Hypercomplex Numbers in APL Bob Smith Sudley Place Software Originally Written 14 Sep 2015 Updated 11 Apr 2018
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Is Ja Dialect of APL? Reported by Jonathan Barman Eugene Mcdonnell - the Question Is Irrelevant
VICTOR Vol.8 No.2 convert the noun back to verb and as a result various interesting control structures can be created. The verbal nouns thus formed can be put into arrays and some of the benefits that various people have suggested might arise from arrays of functions can be obtained. A design goal is to make compilation possible without limiting the expressive power of J. APL Technology of Computer Simulation, by A Boozin and I Popselov reported by Sylvia Camacho Both authors of this paper are academic mathematicians and their use of APL is just where one might expect it to be: in modelling linear differential equations and stochastic Markovian processes. Questions put showed that they first used APL 15 years ago on terminals to what was described as 'a big old computer'. Currently they use APL*PLUS/PC. They have set up an environment in which they can split the screen and display data graphically with some zoom facilities. This environment can be used in conjunction with raw APL while exploring the results of a model, or can be used in conjunction with data derived from APL or some other source. In fact it is often used by Pascal programmers. Most recently they have been setting up economic models to try to come to terms with the consequences of decentralisation and privatisation. They use both Roman and Cyrillic alphabets. Panel: Is Ja Dialect of APL? reported by Jonathan Barman Eugene McDonnell - The question is irrelevant. Surely proponents of J would not be thrown out of the APL community? Garth Foster - J is quite definitely not APL. -
Gauging the Octonion Algebra
UM-P-92/60_» Gauging the octonion algebra A.K. Waldron and G.C. Joshi Research Centre for High Energy Physics, University of Melbourne, Parkville, Victoria 8052, Australia By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. PACS numbers: 11.30.Ly, 12.10.Dm, 12.40.-y. Typeset Using REVTEX 1 L INTRODUCTION The aim of this work is to genuinely gauge the octonion algebra as opposed to relating properties of this algebra back to the well known theory of Lie Groups and fibre bundles. Typically most attempts to utilise the octonion symmetry in physics have revolved around considerations of the automorphism group G2 of the octonions and Jordan matrix representations of the octonions [1]. Our approach is more simple since we provide a spinorial approach to the octonion symmetry. Previous to this work there were already several indications that this should be possible. To begin with the statement of the gauge principle itself uno theory shall depend on the labelling of the internal symmetry space coordinates" seems to be independent of the exact nature of the gauge algebra and so should apply equally to non-associative algebras. The octonion algebra is an alternative algebra (the associator {x-1,y,i} = 0 always) X -1 so that the transformation law for a gauge field TM —• T^, = UY^U~ — ^(c^C/)(/ is well defined for octonionic transformations U. -
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 Octonions
Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x . -
An Octonion Model for Physics
An octonion model for physics Paul C. Kainen¤ Department of Mathematics Georgetown University Washington, DC 20057 USA [email protected] Abstract The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread. Keywords: Geometric algebra, the Four Color Conjecture, rooted cu- bic plane trees, Catalan numbers, quaternions, octaves, quantum alge- bra, gravity, waves, associativity 1 Introduction We explore some consequences of octonion arithmetic for a hidden variables model of quantum theory and ideas regarding the propagation of gravity. This approach may also have utility for number theory and combinatorial topology. Octonion models are currently the focus of much work in the physics com- munity. See, e.g., Okubo [29], Gursey [14], Ward [36] and Dixon [11]. Yaglom [37, pp. 94, 107] referred to an \octonion boom" and it seems to be acceler- ating. Historically speaking, the inventors/discoverers of the quaternions, oc- tonions and related algebras (Hamilton, Cayley, Graves, Grassmann, Jordan, Cli®ord and others) were working from a physical point-of-view and wanted ¤4th Conference on Emergence, Coherence, Hierarchy, and Organization (ECHO 4), Odense, Denmark, 2000 1 their abstractions to be helpful in solving natural problems [37]. Thus, a con- nection between physics and octonions is a reasonable though not yet fully justi¯ed suspicion. It is easy to see the allure of octaves since there are many phenomena in the elementary particle realm which have 8-fold symmetries. -
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. -
K.E.Iverson and APL.J
K.E.Iverson Kenneth Iverson Charismatic mathematician who invented the APL computer programming language A GIFTED mathematician and a charismatic teacher, Ken Iverson made a highly influential contribution to the field of computer science. In the early 1960s a mathematical notation which he had developed as an aide to teaching algebra formed the basis of APL, one of the languages used in programming IBM’s early mainframe computer, the System/360. This concise and powerful language contributed substantially to IBM’s domination of the emerging computer industry during the 1960s and 1970s. Kenneth Eugene Iverson was born in Camrose, Alberta, Canada, in 1920. He demonstrated an early aptitude for mathematics ― he taught himself calculus in his teens. During the Second World War he served in the Royal Canadian Air Force as a flight engineer specialising in reconnaissance. After the war he obtained a degree in mathematics and physics from Queen’s University, Ontario, and went on to postgraduate study at Harvard where, in 1954, he obtained a doctorate in applied mathematics, and from 1955 to 1960 he was assistant professor of applied mathematics. During this period he developed a novel way of teaching algebra to students, the “Iverson notation”. It attracted the interest of IBM, which was already well established in commercial and scientific computing fields and was developing a new mainframe, the System/360. IBM recruited Iverson and three colleagues to turn his teaching notation into a program- ming language which could be used on the System/360. The result, expounded in his book A Programming Language (1962), came to be known as APL. -
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]). -
