 5D Color Space." In WSCG 2023 – 31. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision. University of West Bohemia, Czech Republic, 2023. http://dx.doi.org/10.24132/csrn.3301.1.

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Geometric Algebra (GA) is popular for its immediate geometric interpretations of algebraic objects and operations. It is based on Clifford Algebra on vector spaces and extends linear algebra of vectors by operations such as an invertible product, i.e. divisions by vectors. This formalism allows for a complete algebra on vectors same as for scalar or complex numbers. It is particularly suitable for rotations in arbitrary dimensions. In Euclidean 3D space quaternions are known to be numerically superior to rotation matrices and already widely used in computer graphics. However, their meaning beyond its numerical formalism often remains mysterious. GA allows for an intuitive interpretation in terms of planes of rotations and extends this concept to arbitrary dimensions by embedding vectors into a higher dimensional, but still intuitively graspable space of multi-vectors. However, out intuition of more than three spatial dimensions is deficient. The space of colors forms a vector space as well, though one of non-spatial nature, but spun by the primary colors red, green, blue. The GA formalism can be applied here as well, amalgamating surprisingly with the notion of vectors and co-vectors known from differential geometry: tangential vectors on a manifold correspond to additive colors red/green/blue, whereas co-vectors from the co-tangential space correspond to subtractive primary colors magenta, yellow, cyan. GA in turn considers vectors, bi-vectors and anti-vectors as part of its generalized multi-vector zoo of algebraic objects. In 3D space vectors, anti-vectors, bi-vectors and covectors are all three-dimensional objects that can be identified with each other, so their distinction is concealed. Confusions arise from notions such as “normal vectors” vs. “axial vectors”. Higher dimensional spaces exhibit the differences more clearly. Using colors instead of spatial dimensions we can expand our intuition by considering "transparency" as an independent, four-dimensional property of a color vector. We can thereby explore 4D GA alternatively to spacetime in special/general relativity. However, even in 4D possibly confusing ambiguities remain between vectors, co-vectors, bi-vectors and bi-co-vectors: bi-vectors and bi-co-vectors - both six-dimensional objects - are visually equivalent. They become unequivocal only in five or higher dimensions. Envisioning five-dimensional geometry is even more challenging to the human mind, but in color space we can add another property, "texture" to constitute a five-dimensional vector space. The properties of a bi-vector and a bi-co-vector becomes evident there: We can still study all possible combinations of colors/transparency/texture visually. This higher-dimensional yet intuitive approach demonstrates the need to distinguish among different kinds of vectors before identifying them in special situations, which also clarifies the meanings of algebraic objects in 3D Euclidean space and allows for better formulations of algorithms in 3D.
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Saldias, Daniel P., Luiz A. Radavelli, Carlos R. M. Roesler, and Daniel Martins. "Kinematic synthesis of the passive human knee joint by differential evolution and quaternions algebra: A preliminary study." In 2014 5th IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics (BioRob). IEEE, 2014. http://dx.doi.org/10.1109/biorob.2014.6913759.

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Nelson, Donald D., and Elaine Cohen. "User Interaction With CAD Models With Nonholonomic Parametric Surface Constraints." In ASME 1998 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/imece1998-0260.

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Abstract User manipulation of assembly models can provide insight during the early, formulative design stages into kinematic and dynamic characteristics of a mechanism. We present the advantages of kinematic representation of constraint equations in fully Cartesian coordinates, a departure from standard practice for interactive mechanical assembly at interactive rates. Formulations of a surface rolling contact constraint equation and its Jacobian, defined as a joint between two NURBS surfaces via position, tangency and velocity constraint relations, are derived for use in dynamic simulation and assembly optimization. The constraint equation formulations use quaternions to represent orientation. An appendix develops appropriate differential algebra. In this work we develop the use of constraints in global frame Cartesian coordinates for describing operator-in-the-loop interactions with mechanical assemblies under a unified framework combining lower-pair joints and more general surface contact interactions.
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Müller, Andreas, Zdravko Terze, and Viktor Pandza. "A Non-Redundant Formulation for the Dynamics Simulation of Multibody Systems in Terms of Unit Dual Quaternions." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-60191.

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Quaternions are favorable parameters to describe spatial rotations of rigid bodies since they give rise to simple equations governing the kinematics and attitude dynamics in terms of simple algebraic equations. Dual quaternions are the natural extension to rigid body motions. They provide a singularity-free purely algebraic parameterization of rigid body motions, and thus serve as global parameters within the so-called absolute coordinate formulation of MBS. This attractive feature is owed to the inherent redundancy of these parameters since they must satisfy two quadratic conditions (unit condition and Plcker condition). Formulating the MBS kinematics in terms of dual quaternions leads to a system of differential-algebraic equations (DAE) with index 3. This is commonly transformed to an index 1 DAE system by replacing the algebraic constraints with their time derivative. This leads to the well-known problem of constraint violation. A brute force method, enforcing the unit constraint of quaternions, is to normalize them after each integration step. Clearly this correction affects the overall solution and the dynamic consistency. Moreover, for unit dual quaternions the two conditions cannot simply be enforced in such a way. In this paper a non-redundant formulation of the motion equations in terms of dual quaternions is presented. The dual quaternion constraints are avoided by introducing a local canonical parameterization. The key to this formulation is to treat dual unit quaternions as Lie group. The formulation can be solved with any standard integration scheme. Examples are reported displaying the excellent performance of this formulation regarding the constraint satisfaction as well as the solution accuracy.
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