Hypercomplex Numbers and Their Matrix Representations
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Characterizations of Several Split Regular Functions on Split Quaternion in Clifford Analysis
East Asian Math. J. Vol. 33 (2017), No. 3, pp. 309{315 http://dx.doi.org/10.7858/eamj.2017.023 CHARACTERIZATIONS OF SEVERAL SPLIT REGULAR FUNCTIONS ON SPLIT QUATERNION IN CLIFFORD ANALYSIS Han Ul Kang, Jeong Young Cho, and Kwang Ho Shon* Abstract. In this paper, we investigate the regularities of the hyper- complex valued functions of the split quaternion variables. We define several differential operators for the split qunaternionic function. We re- search several left split regular functions for each differential operators. We also investigate split harmonic functions. And we find the correspond- ing Cauchy-Riemann system and the corresponding Cauchy theorem for each regular functions on the split quaternion field. 1. Introduction The non-commutative four dimensional real field of the hypercomplex num- bers with some properties is called a split quaternion (skew) field S. Naser [12] described the notation and the properties of regular functions by using the differential operator D in the hypercomplex number system. And Naser [12] investigated conjugate harmonic functions of quaternion variables. In 2011, Koriyama et al. [9] researched properties of regular functions in quater- nion field. In 2013, Jung et al. [1] have studied the hyperholomorphic functions of dual quaternion variables. And Jung and Shon [2] have shown hyperholo- morphy of hypercomplex functions on dual ternary number system. Kim et al. [8] have investigated regularities of ternary number valued functions in Clif- ford analysis. Kang and Shon [3] have developed several differential operators for quaternionic functions. And Kang et al. [4] researched some properties of quaternionic regular functions. Kim and Shon [5, 6] obtained properties of hyperholomorphic functions and hypermeromorphic functions in each hyper- compelx number system. -
Arxiv:1001.0240V1 [Math.RA]
Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS). -
On Hypercomplex Number Systems*
ON HYPERCOMPLEX NUMBER SYSTEMS* (FIRST PAPER) BY HENRY TABER Introduction. The method invented by Benjamin Peirce t for treating the problem to determine all hypereomplex number systems (or algebras) in a given number of units depends chiefly, first, upon the classification of hypereomplex number sys- tems into idempotent number systems, containing one or more idempotent num- bers, \ and non-idempotent number systems containing no idempotent number ; and, second, upon the regularizaron of idempotent number systems, that is, the classification of each of the units of such a system with respect to one of the idempotent numbers of the system. For the purpose of such classification and regularizaron the following theorems are required : Theorem I.§ In any given hypereomplex number system there is an idem- potent number (that is, a number I + 0 such that I2 = I), or every number of the system is nilpotent. || Theorem H.*rj In any hypereomplex number system containing an idem- potent number I, the units ex, e2, ■■ -, en can be so selected that, with reference ♦Presented to the Society February 27, 1904. Received for publication February 27, 1904, and September 6, 1904. t American Journal of Mathematics, vol. 4 (1881), p. 97. This work, entitled Linear Associative Algebra, was published in lithograph in 1870. For an estimate of Peirce's work and for its relation to the work of Study, Scheffers, and others, see articles by H. E. Hawkes in the American Journal of Mathematics, vol. 24 (1902), p. 87, and these Transactions, vol. 3 (1902), p. 312. The latter paper is referred to below when reference is made to Hawkes' work. -
Hypercomplex Numbers
Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame. -
Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space
Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space Murat Babaarslan and Yusuf Yayli Abstract. A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results. Mathematics Subject Classification (2010). Primary 53A05; Secondary 53A17, 53A35. Key words: Spacelike constant slope surface, split quaternion, homothetic motion. 1. Introduction Quaternions were discovered by Sir William Rowan Hamilton as an extension to the complex number in 1843. The most important property of quaternions is that every unit quaternion represents a rotation and this plays a special role in the study of rotations in three- dimensional spaces. Also quaternions are an efficient way understanding many aspects of physics and kinematics. Many physical laws in classical, relativistic and quantum mechanics can be written nicely using them. Today they are used especially in the area of computer vision, computer graphics, animations, aerospace applications, flight simulators, navigation systems and to solve optimization problems involving the estimation of rigid body transformations. Ozdemir and Ergin [9] showed that a unit timelike quaternion represents a rotation in Minkowski 3-space. -
