The Culture of Quaternions the Phoenix Bird of Mathematics
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James Clerk Maxwell
James Clerk Maxwell JAMES CLERK MAXWELL Perspectives on his Life and Work Edited by raymond flood mark mccartney and andrew whitaker 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013942195 ISBN 978–0–19–966437–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. -
From Counting to Quaternions -- the Agonies and Ecstasies of the Student Repeat Those of D'alembert and Hamilton
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Keck Graduate Institute Journal of Humanistic Mathematics Volume 1 | Issue 1 January 2011 From Counting to Quaternions -- The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton Reuben Hersh University of New Mexico Follow this and additional works at: https://scholarship.claremont.edu/jhm Recommended Citation Hersh, R. "From Counting to Quaternions -- The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton," Journal of Humanistic Mathematics, Volume 1 Issue 1 (January 2011), pages 65-93. DOI: 10.5642/jhummath.201101.06 . Available at: https://scholarship.claremont.edu/jhm/vol1/ iss1/6 ©2011 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. From Counting to Quaternions { The Agonies and Ecstasies of the Student Repeat Those of D'Alembert and Hamilton Reuben Hersh Department of Mathematics and Statistics, The University of New Mexico [email protected] Synopsis Young learners of mathematics share a common experience with the greatest creators of mathematics: \hitting a wall," meaning, first frustration, then strug- gle, and finally, enlightenment and elation. -
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. -
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]. -
An Idle, Unpopular Student
CHAPTER THREE An Idle, Unpopular Student SLEEPLESS he lay beneath a blanket in the cockpit of the yacht, under the wide and starry sky spread seamlessly from horizon to horizon. Nothing obtruded to betray the presence of 19th‐century man, beyond the small wooden island with sails carrying the 37‐year‐old Louis across the greatness of the Pacific towards the archipelago of the Paumotus. In the glow of the binnacle lamp he could see the silhouette of the helmsman, guiding them towards a tiny tuft of palms and a coastline so low it was almost submerged, like the islands in the milk of Louis’s porridge all those years ago. The Pacific night around him now was warm as milk . and all of a sudden I had a vision of ‐ Drummond Street. It came on me like a flash of lightning: I simply returned thither, and into the past. And when I remember all I hoped and feared as I pickled about Rutherford’s in the rain and the east wind; how I feared I should make a mere shipwreck, and yet timidly hoped not; how I feared I should never have a friend, far less a wife, and yet passionately hoped I might; how I hoped (if I did not take to drink) I should possibly write one little book, etc. etc. And then now ‐ what a change! I feel somehow as if I should like the incident set upon a brass plate at the corner of that dreary thoroughfare for all students to read, poor devils, when their hearts are down.1 THE officious clanging of the bell summoned students from the Pump, as they called Rutherford’s tavern in Drummond Street, a stone’s throw from the great Playfair arch that gave access to the quadrangle of the university. -
The Scottish Irish Mathematical Trail: People, Places, Practices
BSHM-CSHPM/SCHPM Conference at St Andrews The Scottish Irish Mathematical Trail: People, Places, Practices Colm Mulcahy, Spelman College, Atlanta, GA, USA www.mathsireland.ie 14 July 2021 ABSTRACT We survey about 65 of the mathematical people with educational or career ties to both Scotland and Ireland, going back to the 1760s. The scope is broad, included pure and applied maths, astronomy, theoretical physics, maths physics, and statistics. From James & William Thomson and Alice Everitt in distant times, to Philip Gormley, Sheila Tinney, Murray McBeath, Andrew Young, and Alisdair Wood from the past century, we’ll explore many maths connections between Scotland and Ireland. Additional people (and all in more detail) can be found at the new blog at www.mathsireland.ie. Travellers on the Scottish Irish mathematical trail A panorama covering over 250 years Includes mathematical people with educational or career ties to Scotland and Ireland, going back to the 1760s. It’s very broad in scope, included pure and applied maths, astronomy, theoretical physics, maths physics, stats, etc. Travellers on the Scottish Irish mathematical trail Most obviously, we include: 1. Irish scholars who went to Scotland for education and/or employment: from James & William Thomson in the 1800s, to Liam O’Carroll and George Gettingby in modern times. 2. Scottish scholars who went to Ireland for similar reasons: from Peter Tait, George Slesser, and Charles Niven in the 1800s, to Gordon Lessells and Alastair Wood more recently. 3. People from elsewhere who spent significant stretches of time (3 or more years) in both countries: e.g., Joseph Everett, Edmund Whittaker, and Andrew Young. -
Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics
QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D Columbia University 2015 Approved by ______________________________________________________________________ Chairperson of Supervisory Committee _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Program Authorized to Offer Degree ___________________________________________________________________ Date _______________________________________________________________________________ COLUMBIA UNIVERSITY QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS By Johannes C. Familton Chairperson of the Supervisory Committee: Dr. Bruce Vogeli and Dr Henry O. Pollak Department of Mathematics Education TABLE OF CONTENTS List of Figures......................................................................................................iv List of Tables .......................................................................................................vi Acknowledgements .......................................................................................... vii Chapter I: Introduction ......................................................................................... 1 A. Need for Study ........................................................................................ -
Poly Slice Monogenic Functions, Cauchy Formulas and the PS
POLY SLICE MONOGENIC FUNCTIONS, CAUCHY FORMULAS AND THE PS-FUNCTIONAL CALCULUS DANIEL ALPAY, FABRIZIO COLOMBO, KAMAL DIKI, AND IRENE SABADINI Abstract. Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. The notion of poly slice analytic function has been recently introduced for the quaternionic setting. In this paper we study the theory of poly slice monogenic functions and the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-functional calculus. Also for this poly monogenic functional calculus we use the notion of S-spectrum. AMS Classification 47A10, 47A60. Keywords: Poly slice monogenic function, Cauchy formulas, PS-resolvent operators, modified S- resolvent operators, spectral theory on the S-spectrum. Contents 1. Introduction 1 2. Preliminary material 4 3. Poly slice monogenic functions 7 4. Cauchy formulas and product of poly slice monogenic functions 15 −1 −1 4.1. Cauchy formulas with kernels PℓSL and PℓSR 15 −1 −1 4.2. Cauchy formulas with kernels ΠℓSL and ΠℓSR 18 4.3. Poly ⊛L-product and ⊛R-product 21 5. Formulations of the PS-functional calculus via the ΠS-resolvent operators 23 6. Formulations of the PS-functional calculus via the modified S-resolvent operators 29 7. Equivalence of the two definitions of the PS-functional calculus and the product rules 34 References 39 arXiv:2011.13912v1 [math.FA] 27 Nov 2020 1. -
The Vortex Theory of Atoms — Pinnacle of Classical Physics —
The Vortex Theory of Atoms — pinnacle of classical physics — Thesis for the master’s degree in History and Philosophy of Science, by Steven van der Laan1 Supervisor: Dr. Jeroen van Dongen Institute for History and Foundations of Science Utrecht University December 2012 [email protected] i Abstract Investigation into a nineteenth century atomic theory that assumed atoms to be vortices in an ether. Originally an idea of Kelvin, the vortex atom theory enjoyed great popularity in England, roughly form 1870 to 1890. In this thesis, the popular- ity of the vortex atom theory is explained through four values in nineteenth century English physics. Furthermore, it will be argued that the (surprising) fact that the theory was virtually ignored by German scientists can be explained by a German critical attitude towards scientific theories. An attitude that was a reaction to the Romantic philosophy of Naturphilosophie. Acknowledgements Without the support of my supervisor, Jeroen van Dongen, this masterthesis would have been nothing of what it is in its current form. Not only did he teach me a lot about doing history of science and writing academic texts in general, he also broke my habit of cutting corners while doing research. I am furthermore always thankful for Sylvia and Arjo, for they made everything possible. ii Contents 1 Introduction to the vortex theory of atoms 1 1.1 Etherandatoms ................................... 1 1.2 ThevortextheoryofKelvin ............................ 2 1.3 Historyofthevortextheoryofatoms . ..... 3 1.4 Researchquestion.................................. 10 2 The Vortex Theory of Atoms 12 2.1 Propertiesofgasesandmolecules . ... 12 2.2 Requirementsmadetoatoms . .. .. 13 2.3 Vortextheoryofgases .............................. .. 16 2.4 Vortextheoryforchemistry . -
