Introduction to Clifford Analysis
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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. -
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. -
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. -
Hypercomplex Analysis Selected Topics
Hypercomplex Analysis Selected Topics (Habilitation Thesis) Roman Lávička Field: Mathematics — Mathematical Analysis Faculty of Mathematics and Physics Charles University in Prague August 2011 Contents 1 Introduction 5 2 Preliminaries of hypercomplex analysis 11 2.1 Clifford analysis . 11 2.1.1 Euclidean Clifford analysis . 14 2.1.2 Generalized Moisil-Théodoresco systems . 14 2.1.3 Hermitian Clifford analysis . 16 2.2 Quaternionic analysis . 17 3 Complete orthogonal Appell systems 19 3.1 Spherical harmonics . 19 3.1.1 Gelfand-Tsetlin bases for spin modules . 21 3.2 Clifford algebra valued spherical monogenics . 22 3.3 Spinor valued spherical monogenics . 24 3.3.1 The generalized Appell property in dimension 3 . 27 3.4 Hodge-de Rham systems . 29 3.4.1 The Riesz system in dimension 3 . 33 3.5 Hermitian monogenics . 35 4 Finely monogenic functions 39 4.1 Finely holomorphic functions . 39 4.2 Finely monogenic functions . 41 4.3 Finely differentiable monogenic functions . 42 4.4 Open problems . 45 List of reprinted papers 47 Bibliography 49 [L1] Reversible maps in the group of quaternionic Möbius trans- formations 55 [L2] A generalization of monogenic functions to fine domains 71 [L3] A remark on fine differentiability 83 [L4] Finely continuously differentiable functions 91 [L5] The Gelfand-Tsetlin bases for spherical monogenics in di- mension 3 105 3 [L6] Canonical bases for sl(2,C)-modules of spherical monogen- ics in dimension 3 135 [L7] Orthogonal basis for spherical monogenics by step two branch- ing 149 [L8] The Fischer Decomposition for the H-action and Its Appli- cations 177 [L9] The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces 189 [L10] Gelfand-Tsetlin Bases of Orthogonal Polynomials in Her- mitean Clifford Analysis 211 4 Chapter 1 Introduction Without any doubt, complex analysis belongs to the most important areas of mathematics. -
Hypercomplex Numbers and Their Matrix Representations
Hypercomplex numbers and their matrix representations A short guide for engineers and scientists Herbert E. M¨uller http://herbert-mueller.info/ Abstract Hypercomplex numbers are composite numbers that sometimes allow to simplify computations. In this article, the multiplication table, matrix represen- tation and useful formulas are compiled for eight hypercomplex number systems. 1 Contents 1 Introduction 3 2 Hypercomplex numbers 3 2.1 History and basic properties . 3 2.2 Matrix representations . 4 2.3 Scalar product . 5 2.4 Writing a matrix in a hypercomplex basis . 6 2.5 Interesting formulas . 6 2.6 Real Clifford algebras . 8 ∼ 2.7 Real numbers Cl0;0(R) = R ......................... 10 3 Hypercomplex numbers with 1 generator 10 ∼ 3.1 Bireal numbers Cl1;0(R) = R ⊕ R ...................... 10 ∼ 3.2 Complex numbers Cl0;1(R) = C ....................... 11 4 Hypercomplex numbers with 2 generators 12 ∼ ∼ 4.1 Cockle quaternions Cl2;0(R) = Cl1;1(R) = R(2) . 12 ∼ 4.2 Hamilton quaternions Cl0;2(R) = H ..................... 13 5 Hypercomplex numbers with 3 generators 15 ∼ ∼ 5.1 Hamilton biquaternions Cl3;0(R) = Cl1;2(R) = C(2) . 15 ∼ 5.2 Anonymous-3 Cl2;1(R) = R(2) ⊕ R(2) . 17 ∼ 5.3 Clifford biquaternions Cl0;3 = H ⊕ H .................... 19 6 Hypercomplex numbers with 4 generators 20 ∼ ∼ ∼ 6.1 Space-Time Algebra Cl4;0(R) = Cl1;3(R) = Cl0;4(R) = H(2) . 20 ∼ ∼ 6.2 Anonymous-4 Cl3;1(R) = Cl2;2(R) = R(4) . 21 References 24 A Octave and Matlab demonstration programs 25 A.1 Cockle Quaternions . 25 A.2 Hamilton Bi-Quaternions . -
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. -
Algebraic Foundations of Split Hypercomplex Nonlinear Adaptive