Quaternion Algebra and Calculus
Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal. -
Truly Hypercomplex Numbers
TRULY HYPERCOMPLEX NUMBERS: UNIFICATION OF NUMBERS AND VECTORS Redouane BOUHENNACHE (pronounce Redwan Boohennash) Independent Exploration Geophysical Engineer / Geophysicist 14, rue du 1er Novembre, Beni-Guecha Centre, 43019 Wilaya de Mila, Algeria E-mail: [email protected] First written: 21 July 2014 Revised: 17 May 2015 Abstract Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This paper provides a definitive solution to this problem by defining the truly hypercomplex numbers of dimension N ≥ 3. The secret lies in the definition of the multiplicative law and its properties. This law is based on spherical and hyperspherical coordinates. These numbers which I call spherical and hyperspherical hypercomplex numbers define Abelian groups over addition and multiplication. Nevertheless, the multiplicative law generally does not distribute over addition, thus the set of these numbers equipped with addition and multiplication does not form a mathematical field. However, such numbers are expected to have a tremendous utility in mathematics and in science in general. Keywords Hypercomplex numbers; Spherical; Hyperspherical; Unification of numbers and vectors Note This paper (or say preprint or e-print) has been submitted, under the title “Spherical and Hyperspherical Hypercomplex Numbers: Merging Numbers and Vectors into Just One Mathematical Entity”, to the following journals: Bulletin of Mathematical Sciences on 08 August 2014, Hypercomplex Numbers in Geometry and Physics (HNGP) on 13 August 2014 and has been accepted for publication on 29 April 2015 in issue No. -
Coset Group Construction of Multidimensional Number Systems
Symmetry 2014, 6, 578-588; doi:10.3390/sym6030578 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Article Coset Group Construction of Multidimensional Number Systems Horia I. Petrache Department of Physics, Indiana University Purdue University Indianapolis, Indianapolis, IN 46202, USA; E-Mail: [email protected]; Tel.: +1-317-278-6521; Fax: +1-317-274-2393 Received: 16 June 2014 / Accepted: 7 July 2014 / Published: 11 July 2014 Abstract: Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature. Keywords: complex numbers; quaternions; representations 1. Introduction While the utility of the familiar complex numbers in physics and applied sciences is not questionable, one important question still stands: where do they come from? What are complex numbers fundamentally? This question becomes even more important considering that complex numbers in more than two dimensions such as quaternions and octonions are gaining renewed interest in physics. This manuscript demonstrates that multidimensional numbers systems are representations of small group symmetries. Although measurements in the laboratory produce real numbers, complex numbers in two dimensions are essential elements of mathematical descriptions of physical systems. -
A Tutorial on Euler Angles and Quaternions
A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ Version 2.0.1 c 2014–17 by Moti Ben-Ari. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-sa/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Chapter 1 Introduction You sitting in an airplane at night, watching a movie displayed on the screen attached to the seat in front of you. The airplane gently banks to the left. You may feel the slight acceleration, but you won’t see any change in the position of the movie screen. Both you and the screen are in the same frame of reference, so unless you stand up or make another move, the position and orientation of the screen relative to your position and orientation won’t change. The same is not true with respect to your position and orientation relative to the frame of reference of the earth. The airplane is moving forward at a very high speed and the bank changes your orientation with respect to the earth. The transformation of coordinates (position and orientation) from one frame of reference is a fundamental operation in several areas: flight control of aircraft and rockets, move- ment of manipulators in robotics, and computer graphics. This tutorial introduces the mathematics of rotations using two formalisms: (1) Euler angles are the angles of rotation of a three-dimensional coordinate frame. -
Classification of Left Octonion Modules