Hypercomplex Analysis and Applications
Trends in Mathematics Hypercomplex Analysis and Applications Bearbeitet von Irene Sabadini, Franciscus Sommen 1. Auflage 2010. Buch. vIII, 284 S. Hardcover ISBN 978 3 0346 0245 7 Format (B x L): 15,5 x 23,5 cm Gewicht: 602 g Weitere Fachgebiete > Mathematik > Mathematische Analysis Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Bounded Perturbations of the Resolvent Operators Associated to the F-Spectrum Fabrizio Colombo and Irene Sabadini Abstract. Recently,wehaveintroducedtheF-functional calculus and the SC-functional calculus. Our theory can be developed for operators of the form T = T0 + e1T1 + ...+ enTn where (T0,T1,...,Tn)isan(n + 1)-tuple of linear commuting operators. The SC-functional calculus, which is defined for bounded but also for unbounded operators, associates to a suitable slice monogenic function f with values in the Clifford algebra Rn the operator f(T ). The F-functional calculus has been defined, for bounded operators T , by an integral transform. Such an integral transform comes from the Fueter’s mapping theorem and it associates to a suitable slice monogenic function f n−1 the operator f˘(T ), where f˘(x)=Δ 2 f(x) and Δ is the Laplace operator. Both functional calculi are based on the notion of F-spectrum that plays the role that the classical spectrum plays for the Riesz-Dunford functional calculus. -
Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home
Clyde Davenport's Commutative Hypercomplex Math Page Page 1 of 11 Commutative Hypercomplex Mathematics Hypercomplex Algebra 4-D Function Theory Hypercomplex Analysis Links Home Introduction Below, we shall give a comprehensive sketch of the 4-D commutative hypercomplex algebra ( not quaternions) and its associated function theory and analysis. The great advantage is a complete, classical 4-D function theory, something that is impossible with quaternions and other noncommutative or nonassociative systems. Fortunately, a wide audience should be able to follow the discussion, because the commutative hypercomplex math is derived directly from well-known, fundamental concepts, such as groups, rings, calculus, complex variables, matrices, complex function theory, and vector analysis. For a discussion of elementary group and ring theory, see any good introductory text on abstract algebra, such as Herstein, 1986 . There will be no occasion for deep theorems and complicated proofs. Top First, some background. How many of us know that Sir William Rowan Hamilton developed the quaternions (4-D numbers similar to those that we will discuss here, except noncommutative) in the 1830s specifically for field calculations? And that scientists and engineers of the time vehemently resisted their use? Something in the mindset at the time simply could not accept the notion that there could be a "fourth dimension," especially if it was claimed to be time. They clung tenaciously to a primitive combination of component- by-component calculation and extensive use of geometry. By the late 1800s, their calculations were accompanied by elaborate geometrical figures that looked like the 2 X 4 framing of a house. To allay their aversions, around 1880, J. -
The H∞ Functional Calculus Based on the S- Spectrum for Quaternionic
Chapman University Chapman University Digital Commons Mathematics, Physics, and Computer Science Science and Technology Faculty Articles and Faculty Articles and Research Research 2016 The H∞ uncF tional Calculus Based on the S- Spectrum for Quaternionic Operators and for N- Tuples of Noncommuting Operators Daniel Alpay Chapman University, [email protected] Fabrizio Colombo Politecnico di Milano Tao Qian University of Macau Irene Sabadini Politecnico di Milano Tao Qian University of Macau Follow this and additional works at: http://digitalcommons.chapman.edu/scs_articles Part of the Algebra Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation D. Alpay, F. Colombo, I. Sabadini and T. Qian. The $H^\infty$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators. Journal of functional analysis, vol. 271, (2016), 1544-1584. This Article is brought to you for free and open access by the Science and Technology Faculty Articles and Research at Chapman University Digital Commons. It has been accepted for inclusion in Mathematics, Physics, and Computer Science Faculty Articles and Research by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. The H∞ uncF tional Calculus Based on the S-Spectrum for Quaternionic Operators and for N-Tuples of Noncommuting Operators Comments NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document.