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What Are Lyapunov Exponents, and Why Are They Interesting?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016. -
Twenty Female Mathematicians Hollis Williams
Twenty Female Mathematicians Hollis Williams Acknowledgements The author would like to thank Alba Carballo González for support and encouragement. 1 Table of Contents Sofia Kovalevskaya ................................................................................................................................. 4 Emmy Noether ..................................................................................................................................... 16 Mary Cartwright ................................................................................................................................... 26 Julia Robinson ....................................................................................................................................... 36 Olga Ladyzhenskaya ............................................................................................................................. 46 Yvonne Choquet-Bruhat ....................................................................................................................... 56 Olga Oleinik .......................................................................................................................................... 67 Charlotte Fischer .................................................................................................................................. 77 Karen Uhlenbeck .................................................................................................................................. 87 Krystyna Kuperberg ............................................................................................................................. -
Continuity of the Lyapunov Exponents of Linear Cocycles
Continuity of the Lyapunov Exponents of Linear Cocycles Publicações Matemáticas Continuity of the Lyapunov Exponents of Linear Cocycles Pedro Duarte Universidade de Lisboa Silvius Klein PUC-Rio 31o Colóquio Brasileiro de Matemática Copyright 2017 by Pedro Duarte e Silvius Klein Direitos reservados, 2017 pela Associação Instituto Nacional de Matemática Pura e Aplicada - IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz 31o Colóquio Brasileiro de Matemática Álgebra e Geometria no Cálculo de Estrutura Molecular - C. Lavor, N. Maculan, M. Souza e R. Alves Continuity of the Lyapunov Exponents of Linear Cocycles - Pedro Duarte e Silvius Klein Estimativas de Área, Raio e Curvatura para H-superfícies em Variedades Riemannianas de Dimensão Três - William H. Meeks III e Álvaro K. Ramos Introdução aos Escoamentos Compressíveis - José da Rocha Miranda Pontes, Norberto Mangiavacchi e Gustavo Rabello dos Anjos Introdução Matemática à Dinâmica de Fluídos Geofísicos - Breno Raphaldini, Carlos F.M. Raupp e Pedro Leite da Silva Dias Limit Cycles, Abelian Integral and Hilbert’s Sixteenth Problem - Marco Uribe e Hossein Movasati Regularization by Noise in Ordinary and Partial Differential Equations - Christian Olivera Topological Methods in the Quest for Periodic Orbits - Joa Weber Uma Breve Introdução à Matemática da Mecânica Quântica - Artur O. Lopes Distribuição: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ e-mail: [email protected] http://www.impa.br ISBN: 978-85-244-0433-7 i \notes" | 2017/5/29 | 19:08 | page i | #1 i i i Contents Preface 1 1 Linear Cocycles 7 1.1 The definition and examples of ergodic systems . -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Alwyn C. Scott
the frontiers collection the frontiers collection Series Editors: A.C. Elitzur M.P. Silverman J. Tuszynski R. Vaas H.D. Zeh The books in this collection are devoted to challenging and open problems at the forefront of modern science, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science. Furthermore, it is intended to encourage active scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, consciousness and complex systems – the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. Other Recent Titles The Thermodynamic Machinery of Life By M. Kurzynski The Emerging Physics of Consciousness Edited by J. A. Tuszynski Weak Links Stabilizers of Complex Systems from Proteins to Social Networks By P. Csermely Quantum Mechanics at the Crossroads New Perspectives from History, Philosophy and Physics Edited by J. Evans, A.S. Thorndike Particle Metaphysics A Critical Account of Subatomic Reality By B. Falkenburg The Physical Basis of the Direction of Time By H.D. Zeh Asymmetry: The Foundation of Information By S.J. Muller Mindful Universe Quantum Mechanics and the Participating Observer By H. Stapp Decoherence and the Quantum-to-Classical Transition By M. Schlosshauer For a complete list of titles in The Frontiers Collection, see back of book Alwyn C. -
A Fairly Complete History and Tour of Aynho Village – Updated January 2017 Aynho Is a Two-Part Name