Classification of left octonion modules Qinghai Huo, Yong Li, Guangbin Ren Abstract It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element m may be the whole module and may be not in the form Om. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions. Keywords: Octonion module; associative element; cyclic element; Cℓ7-module. AMS Subject Classifications: 17A05 Contents 1 introduction 1 2 Preliminaries 3 2.1 The algebra of the octonions O .............................. 3 2.2 Universal Clifford algebra . ... 4 3 O-modules 6 4 The structure of left O-moudles 10 4.1 Finite dimensional O-modules............................... 10 4.2 Structure of general left O-modules............................ 13 4.3 Cyclic elements in left O-module ............................. 15 arXiv:1911.08282v2 [math.RA] 21 Nov 2019 1 introduction The theory of quaternion Hilbert spaces brings the classical theory of functional analysis into the non-commutative realm (see [10, 16, 17, 20, 21]). It arises some new notions such as spherical spectrum, which has potential applications in quantum mechanics (see [4, 6]). -
Geodesics in Hypercomplex Number Systems. Application to Commutative Quaternions
Dipartimento Energia GEODESICS IN HYPERCOMPLEX NUMBER SYSTEMS. APPLICATION TO COMMUTATIVE QUATERNIONS FRANCESCO CATONI, PAOLO ZAMPETTI ENEA - Dipartimento Energia Centro Ricerche Casaccia, Roma ROBERTO CANNATA ENEA - Funzione Centrals INFO Centro Ricerche Casaccia, Roma LUCIANA BORDONI ENEA - Funzione Centrals Studi Centro Ricerche Casaccia, Roma R eCE#VSQ J foreign SALES PROHIBITED RT/EFtG/97/11 ENTE PER LE NUOVE TECNOLOGIE, L'ENERGIA E L'AMBIENTE Dipartimento Energia GEODESICS IN HYPERCOMPLEX NUMBER SYSTEMS. APPLICATION TO COMMUTATIVE QUATERNIONS FRANCESCO CATONI, PAOLO ZAMPETTI ENEA - Dipartimento Energia Centro Ricerche Casaccia, Roma ROBERTO CANNATA ENEA - Funzione Centrale INFO Centro Ricerche Casaccia, Roma LUCIANA GORDON! ENEA - Funzione Centrale Studi Centro Ricerche Casaccia, Roma RT/ERG/97/11 Testo pervenuto nel giugno 1997 I contenuti tecnico-scientifici del rapporti tecnici dell'ENEA rispecchiano I'opinione degli autori e non necessariamente quella dell'Ente. DISCLAIMER Portions of this document may be illegible electronic image products. Images are produced from the best available original document. RIASSUNTO Geodetiche nei sistemi di numeri ipercomplessi. Appli cations ai quaternion! commutativi Alle funzioni di variabili ipercomplesse sono associati dei campi fisici. Seguendo le idee che hanno condotto Einstein alia formulazione della Relativita Generale, si intende verificare la possibility di descrivere il moto di un corpo in un campo gravitazionale, mediante le geodetiche negli spazi ’’deformati ” da trasformazioni funzionali di variabili ipercomplesse che introducono nuove simmetrie dello spazio. II presente lavoro rappresenta la fase preliminare di questo studio pin ampio. Come prima applicazione viene studiato il sistema particolare di numeri, chiamato ”quaternion! commutativi ”, che possono essere considerati come composizione dei sistemi bidimensionali dei numeri complessi e dei numeri iperbolici. -
Working with 3D Rotations
Working With 3D Rotations Stan Melax Graphics Software Engineer, Intel Human Brain is wired for Spatial Computation Which shape is the same: a) b) c) “I don’t need to ask A childhood IQ test question for directions” Translations Rotations Agenda ● Rotations and Matrices (hopefully review) ● Combining Rotations ● Matrix and Axis Angle ● Challenges of deep Space (of Rotations) ● Quaternions ● Applications Terminology Clarification Preferred usages of various terms: Linear Angular Object Pose Position (point) Orientation A change in Pose Translation (vector) Rotation Rate of change Linear Velocity Spin also: Direction specifies 2 DOF, Orientation specifies all 3 angular DOF. Rotations Trickier than Translations Translations Rotations a then b == b then a x then y != y then x (non-commutative) ● Programming with rotations also more challenging! 2D Rotation θ Rotate [1 0] by θ about origin 1,1 [ cos(θ) sin(θ) ] θ θ sin cos θ 1,0 2D Rotation θ Rotate [0 1] by θ about origin -1,1 0,1 sin θ [-sin(θ) cos(θ)] θ θ cos 2D Rotation of an arbitrary point Rotate about origin by θ = cos θ + sin θ 2D Rotation of an arbitrary point 푥 Rotate 푦 about origin by θ 푥′, 푦′ 푥′ = 푥 cos θ − 푦 sin θ 푦′ = 푥 sin θ + 푦 cos θ 푦 푥 2D Rotation Matrix 푥 Rotate 푦 about origin by θ 푥′, 푦′ 푥′ = 푥 cos θ − 푦 sin θ 푦′ = 푥 sin θ + 푦 cos θ 푦 푥′ cos θ − sin θ 푥 푥 = 푦′ sin θ cos θ 푦 cos θ − sin θ Matrix is rotation by θ sin θ cos θ 2D Orientation 풚 Yellow grid placed over first grid but at angle of θ cos θ − sin θ sin θ cos θ 풙 Columns of the matrix are the directions of the axes.