Mathematical Methods in the Research Article Applied Sciences Received XXXX (www.interscience.wiley.com) DOI: 10.1002/sim.0000 MOS subject classification: 60G35; 15A66 Algebraic foundations of split hypercomplex nonlinear adaptive filtering E. Hitzer∗ A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Copyright c 2011 John Wiley & Sons, Ltd. Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering, Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford geometric algebra 1. Introduction Split quaternion nonlinear adaptive filtering has recently been treated by [23], who showed its superior performance for Saito’s Chaotic Signal and for wind forecasting. The quaternionic methods constitute a generalization of complex valued adaptive filters, treated in detail in [19]. A method of quaternionic least mean square algorithm for adaptive quaternionic has previously been developed in [22]. Additionally, [24] successfully proposes the usage of local analytic fully quaternionic functions in the Quaternion Nonlinear Gradient Descent (QNGD). Yet the unconditioned use of analytic fully quaternionic activation functions in neural networks faces problems with poles due to the Liouville theorem [3]. The quaternion algebra of Hamilton is a special case of the higher dimensional Clifford algebras [10]. The problem with poles in nonlinear analytic functions does not generally occur for hypercomplex activation functions in Clifford algebras, where the Dirac and the Cauchy-Riemann operators are not elliptic [21], as shown, e.g., for hyperbolic numbers in [17]. -
Hypercomplex Number in Three Dimensional Spaces. Abdelkarim Assoul
Hypercomplex number in three dimensional spaces. Abdelkarim Assoul To cite this version: Abdelkarim Assoul. Hypercomplex number in three dimensional spaces.. 2016. hal-01686021v2 HAL Id: hal-01686021 https://hal.archives-ouvertes.fr/hal-01686021v2 Preprint submitted on 4 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright Hypercomplex number in three dimensional spaces Article: written by Assoul AbdelKarim Secondry School Maths teacher Summary Any point of the real line is the real number image and any point of the R² plane is the complex number image. Is any point of space R3 a number image, if this number exists is it unique and what is its figure and properties? In this article we are going to build an algebra in a commutative field R and demonstrate that this one is isomorphic to R so that at the end this lead to the existence of this number its uniqueness also its figure and properties. The succession of this work wille be the elaboration of this theory so that it will be usefull in applied mathematics,theoretical and physics quantum and espeially using this hypercomplex number for calculating the amplitude and the substitution of the (bit) by the (qubit) in order to have a faster and powerful quantum computer which has also much more capacity. -
Basic Sets of Special Monogenic Polynomials in Fréchet Modules
Hindawi Journal of Complex Analysis Volume 2017, Article ID 2075938, 11 pages https://doi.org/10.1155/2017/2075938 Research Article Basic Sets of Special Monogenic Polynomials in Fréchet Modules Gamal Farghaly Hassan,1,2 Lassaad Aloui,3 and Allal Bakali1 1 Department of Mathematics, Faculty of Sciences, Northern Border University, P.O. Box 1321, Arar, Saudi Arabia 2Faculty of Science, University of Assiut, Assiut 71516, Egypt 3Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis-El Manar, Tunis, Tunisia Correspondence should be addressed to Gamal Farghaly Hassan; [email protected] Received 20 June 2016; Revised 31 October 2016; Accepted 18 December 2016; Published 14 February 2017 Academic Editor: Konstantin M. Dyakonov Copyright © 2017 Gamal Farghaly Hassan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is concerned with the study of the theory of basic sets inFrechet´ modules in Clifford analysis. The main aim of this account, which is based on functional analysis consideration, is to formulate criteria of general type for the effectiveness (convergence properties) of basic sets either in the space itself or in a subspace of finer topology. By attributing particular forms for the Frechet´ module of different classes of functions, conditions are derived from the general criteria for the convergence properties in open and closed balls. Our results improve and generalize some known results in complex and Clifford setting concerning the effectiveness of basic sets. 1. Introduction aboutthestudyofbasicsetsofpolynomialsincomplexanal- ysis, we refer to [17–20]. -
