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## Timelike Christoffel pairs in the split-quaternions

DUSSAN, M. P.; MAGID, M.
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
37.31%
We characterize the Christoffel pairs of timelike isothermic surfaces in the four-dimensional split-quaternions. When restricting the receiving space to the three-dimensional imaginary split-quaternions, we establish an equivalent condition for a timelike surface in R(2)(3) to be real or complex isothermic in terms of the existence of integrating factors.; Institute of Mathematics and Statistics (IME) at the University of Sao Paulo (USP); Institute of Mathematics and Statistics (IME) at the University of Sao Paulo (USP); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); State of Sao Paulo Research Foundation (FAPESP)

## Quatérnios, operadores de Fueter e relações quaterniônicas transcendentais

Oliveira, Ana Carolina de
Tipo: Dissertação de Mestrado Formato: 76 f.
POR
Relevância na Pesquisa
27.43%
Pós-graduação em Matemática - IBILCE; 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.

## Rotações tridimensionais em biomecânica via quatérnions: aplicações na análise dos movimentos esportivos

Santiago, Paulo Roberto Pereira
Tipo: Tese de Doutorado Formato: 95 f. : il., tabs., gráfs.
POR
Relevância na Pesquisa
37.86%

## Análise da estabilidade do movimento rotacional de satélites artificiais com quatérnions e sob a influência de torques externos

Santos, Josué Cardoso dos
Tipo: Trabalho de Conclusão de Curso
POR
Relevância na Pesquisa
37.43%
The aim of this work is to analyze the stability of the rotational motion’s artificial satellite using the Routh Hurwitz Algorithm (CRH) and the quaternions to describe the satellite’s attitude. This algorithm allows the investigation of the stability of the motion using the coefficients of the characteristic equation associated with the equation of the rotational motion in the linear form. The equations of the rotational motion are given by the four cinematic equations for the quaternion and the three equations of Euler for the spin velocity’s components. In the Euler equations are included the components of the gravity gradient torque (TGG) and the solar radiation torque (TRS). The TGG is generated by the difference of the Earth gravity force direction and intensity actuating on each satellite mass element and it depends on the mass distribution and the form of the satellite. The TRS is created by changing of the linear momentum, which happens due to the interactions of solar photons with the satellite surface. The equilibrium points are gotten by the equation of rotational motion and the CRH is applied in the linear form of these equations. Simulations are developed for small and medium satellites, but the gotten equilibrium points are not stable by CRH. However...

## Fourier series for quaternions and the square of the error theorem

Martinez, Cristiane Aparecida Pendeza; Borges Neto, Manoel Ferreira; Martinez, André L.M.; Castelani, Emerson V.
Tipo: Artigo de Revista Científica Formato: 557-568
ENG
Relevância na Pesquisa
27.51%
In this paper we introduce a type of Hypercomplex Fourier Series based on Quaternions, and discuss on a Hypercomplex version of the Square of the Error Theorem. Since their discovery by Hamilton (Sinegre [1]), quaternions have provided beautifully insights either on the structure of different areas of Mathematics or in the connections of Mathematics with other fields. For instance: I) Pauli spin matrices used in Physics can be easily explained through quaternions analysis (Lan [2]); II) Fundamental theorem of Algebra (Eilenberg [3]), which asserts that the polynomial analysis in quaternions maps into itself the four dimensional sphere of all real quaternions, with the point infinity added, and the degree of this map is n. Motivated on earlier works by two of us on Power Series (Pendeza et al. [4]), and in a recent paper on Liouville’s Theorem (Borges and Mar˜o [5]), we obtain an Hypercomplex version of the Fourier Series, which hopefully can be used for the treatment of hypergeometric partial differential equations such as the dumped harmonic oscillation.

## Números complexos, quatérnions e rotações

Santos, Michel Valmor dos
Tipo: Trabalho de Conclusão de Curso Formato: 58 f.
PT_BR
Relevância na Pesquisa
37.43%
TCC (graduação) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Curso de Matemática.; O presente trabalho visa discutir os conjuntos dos números complexos, dos quatérnions, e sua relação com rotações no plano e no espaço. Nos capítulos acerca destes conjuntos discorreremos sobre noções básicas, as quais serão fundamentais para que possamos embasar a teoria sobre o uso dos complexos para realizarmos rotações no plano e dos quatérnions para rotações no espaço. Também faremos a dedução de matrizes de rotação tendo como ponto de partida as base canônicas. Visitaremos noções da teoria de anéis, grupos e mais especificamente grupos lineares com a finalidade de apresentarmos as rotações sob esta ótica. Finalizaremos deduzindo matrizes de rotação no espaço, a partir dos números quatérnions.

