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The 3X + 1 Problem: an Annotated Bibliography
The 3x +1 Problem: An Annotated Bibliography (1963–1999) (Sorted by Author) Jeffrey C. Lagarias Department of Mathematics University of Michigan Ann Arbor, MI 48109–1109 [email protected] (January 1, 2011 version) ABSTRACT. The 3x + 1 problem concerns iteration of the map T : Z Z given by → 3x + 1 if x 1 (mod 2) . 2 ≡ T (x)= x if x 0 (mod 2) . 2 ≡ The 3x + 1 Conjecture asserts that each m 1 has some iterate T (k)(m) = 1. This is an ≥ annotated bibliography of work done on the 3x + 1 problem and related problems from 1963 through 1999. At present the 3x + 1 Conjecture remains unsolved. This version of the 3x + 1 bibliography sorts the papers alphabetically by the first author’s surname. A different version of this bibliography appears in the recent book: The Ultimate Challenge: The 3x + 1 Problem, Amer. Math. Soc, Providence, RI 2010. In it the papers are sorted by decade (1960-1969), (1970-1979), (1980-1989), (1990-1999). 1. Introduction The 3x + 1 problem is most simply stated in terms of the Collatz function C(x) defined on integers as “multiply by three and add one” for odd integers and “divide by two” for even integers. That is, arXiv:math/0309224v13 [math.NT] 11 Jan 2011 3x +1 if x 1 (mod 2) , ≡ C(x)= x if x 0 (mod 2) , 2 ≡ The 3x + 1 problem, or Collatz problem , is to prove that starting from any positive integer, some iterate of this function takes the value 1. Much work on the problem is stated in terms of the 3x + 1 function 3x + 1 if x 1 (mod 2) 2 ≡ T (x)= x if x 0 (mod 2) . -
The Other Authors of Counting Australia in Graeme Cohen
The other authors of Counting Australia In Graeme Cohen I happen to know that Ernie Tuck has a review of Counting Australia In in this issue of the Gazette. He tells us that there are about 1200 names in the Names Index. This is a count which I had not carried out myself but I have been cataloguing the material collected over five years, a job not yet finished, and have about 320 files of information on Australian mathematicians, dead and alive, here and abroad. These include published obituaries and other biographical reprints, but mostly consist of letters and emailed notes. My intention here is to thank all of those who wrote to me, or emailed me, or spoke to me in person or by phone, by writingof some of the sto- ries that arose that way. Motivation My time involved more than research directed to- wards the writingof a history of Australian math- ematics. The work as a whole was dubbed the Australian Mathematical Society History Project and an important aspect was the tapingof inter- views with 25 mathematicians who were founda- tion members of the Society or who had gained generally acknowledged eminence for their service to mathematics and the profession of mathemati- cian. (By ‘mathematics’ and ‘mathematician’ I am coveringthe gamut of the mathematical sciences. I found that, in nearly all cases, statisticians and mathematical physicists and the rest were pleased to be known generally as mathematicians.) Of those with whom I taped my interviews, five have since died: Oliver Lancaster, whose interview was sadly too late in his life to be useful for me, Fenton Pillow, Bernhard Neu- mann, Ren Potts and George Szekeres. -
Fysica, Vriend of Vijand?
Fysica, vriend of vijand? Bachelorscriptie Natuur- en Wiskunde Universiteit Utrecht Lisette Graafland Begeleider: Steven Wepster 20 augustus 2013 Inhoudsopgave 1 Inleiding 4 2 Ontdekking en ontwikkeling van de quaternionen: Hamilton 6 2.1 Hamiltons ontdekking van de Quaternionen . 8 2.2 Conclusie ontdekking Hamilton . 13 2.3 Hamilton en de quaternionen aan het einde van zijn leven . 14 3 Ontwikkeling van de quaternionen: Peter Guthrie Tait en Ja- mes Clerck Maxwell 15 3.1 Peter Guthrie Tait, advocaat van de quaternionen . 15 3.2 Maxwell en zijn Treatise on Electricity and Magnetism . 17 3.3 Conclusie ontwikkeling van de quaternionen door Tait en Maxwell 20 4 De vectoranalyse van Gibbs en Heaviside 21 4.1 Josiah Willard Gibbs . 22 4.2 Oliver Heaviside . 23 4.3 Conclusie opkomst Gibbs-Heaviside systeem . 24 5 Debat tussen de quaternionisten en de aanhangers van het Gibbs-Heaviside systeem 25 5.1 Taits voorwoord in de derde editie van An Elementairy Treatise on Quaternions ............................ 26 5.2 Gibbs' antwoord op Tait: On the R^oleof Quaternions in the Algebra of Vectors .......................... 30 5.3 Taits reactie op Gibbs: The R^oleof Quaternions in the Algebra of Vectors ............................... 36 5.4 Gibbs tweede reactie op Tait: Quaternions and the Ausdehnungs- lehre .................................. 38 5.5 Taits reactie op Quaternions and the Ausdehungslehre . 40 5.6 Bijdragen aan het debat van andere wetenschappers . 41 5.6.1 Alexander McAulay, een enthousiaste quaternionist . 41 5.6.2 Alexander Macfarlane, een combinateur van het beste . 42 5.6.3 Cargill Gilston Knott, een wijze quaternionist . 44 5.6.4 Arthur Cayley, een conservatieve deelnemer . -