Solomon Lefschetz
NATIONAL ACADEMY OF SCIENCES S O L O M O N L EFSCHETZ 1884—1972 A Biographical Memoir by PHILLIP GRIFFITHS, DONALD SPENCER, AND GEORGE W HITEHEAD Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1992 NATIONAL ACADEMY OF SCIENCES WASHINGTON D.C. SOLOMON LEFSCHETZ September 3, 1884-October 5, 1972 BY PHILLIP GRIFFITHS, DONALD SPENCER, AND GEORGE WHITEHEAD1 OLOMON LEFSCHETZ was a towering figure in the math- Sematical world owing not only to his original contribu- tions but also to his personal influence. He contributed to at least three mathematical fields, and his work reflects throughout deep geometrical intuition and insight. As man and mathematician, his approach to problems, both in life and in mathematics, was often breathtakingly original and creative. PERSONAL AND PROFESSIONAL HISTORY Solomon Lefschetz was born in Moscow on September 3, 1884. He was a son of Alexander Lefschetz, an importer, and his wife, Vera, Turkish citizens. Soon after his birth, his parents left Russia and took him to Paris, where he grew up with five brothers and one sister and received all of his schooling. French was his native language, but he learned Russian and other languages with remarkable fa- cility. From 1902 to 1905, he studied at the Ecole Centrale des Arts et Manufactures, graduating in 1905 with the de- gree of mechanical engineer, the third youngest in a class of 220. His reasons for entering that institution were com- plicated, for as he said, he had been "mathematics mad" since he had his first contact with geometry at thirteen. -
(Communicated by W. BURNSIDE. ) the Object of This Paper Is in the First Place to Set the Theory of Hypercomplex Numbers on a Ra
HYPERCOMPLEX NUMBERS. 77 ON HYPERCOMPLEX NUMBERS By J. H. MACLAGAN WEDDERBURN. (Communicated by W. BURNSIDE. ) [Received July 7th, 1907.—Read November 14th, 1907.] INDEX OF TERMS. Pago Algebra . ... 79 A + B 79 A^B 80 AB 80 Complex 79 Composition series ... ... ... ... ... ... ... ... 83 Difference algebra 82 Difference series ... ... ... ... ... ... ... ... 83 Direct product 99 Direct sum ... ... ... ... ... ... ... ... ... 84 Idempotent ... ... ... ... ... ... ... ... ... 90 Identical equation ... ... ... ... ... ... ... ... 101 Index 87 Integral sub-algebra ... ... ... ... ... ... 84 Invariant... ... ... ... ... ... ... ... ... ... 81 Matric algebra ... ... ... ... ... ... ... ... ... 98 Modular sub-algebra ... ... ... ... ... ... ... ... 112 Modulus 84 Nilpotent 87 Order 79 Potent Algebra 89 Primitive ... ... ... ... ... ... ... 91 Principal idempotent element... ... ... ... ... 92 Quadrate algebra ... ... ... ... -. • ... ... ... 98 Reduced equation ... ... ... ... ... ... ... ... 101 Reducible 84 Reduction series ... ... ... ••• ••• ••• ... ••• 86 Semi-invariant ... ... ... ... ... ... ... ... ... 113 Semi-simple ... ... ... ... ... ••• ••• ••• ... 94 Simple 81 Supplement ... ... ... ... ... ... -. ... 79 Zero algebra 88 THE object of this paper is in the first place to set the theory of hypercomplex numbers on a rational basis. The methods usually employed in treating the parts of the subject here taken up are, as a 78 MR. J, H. MACLAGAN WEDDERBURN [NOV. 14, rule, dependent on the theory of the characteristic equation, and are for this reason often valid only for a particular field or class of fields. Such, for instance, are the methods used by Cartan in his fundamental and far-reaching memoir, Sur les groupes bilineaires et les systemes com- plexes. It is true that the methods there used are often capable of generalisation to any field ; but I do not think that this is by any means always the case. My object throughout has been to develop a treatment analogous to "that Tvhich has been so successful in the theory of finite groups.