## Números complexos, quatérnions e rotações

Santos, Michel Valmor dos
Tipo: Trabalho de Conclusão de Curso Formato: 58 f.
PT_BR
Relevância na Pesquisa
37.31%
TCC (graduação) - Universidade Federal de Santa Catarina. Centro de Ciências Físicas e Matemáticas. Curso de Química.; ns, e sua relação com rotações no plano e no espaço. Nos capítulos acerca destes conjuntos discorreremos sobre noções básicas, as quais serão fundamentais para que possamos embasar a teoria sobre o uso dos complexos para realizarmos rotações no plano e dos quatérnions para rotações no espaço. Também faremos a dedução de matrizes de rotação tendo como ponto de partida as base canônicas. Visitaremos noções da teoria de anéis, grupos e mais especificamente grupos lineares com a finalidade de apresentarmos as rotações sob esta ótica. Finalizaremos deduzindo matrizes de rotação no espaço, a partir dos números quatérnions.

## Números complexos, quatérnions e rotações

Santos, Michel Valmor dos
Fonte: Florianópolis, SC Publicador: Florianópolis, SC
Tipo: Trabalho de Conclusão de Curso Formato: 58 f.
PT_BR
Relevância na Pesquisa
27.31%
TCC (graduação) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Curso de Matemática.; O presente trabalho visa discutir os conjuntos dos números complexos, dos quatérnions, e sua relação com rotações no plano e no espaço. Nos capítulos acerca destes conjuntos discorreremos sobre noções básicas, as quais serão fundamentais para que possamos embasar a teoria sobre o uso dos complexos para realizarmos rotações no plano e dos quatérnions para rotações no espaço. Também faremos a dedução de matrizes de rotação tendo como ponto de partida as base canônicas. Visitaremos noções da teoria de anéis, grupos e mais especificamente grupos lineares com a finalidade de apresentarmos as rotações sob esta ótica. Finalizaremos deduzindo matrizes de rotação no espaço, a partir dos números quatérnions.

## Quaterniões : cálculo numérico e simbólico; Quaternions : numerical and symbolical calculus

Pinheiro, Maria Lúcia Gonçalves
Relevância na Pesquisa
27.57%

## Dos números complexos aos quatérnions: desenvolvimento algébrico, interpretação geométrica e aplicações

Santos, Marcos André dos
POR
Relevância na Pesquisa
37.15%
This work was developed after observing the difﬁculties and unmotivated of the high school students to learning complex numbers. The development consisted in create a timeline in the study of complex numbers since Cardano at Sir Hamilton, expecting to contribute to understanding of this subject, associating algebraic properties and geometric interpretation, seeing to improve the understanding of the use of complex numbers to solve problems. Also, the history of imaginary unit i introduction and representation two-dimensional (2D) complex numbers a+bi, extending this for four-dimensions(4D) quaternions numbers a+bi+cj+dk, and its less usual forms like matrix form, vector form, including the procedure used in the rotation, showing your importance as motivation in the teaching of geometry, in physics and graphic computation.; Capes; Este trabalho foi desenvolvido a partir da constatação das diﬁculdades e falta de motivação dos alunos do ensino médio no aprendizado de números complexos. O desenvolvimento consistiu em realizar uma linha do tempo no estudo dos números complexos desde Cardano até Sir Hamilton, buscando contribuir para sua melhor compreensão, associando as propriedades algébricas com a interpretação geométrica visando melhorar o entendimento do uso dos números complexos na resolução de problemas. Ainda...

## NPSNET: flight simulation dynamic modeling using quaternions

Cooke, Joseph M.
EN_US
Relevância na Pesquisa
37.15%
Approved for public release; distribution is unlimited; The Naval Postgraduate School (NPS) has actively explored the design and implementation of networked, realtime, three-dimensional battlefield simulations on low cost, commercially available graphics workstations. The most recent system, NPSNET, has improved in functionality to such an extent, that it is considered a low cost version of the Defense Advanced Research Project Agency's (DARPA) SIMNET system. In order to reach that level, it was necessary to economize in certain areas of the code so that real time performance occurred at an acceptable level. One of those areas was in aircraft dynamics. However, with 'off-the-shelf' computers becoming faster and cheaper, real-time and realistic dynamics are no longer an expensive option. The realistic behavior can now be enhanced through the incorporation of an aerodynamic model. To accomplish this task, a prototype flight simulator was built that is capable of simulating numerous types of aircraft simultaneously within a virtual world. Beside being easily incorporated into NPSNET, such a simulator will also provide the base functionality for the creation of a general purpose aerodynamic simulator that is particularly useful to aerodynamic students for graphically analyzing differing aircraft's stability and control characteristics. This system is designed for use on a Silicon Graphics workstation and uses the GL libraries. Computer Graphics...

## Quatérnios, um ensaio sobre a regularidade e hiperperiodicidade de funções quaterniônicas, e o Teorema de Cauchy

Barreiro, Rodrigo Cardoso
Tipo: Dissertação de Mestrado Formato: 76 f. : il. color.
POR
Relevância na Pesquisa
27.31%
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Pós-graduação em Matemática - IBILCE; O objetivo deste trabalho ée estabelecer similaridades entre a análise complexa e os quatérnios. Nele é feito um estudo da regularidade de funções quaterniônicas e são estabelecidas as funções exponencial e logarítmica para os quatérnios sendo feito um estudo da hiperpe- riodicidade dessas funções. Outro resultado apresentado é a generalização quaterniônica da fórmula integral de Cauchy um dos principais teoremas da análise complexa.; The objective of this work is to establish similarities between the complex analysis and the quaternions. In it is made a study of the regularity of quaternionic functions and are established the exponential and logarithmic functions for the quaternions being made a study of the hiperperiodicity of these functions. Another presented result is the quater- nionic generalization of the Cauchy's integral formula one of the main theorems of the complex analysis.