Da Força Ao Tensor: Evolução Do Conceito Físico E Da Representação Matemática Do Campo Eletromagnético
DA FORÇA AO TENSOR: EVOLUÇÃO DO CONCEITO FÍSICO E DA REPRESENTAÇÃO MATEMÁTICA DO CAMPO ELETROMAGNÉTICO Cibelle Celestino Silva Orientador: Prof. Dr. Roberto de Andrade Martins Tese apresentada ao Instituto de Física Gleb Wataghin - UNICAMP para obtenção do grau de Doutor em Ciências. Campinas, 11 de outubro de 2002. Para Francisca e Francisco, minhas raízes e frutos. iii AGRADECIMENTOS Ao Professor Roberto Martins pela confiança, dedicação, paciência, estímulo e por tudo que tenho aprendido nesses anos de convivência. Ao Luciano pelo seu amor e companheirismo. À minha família por todo o amor e alegria. A todos os amigos e colegas que de alguma maneira ou outra estiveram ao meu lado nesta caminhada. E finalmente agradeço à FAPESP pelo apoio financeiro. iv Resumo: O objetivo deste trabalho é estudar o desenvolvimento paralelo dos conceitos físicos e do formalismo matemático usado para descrever o conhecimento eletromagnético no século XIX e início do século XX. Este estudo considera questões como os modelos de éter usados para descrever o campo eletromagnético, a simetria associada com os campos, as dimensões das grandezas físicas. Estudamos também a escolha entre o cálculo de quatérnions e vetores e o desenvolvimento do formalismo quadridimensional como a melhor maneira de descrever os fenômenos eletromagnéticos. Abstract: The aim of this work is to study the parallel development of the physical concepts and the mathematical formalism used to describe the electromagnetic knowledge in nineteenth and early twentieth century. This study takes into account issues such as the ether models used to describe the electromagnetic field, the symmetries associated with the fields, the dimensions of physical quantities. -
Robust Tracking of Dynamic Targets with Aerial Vehicles Using Quaternion-Based Techniques Hernan Abaunza Gonzalez
Robust tracking of dynamic targets with aerial vehicles using quaternion-based techniques Hernan Abaunza Gonzalez To cite this version: Hernan Abaunza Gonzalez. Robust tracking of dynamic targets with aerial vehicles using quaternion- based techniques. Automatic. Université de Technologie de Compiègne, 2019. English. NNT : 2019COMP2480. tel-02155857 HAL Id: tel-02155857 https://tel.archives-ouvertes.fr/tel-02155857 Submitted on 14 Jun 2019 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. Par Hernán ABAUNZA GONZALEZ Robust tracking of dynamic targets with aerial vehicles using quaternion-based techniques Thèse présentée pour l’obtention du grade de Docteur de l’UTC Soutenue le 26 avril 2019 Spécialité : Automatique et Robotique : Unité de recherche Heudyasic (UMR-7253) D2480 SORBONNE UNIVERSITES, UNIVERSITE DE TECHNOLOGIE DE COMPIEGNE HEUDIASYC, UMR 7253 CNRS Ecole Doctorale “Sciences Pour l’Ingénieur” ROBUST TRACKING OF DYNAMIC TARGETS WITH AERIAL VEHICLES USING QUATERNION‐BASED TECHNIQUES T H E S I S presented by Hernan ABAUNZA GONZALEZ´ to obtain -
Hutchins School Magazine, №47, June 1931
VOL. XII., No. S. I Whony set up and printed in Australia by "The Critic" Pty. Ltd., 125 Collins Street ••••• Hobart iJuut. 1931 I;nburt. WUli. THE i!;utr4tu!i ~r~nnl VOL. XII. JUNE, 1931. No.5. Editorial. -§- HE most reliable economists and financiers in Australia T are agreed that the present financial crisis is due chiefly to the fact that Australia has for many years past been bor rowing from abroad, principally from England, and has been spending this borrowed money in ways that have not been productive of much revenue. During the war, the country entered on a period of extravagant borrowing and spending. A loan would be raised, and subscriptions called for in Aus tralia and elsewhere, and when the repayment of that loan was due, the simple expedient was adopted of raising another loan to pay for the previous one. And so it has gone on. The extravagance of Governments, amounting practically to dishonesty, had its effects on prices and wages which con tinued to rise to absurd heights. Governments and unions boasted much of our Australian standard of living as if it were something to be proud of, but it was built on false foundations and was bound to collapse. The foolish attempt to make Australia a manufacturing country before its primary industries were properly developed, took the form of a complicated system of tariffs and bounties, which bolstered up some industries at the expense of others, and which assisted one State to the detriment of others, in stead of allowing industries to develop naturally as the demand for them arose. -
HPS: Annual Report 2002-03