A Fairly Complete History and Tour of Aynho Village – updated January 2017 Aynho is a two-part name - ‘Ayn’ is either a corruption of a Saxon personal name, or more likely the Saxon word for a spring or stream. The ‘Hoh’ is a Saxon word for a promontory/projecting ridge of land standing on a plain as Aynho does. The earliest mention (in the Domesday Book) of an owner of the manor of Aynho is Asgar - a Danish thane (knight). He was standard bearer for Edward the Confessor who reigned from 1042 to 1066. (Edward was born at Islip about fifteen miles south east of Aynho, so he probably knew Asgar). The entry showed 3¼ hides (about 400 acres altogether), land for 8 ploughs, a mill and 20 acres of meadow. Why was Aynho so relatively important in the mid-ten hundreds? Probably because of its location high up overlooking the whole Cherwell valley. There were very few significant houses in existence within a radius of twenty miles at that time, and it is believed that Aynho had a substantial wooden Saxon manor house then. For example Oxford Castle was not built until 1073, Banbury Castle 1135, Broughton Castle 1300, Rousham House 1635 and Upton House 1695. The first proper Oxford College, University College, wasn’t founded until1249. Apart from Aynho north of Oxford only Sulgrave Manor is recorded as having an Anglo-Saxon Manor House around the late 9th century. William the Conqueror gave the village to one of his barons, Geoffrey de Mandeville, for helping him win the Battle of Hastings in 1066. -
Interview with Mathmedia
i “Taiwan-I-12-transcript2” — 2013/11/7 — 15:23 — page 1 — #1 i i i Interview with MathMedia Interviewee: Frans Oort Interviewer: Ching-Li Chai Venue: Institute of Mathematics Academia Sinica Date: December 3rd, 2012 Ching-Li Chai(CHAI): Good morning, Frans. Doing this interview is a pleasure. I didn’t expect this. Frans Oort (OORT): The pleasure is mine. It’s a surprise for both of us. CHAI: There is a ritual that we first invite people to talk about their formative years which is always interesting because people’s backgrounds are all different. Would you like to say something about your formative years or when you are younger and decided to go to mathematics? Some people believe that mathematicians are born. The beginning1 OORT: Let me first say something about life as a mathematician, and later I will tell some stories about my personal life. For me it is a surprise you can be together with other people, with different cultural backgrounds, in history, in bringing up, with parents on the one hand, while on the other hand you can be so close to other people. With other mathematicians all of a sudden you understand each other on a much higher level than you ever thought possible. This is one of the most exciting things I have seen in my life. Certainly this is true with my collaborator here; with Ching-Li I have so much in common, and we understand each other. I think mathematically we have a deep contact and I am very grateful for that. -
Notices of the American Mathematical Society
• ISSN 0002-9920 March 2003 Volume 50, Number 3 Disks That Are Double Spiral Staircases page 327 The RieITlann Hypothesis page 341 San Francisco Meeting page 423 Primitive curve painting (see page 356) Education is no longer just about classrooms and labs. With the growing diversity and complexity of educational programs, you need a software system that lets you efficiently deliver effective learning tools to literally, the world. Maple® now offers you a choice to address the reality of today's mathematics education. Maple® 8 - the standard Perfect for students in mathematics, sciences, and engineering. Maple® 8 offers all the power, flexibility, and resources your technical students need to manage even the most complex mathematical concepts. MapleNET™ -- online education ,.u A complete standards-based solution for authoring, nv3a~ _r.~ .::..,-;.-:.- delivering, and managing interactive learning modules \~.:...br *'r¥'''' S\l!t"AaITI(!\pU;; ,"", <If through browsers. Derived from the legendary Maple® .Att~~ .. <:t~~::,/, engine, MapleNefM is the only comprehensive solution "f'I!hlislJer~l!'Ct"\ :5 -~~~~~:--r---, for distance education in mathematics. Give your institution and your students cornpetitive edge. For a FREE 3D-day Maple® 8 Trial CD for Windows®, or to register for a FREE MapleNefM Online Seminar call 1/800 R67.6583 or e-mail [email protected]. ADVANCING MATHEMATICS WWW.MAPLESOFT.COM I [email protected]\I I WWW.MAPLEAPPS.COM I NORTH AMERICAN SALES 1/800 267. 6583 © 2003 Woter1oo Ma')Ir~ Inc Maple IS (J y<?glsterc() crademork of Woterloo Maple he Mar)leNet so troc1ema'k of Woter1oc' fV'lop'e Inr PII other trcde,nork$ (ye property o~ their respective ('wners Generic Polynomials Constructive Aspects of the Inverse Galois Problem Christian U. -