## Quaternions : a mathematica package for quaternionic analysis

Falcão, M. I.; Miranda, Fernando
Tipo: Conferência ou Objeto de Conferência
Relevância na Pesquisa
37.15%
This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton's Quaternion Algebra. This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions. A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with regular functions in $\mathbb{R}^{n+1}$, for $n\geq2$. The performance of the package is illustrated by presenting several examples and applications.; Fundação para a Ciência e a Tecnologia (FCT); Centro de Matemática da Universidade do Minho; Centro de Investigação e Desenvolvimento em Matemática e Aplicações da Universidade de Aveiro

## The representation of physical motions by various types of quaternions

Delphenich, D. H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.31%
It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex, and complex dual quaternions, respectively. It is shown how someof the physically-useful tensors and spinors can be represented by the various kinds of quaternions. The basic notions of kinematical states are described in each case, except complex dual quaternions, where their possible role in describing the symmetries of the Maxwell equations is discussed.; Comment: 127 pages

## Split Quaternions and Particles in (2+1)-Space

Gogberashvili, Merab
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.51%
It is known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply by half-angle quaternions twice from the left and on the right (with its inverse). This 'double cover' property is potential problem in geometrical application of split quaternions, since (2+2)-signature of their norms should not be changed for each product. If split quaternions form proper algebraic structure for microphysics, representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as wavelength of particles. Invariance of space-time intervals and some quantum behavior, like noncommutativity and fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the Cauchy-Riemann analyticity condition and the two fundamental physical parameters (speed of light and Planck's constant) appear from the requirement of positive definiteness of quaternionic norms.; Comment: The version published in Eur. Phys. J. C

## Quaternions in molecular modeling

Karney, Charles F. F.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.31%
Quaternions are an important tool to describe the orientation of a molecule. This paper considers the use of quaternions in matching two conformations of a molecule, in interpolating rotations, in performing statistics on orientational data, in the random sampling of rotations, and in establishing grids in orientation space. These examples show that many of the rotational problems that arise in molecular modeling may be handled simply and efficiently using quaternions.; Comment: ReVTeX, 10 pages. Provides examples of grids with good covering of orientation space (see http://charles.karney.info/orientation/). If necessary, additional information will be provided at http://charles.karney.info/biblio/quat.html

## Grand Antiprism and Quaternions

Koca, Mehmet; Al-Ajmi, Mudhahir; Koca, Nazife Ozdes
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.31%
Vertices of the 4-dimensional semi-regular polytope, the \textit{grand antiprism} and its symmetry group of order 400 are represented in terms of quaternions with unit norm. It follows from the icosian representation of the \textbf{$E_{8}$} root system which decomposes into two copies of the root system of $H_{4}$. The symmetry of the \textit{grand antiprism} is a maximal subgroup of the Coxeter group $W(H_{4})$. It is the group $Aut(H_{2} \oplus H'_{2})$ which is constructed in terms of 20 quaternionic roots of the Coxeter diagram $H_{2} \oplus H'_{2}$. The root system of $H_{4}$ represented by the binary icosahedral group \textit{I}of order 120, constitutes the regular 4D polytope 600-cell. When its 20 quaternionic vertices corresponding to the roots of the diagram $H_{2} \oplus H'_{2}$ are removed from the vertices of the 600-cell the remaining 100 quaternions constitute the vertices of the\textit{grand antiprism}. We give a detailed analysis of the construction of the cells of the\textit{grand antiprism} in terms of quaternions. The dual polytope of the \textit{grand antiprism} has been also constructed.; Comment: 21 pages, 12 Figures

## On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions

Flaut, Cristina; Shpakivskyi, Vitalii
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.31%
In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.; Comment: Accepted in Adv. in Appl. Clifford Algebras

## Understanding Quaternions and the Dirac Belt Trick

Staley, Mark
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.43%
The Dirac belt trick is often employed in physics classrooms to show that a $2\pi$ rotation is not topologically equivalent to the absence of rotation whereas a $4\pi$ rotation is, mirroring a key property of quaternions and their isomorphic cousins, spinors. The belt trick can leave the student wondering if a real understanding of quaternions and spinors has been achieved, or if the trick is just an amusing analogy. The goal of this paper is to demystify the belt trick and to show that it implies an underlying \emph{four-dimensional} parameter space for rotations that is simply connected. An investigation into the geometry of this four-dimensional space leads directly to the system of quaternions, and to an interpretation of three-dimensional vectors as the generators of rotations in this larger four-dimensional world. The paper also shows why quaternions are the natural extension of complex numbers to four dimensions. The level of the paper is suitable for undergraduate students of physics.; Comment: 19 pages, 4 figures

## Quaternions: sucessos e insucessos de um projeto de pesquisa

Dion, Sonia Maria; Pacca, Jesuína Lopes de Almeida; Machado, Nilson José