Contents The Department Introduction .............................................................................................................................. 2 Staff and affiliates .................................................................................................................... 3 Visitors and students ................................................................................................................ 4 Comings and goings................................................................................................................. 5 Roles and responsibilities......................................................................................................... 6 Prizes, projects and honours..................................................................................................... 7 Seminars and special lectures...................................................................................................8 Students Student statistics..................................................................................................................... 10 Part II dissertation titles ......................................................................................................... 11 Part II primary sources essay titles......................................................................................... 12 MPhil essay and dissertation titles ......................................................................................... 15 PhD theses............................................................................................................................. -
Quaternion Equations and Quaternion Polynomial Matrices
Quaternion Equations and Quaternion Polynomial Matrices by Liji Huang A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of MASTER OF SCIENCE Department of Mathematics University of Manitoba Winnipeg Copyright © 2012 by Liji Huang Contents Abstract i Acknowledgments iii 1. Introduction 1 1.1. History of Quaternions . 1 1.2. Background for Quaternion Equations . 5 1.3. Background for Quaternion Polynomial Matrices . 8 2. Quaternion Equations 9 2.1. Quaternions and Equivalence Classes . 9 2.2. General Quaternion Quadratic Equations . 15 2.3. Special Quaternion Equations and problems . 20 2.3.1. A quadratic equation . 20 2.3.2. Another quadratic equation . 36 2.3.3. A homogeneous quadratic equation . 40 2.3.4. A pair of quaternion equations . 48 2.3.5. Another pair of quaternion equations . 52 2.3.6. A least norm problem . 55 2.3.7. Another least norm problem . 58 3. Quaternion Polynomial Matrices 61 3.1. Generalized Inverse . 61 3.2. Leverrier-Faddeev Method . 74 3.3. Generalized Inversion by Interpolation . 82 A. Maple Codes 89 A.1. Overview . 89 A.2. The Codes . 90 Bibliography 119 Abstract In this thesis, we mainly consider finding formula solutions to quaternion equations and calculating the generalized inverses of quaternion polynomial matrices. In the first chapter, we give a condensed review of the history of quaternions. The topic of the second chapter is solving quadratic and other quaternion equations. Many mathematicians have worked on solving various quaternion equations. How- ever, due to the complex nature of quaternions, the current "best" non-numerical result is a special case of quadratic quaternion equation presented in (Au-Yeung, 2003), which pales in comparison to what is known about equations in complex numbers. -
Phil Howlett
A famous packing problem|if you are not smart enough to count the dots then use applied probability Phil. Howlett E-mail: [email protected] (with some help from Peter Pudney) Scheduling and Control Group (SCG), University of South Australia. 1 Three people that motivated my talk Phil Pollett (the reason we are all here today at Uluru), Charles Pearce (who helped me to a deeper understanding of applied probability) and Jon Borwein (a legend who inspired my interest in convex analysis). 2 Problem description Problem 1 Let N 2 N be a natural number. For each subset S ⊆ N write jSj to denote the number of elements in S. Consider all subsets S ⊆ f1; 2;:::;Ng that contain no3-term arithmetic progression in the form fp − q; p; p + qg. What is the largest possible value for jSj? 2 This problem has not been solved. We will write ν(N) to denote the size of the largest subset of f1; 2;:::;Ng that contains no3-term arithmetic progression. 3 George Szekeres (1911{2005) George Szekeres received his degree in chemistry at the Technical University of Budapest. He moved to Shanghai prior to WWII to escape Nazi persecution of the Jews and later to Australia where he is now recognised as one of our greatest mathematicians. 4 The Szekeres conjecture r Conjecture 1 Let Nr = (3 + 1)=2 where r 2 N. The largest subset Sr ⊆ f1; 2;:::;Nrg which contains no three-term arithmetic progres- r sion has size jSrj = 2 . 2 This conjecture is false but when you look at the supporting evidence you may well be convinced (as I was at first) that it is true. -
NZMS Newsletter