ONE HUNDRED YEARS of COMPLEX DYNAMICS the Subject of Complex Dynamics, That Is, the Behaviour of Orbits of Holomorphic Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Liverpool Repository ONE HUNDRED YEARS OF COMPLEX DYNAMICS MARY REES The subject of Complex Dynamics, that is, the behaviour of orbits of holomorphic functions, emerged in the papers produced, independently, by Fatou and Julia, almost 100 years ago. Although the subject of Dynami- cal Systems did not then have a name, the dynamical properties found for holomorphic systems, even in these early researches, were so striking, so unusually comprehensive, and yet so varied, that these systems still attract widespread fascination, 100 years later. The first distinctive feature of iter- ation of a single holomorphic map f is the partition of either the complex plane or the Riemann sphere into two sets which are totally invariant under f: the Julia set | closed, nonempty, perfect, with dynamics which might loosely be called chaotic | and its complement | open, possibly empty, but, if non-empty, then with dynamics which were completely classified by the two pioneering researchers, modulo a few simply stated open questions. Before the subject re-emerged into prominence in the 1980's, the Julia set was alternately called the Fatou set, but Paul Blanchard introduced the idea of calling its complement the Fatou set, and this was immediately universally accepted. Probably the main reason for the remarkable rise in interest in complex dynamics, about thirty-five years ago, was the parallel with the subject of Kleinian groups, and hence with the whole subject of hyperbolic geome- try. A Kleinian group acting on the Riemann sphere is a dynamical system, with the sphere splitting into two disjoint invariant subsets, with the limit set and its complement, the domain of discontinuity, having exactly similar properties to the Julia and Fatou sets. -
Front Matter
Cambridge University Press 978-1-107-12696-1 - Foundations of Ergodic Theory Marcelo Viana and Krerley Oliveira Frontmatter More information CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 151 Editorial Board B. BOLLOBAS,´ W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO Foundations of Ergodic Theory Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory suitable for a variety of one- or two-semester courses. The authors’ clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle, and thermodynamic formalism. The 400 exercises increase in difficulty through the text and test the reader’s understanding of the whole theory. Hints and solutions are provided at the end of the book. Marcelo Viana is Professor of Mathematics at Instituto Nacional de Matematica´ Pura e Aplicada (IMPA), Rio de Janeiro and a leading research expert in ergodic theory and dynamical systems. He has served in several academic organizations, such as the International Mathematical Union (Vice-president 2011–2014), the Brazilian Mathematical Society (President 2013–2015), the Latin American Mathematical Union (Scientific Coordinator, 2001–2008) and the newly founded Mathematical Council of the Americas. -
PUBLICATIONS of the American Mathematical Society up to AMS Individual 20% Member Discount Receive Substantial Discounts on All AMS Published and Co-Published Books!
PUBLICATIONS of the American Mathematical Society up to AMS Individual 20% Member Discount Receive substantial discounts on all AMs published and co-published books! AMS Textbooks Find the right textbook for your course! The AMs publishes many high-quality books for use in the classroom. to view a comprehensive list of our most widely adopted textbooks, please visit www.ams.org/bookstore/textbooks To Prospective Contents Authors 3 Featured Selections If you would like to submit a manuscript to the AMs, please visit www.ams.org/authors 13 Algebra and Algebraic Geometry 15 Analysis 16 Applications 16 Differential Equations Applied Mathematics The AMs book publication program on 17 Discrete Mathematics and Combinatorics applied and interdisciplinary mathematics 18 Geometry and Topology strengthens the connections between white --> mathematics and other disciplines, 19 Logic and Foundations highlighting the areas where mathematics 20 Mathematical Physics is most relevant. These publications help mathematicians understand how 21 number Theory mathematical ideas may benefit other 22 Applied Mathematics sciences, while offering researchers outside of mathematics important tools to advance 24 Memoirs of the AMs their profession. to view all of our applied 25 AMs-Distributed Publications mathematics publications, go to: www.ams.org/bookstore/appliedmath 26 Index 30 ordering Information Order Online | www.ams.org/bookstore fE aturED Selections Elliptic Partial Differential Equations TEXTBOOK TEXTBOOKS 1 Second Edition FROM THE AMS Q I N G H A N Qing Han, University of Notre Dame, IN, and Fanghua Lin, Courant Institute, New York F A N G H U A L I N University, NY TEXTBOOK Elliptic Partial This volume is based on PDE courses given by the authors at the Courant Institute and at the Differential Equations University of Notre Dame, Indiana. -
Uva-DARE (Digital Academic Repository)
UvA-DARE (Digital Academic Repository) Optimality properties of curves over finite fields Zaitsev, A.I. Publication date 2008 Link to publication Citation for published version (APA): Zaitsev, A. I. (2008). Optimality properties of curves over finite fields. Thomas Stieltjes Institute for Mathematics. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:27 Sep 2021 Bibliography [1] Aldo Andreotti. On a theorem of Torelli. Amer.J.Math., 80:801–828, 1958. [2] Juscelino Bezerra, Arnaldo Garcia, and Henning Stichtenoth. An explicit tower of function fields over cubic finite fields and Zink’s lower bound. J. Reine Angew. Math., 589:159–199, 2005. [3] Pierre Deligne. Vari´et´es ab´eliennes ordinaires sur un corps fini. Invent. Math., 8:238–243, 1969.