The Computational Attitude in Music Theory
The Computational Attitude in Music Theory Eamonn Bell Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2019 © 2019 Eamonn Bell All rights reserved ABSTRACT The Computational Attitude in Music Theory Eamonn Bell Music studies’s turn to computation during the twentieth century has engendered particular habits of thought about music, habits that remain in operation long after the music scholar has stepped away from the computer. The computational attitude is a way of thinking about music that is learned at the computer but can be applied away from it. It may be manifest in actual computer use, or in invocations of computationalism, a theory of mind whose influence on twentieth-century music theory is palpable. It may also be manifest in more informal discussions about music, which make liberal use of computational metaphors. In Chapter 1, I describe this attitude, the stakes for considering the computer as one of its instruments, and the kinds of historical sources and methodologies we might draw on to chart its ascendance. The remainder of this dissertation considers distinct and varied cases from the mid-twentieth century in which computers or computationalist musical ideas were used to pursue new musical objects, to quantify and classify musical scores as data, and to instantiate a generally music-structuralist mode of analysis. I present an account of the decades-long effort to prepare an exhaustive and accurate catalog of the all-interval twelve-tone series (Chapter 2). This problem was first posed in the 1920s but was not solved until 1959, when the composer Hanns Jelinek collaborated with the computer engineer Heinz Zemanek to jointly develop and run a computer program. -
On the Tensor Product of Two Composition Algebras
On The Tensor Product of Two Composition Algebras Patrick J. Morandi S. Pumplun¨ 1 Introduction Let C1 F C2 be the tensor product of two composition algebras over a field F with char(F ) =¡ 2. R. Brauer [7] and A. A. Albert [1], [2], [3] seemed to be the first math- ematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general situation by B. N. Allison [4], [5], [6] and to biquaternion algebras over rings by Knus [12]. In the second section we give some new results on the Albert form of these algebras. We also investigate the F -quadric defined by this Albert form, generalizing a result of Knus ([13]). £ £ Since Allison regarded the involution ¢ = 1 2 as an essential part of the algebra C = C1 C2, he only studied automorphisms of C which are compatible with ¢ . In the last section we show that any automorphism of C that preserves a certain biquaternion subalgebra also is compatible with ¢ . As a consequence, if C is the tensor product of two octonion algebras, we show that C does not satisfy the Skolem-Noether Theorem. Let F be a field and C a unital, nonassociative F -algebra. Then C is a composition algebra if there exists a nondegenerate quadratic form n : C ¤ F such that n(x · y) = n(x)n(y) for all x, y ¥ C. The form n is uniquely determined by these conditions and is called the norm of C. We will write n = nC . Composition algebras only exist in ranks 1, 2, 4 or 8 (see [10]). -
Similarity and Consimilarity of Elements in Real Cayley-Dickson Algebras
SIMILARITY AND CONSIMILARITY OF ELEMENTS IN REAL CAYLEY-DICKSON ALGEBRAS Yongge Tian Department of Mathematics and Statistics Queen’s University Kingston, Ontario, Canada K7L 3N6 e-mail:[email protected] October 30, 2018 Abstract. Similarity and consimilarity of elements in the real quaternion, octonion, and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equations ax = xb and ax = xb in these algebras. Some consequences are also presented. AMS mathematics subject classifications: 17A05, 17A35. Key words: quaternions, octonions, sedenions, Cayley-Dickson algebras, equations, similarity, consim- ilarity. 1. Introduction We consider in the article how to establish the concepts of similarity and consimilarity for ele- n arXiv:math-ph/0003031v1 26 Mar 2000 ments in the real quaternion, octonion and sedenion algebras, as well as in the 2 -dimensional real Cayley-Dickson algebras. This consideration is motivated by some recent work on eigen- values and eigenvectors, as well as similarity of matrices over the real quaternion and octonion algebras(see [5] and [18]). In order to establish a set of complete theory on eigenvalues and eigenvectors, as well as similarity of matrices over the quaternion, octonion, sedenion alge- bras, as well as the general real Cayley-Dickson algebras, one must first consider a basic problem— how to characterize similarity of elements in these algebras, which leads us to the work in this article. Throughout , , and denote the real quaternion, octonion, and sedenion algebras, H O S n respectively; n denotes the 2 -dimensional real Cayley-Dickson algebra, and denote A R C the real and complex number fields, respectively. -
How We Got to APL\1130
How We Got To APL\1130 by Larry Breed APLBUG at CHM, 10 May 2004 APL\1130 was implemented overnight in the autumn of 1967, but it took years of effort to make that possible. In 1965, Ken Iverson’s group in Yorktown was wrestling with the transition from Iverson Notation, a notation suited for blackboards and printed pages, to a machine-executable programming environment. Iverson had already developed a Selectric typeball and was writing a high-school math textbook. By autumn 1965, Larry Breed and Stanford grad student Philip Abrams had written the first implementation in 7090 Fortran (with batch execution). Eugene McDonnell was looking for applications to run on TSM, his project’s experimental time-sharing system on a virtual-memory 7090. With his help, Breed ported the Fortran code, added I/O routines for the 1050 terminals, called the result IVSYS; and Iverson Notation went interactive. Iverson, Breed and other associates now faced issues of input and output, function definition, error handling, suspended execution, and workspaces. They also struggled with both extending the notation to multidimensional arrays and new primitives, and limiting it to what could reasonably be implemented. Whatever it was, it wasn’t “APL”; that name came later. Iverson was also collaborating with John Lawrence of Science Research Associates (SRA), an IBM subsidiary in Chicago aimed at educational markets. Lawrence had been editor of the IBM Systems Journal when it published the Iversonian tour de force “A Formal Description of System/360.” At SRA he was leading a project to make Iverson’s notation the heart of a line of instructional materials, both books and interactive computer applications.