THE NEW ZEALAND MATHEMATICAL SOCIETY (INC.) NEWSLETTER CONTENTS NZMS Council and Officers; Newsletter contributors Editorial NZMS Notices Local News New Colleague Mathematics research graduates; and sculptures Book Review Conferences Visitors Centrefold: John Turner Notices, Grantee reports, Positions available Crossword No 46 NUMBER 64 August 1995 ISSN 0110-0025 PUBLISHER'S NOTICE The Newsletter is the official organ of the New Zealand Mathematical Society Inc. This issue was assembled and printed at Massey University. The official address of the Society is: The New Zealand Mathematical Society, c/- The Royal Society of New Zealand, P O Box 598, Wellington, New Zealand. However, correspondence should normally be sent directly to the Secretary: Dr Margaret Morton, Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand. NZMS COUNCIL AND OFFICERS President Prof Marston Conder (University of Auckland) Incoming Vice President Prof Douglas Bridges (University of Waikato) Secretary Dr Margaret Morton (University of Auckland) Treasurer Dr Mark McGuiness (Victoria University) Councillors Dr Robert Chan (University of Auckland), to 1995 Prof Michael Hendy (Massey University), to 1995 Dr Rick Beatson (Canterbury University), to 1996 Assoc-Prof Ernie Kalnins (Waikato University), to 1996 Dr Mark McGuinness (Victoria University), to 1996 Dr Mick Roberts (AgResearch) to 1997 Dr Dennis McCaughan (University of Otago) to 1997 Membership Secretary Dr John Shanks (University of Otago) Newsletter Editor Prof Michael Hendy -
Butler S., Et Al. (Eds.) Connections in Discrete Mathematics (CUP
Connections in Discrete Mathematics Discrete mathematics has been rising in prominence in the past fifty years, both asa tool with practical applications and as a source of new and interesting mathematics. The topics in discrete mathematics have become so well developed that it is easy to forget that common threads connect the different areas, and it is through discovering and using these connections that progress is often made. For more than fifty years, Ron Graham has been able to illuminate some ofthese connections and has helped bring the field of discrete mathematics to where it is today. To celebrate his contribution, this volume brings together many of the best researchers working in discrete mathematics, including Fan Chung, Erik Demaine, Persi Diaconis, Peter Frankl, Al Hales, Jeffrey Lagarias, Allen Knutson, Janos Pach, Carl Pomerance, Neil Sloane, and of course Ron Graham himself. steve butler is the Barbara J Janson Professor of Mathematics at Iowa State University. His research interests include spectral graph theory, enumerative combinatorics, mathematics of juggling, and discrete geometry. He is the coeditor of The Mathematics of Paul Erdos˝ . joshua cooper is Professor of Mathematics at the University of South Carolina. He currently serves on the editorial board of Involve. His research interests include spectral hypergraph theory, linear and multilinear algebra, probabilistic combinatorics, quasirandomness, combinatorial number theory, and computational complexity. glenn hurlbert is Professor and Chair of the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. His research interests include universal cycles, extremal set theory, combinatorial optimization, combinatorial bijections, and mathematical education, and he is recognized as a leader in the field of graph pebbling. -
Australia's Fertility Transition
AUSTRALIA’S FERTILITY TRANSITION A STUDY OF 19TH-CENTURY TASMANIA AUSTRALIA’S FERTILITY TRANSITION A STUDY OF 19TH-CENTURY TASMANIA HELEN MOYLE Published by ANU Press The Australian National University Acton ACT 2601, Australia Email: [email protected] Available to download for free at press.anu.edu.au ISBN (print): 9781760463366 ISBN (online): 9781760463373 WorldCat (print): 1137826491 WorldCat (online): 1137860712 DOI: 10.22459/AFT.2020 This title is published under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International (CC BY-NC-ND 4.0). The full licence terms are available at creativecommons.org/licenses/by-nc-nd/4.0/legalcode This publication was awarded a College of Arts and Social Sciences PhD Publication Prize in 2019. The prize contributes to the cost of professional copyediting. Cover design and layout by ANU Press. Cover photograph: NS5098/1/88, ‘Woman with two children and an infant sitting outside’, Tasmanian Archives. This edition © 2020 ANU Press Contents Abbreviations . vii Acknowledgements . ix List of figures . xi List of tables . xv 1 . Introduction: Theories of the historical fertility decline . 1 2 . The Australian fertility decline . 27 3 . Tasmania in the 19th and early 20th centuries . 41 4 . Research design and data sources . 63 5 . Characteristics of the Tasmanian marriage cohorts . .. 83 6 . Fertility patterns of the couples in the marriage cohorts . 111 7 . How, when and why did fertility decline? . 145 8 . Why did fertility fall in Tasmania during this period? Qualitative insights . 185 9. Conclusion . 221 Appendix A: Appendix tables . 231 Appendix B: Data sources used for family reconstitution . 253 Appendix C: Local history museums .