Interview with Herbert B. Keller
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The Cell Method: a Purely Algebraic Computational Method in Physics and Engineering Copyright © Momentum Press®, LLC, 2014
THE CELL METHOD THE CELL METHOD A PURELY ALGEBRAIC COMPUTATIONAL METHOD IN PHYSICS AND ENGINEERING ELENA FERRETTI MOMENTUM PRESS, LLC, NEW YORK The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering Copyright © Momentum Press®, LLC, 2014. 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—electronic, mechanical, photocopy, recording, or any other—except for brief quotations, not to exceed 400 words, without the prior permission of the publisher. First published by Momentum Press®, LLC 222 East 46th Street, New York, NY 10017 www.momentumpress.net ISBN-13: 978-1-60650-604-2 (hardcover) ISBN-10: 1-60650-604-8 (hardcover) ISBN-13: 978-1-60650-606-6 (e-book) ISBN-10: 1-60650-606-4 (e-book) DOI: 10.5643/9781606506066 Cover design by Jonathan Pennell Interior design by Exeter Premedia Services Private Ltd. Chennai, India 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America CONTENTS ACKNOWLEDGMENTS vii PREFACE ix 1 A COMpaRISON BETWEEN ALGEBRAIC AND DIFFERENTIAL FORMULATIONS UNDER THE GEOMETRICAL AND TOPOLOGICAL VIEWPOINTS 1 1.1 Relationship Between How to Compute Limits and Numerical Formulations in Computational Physics 2 1.1.1 Some Basics of Calculus 2 1.1.2 The e − d Definition of a Limit 4 1.1.3 A Discussion on the Cancelation Rule for Limits 8 1.2 Field and Global Variables 15 1.3 Set Functions in Physics 20 1.4 A Comparison Between the Cell Method and the Discrete Methods 21 2 ALGEBRA AND THE GEOMETRIC INTERPRETATION -
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 -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Sinai Awarded 2014 Abel Prize
Sinai Awarded 2014 Abel Prize The Norwegian Academy of Sci- Sinai’s first remarkable contribution, inspired ence and Letters has awarded by Kolmogorov, was to develop an invariant of the Abel Prize for 2014 to dynamical systems. This invariant has become Yakov Sinai of Princeton Uni- known as the Kolmogorov-Sinai entropy, and it versity and the Landau Insti- has become a central notion for studying the com- tute for Theoretical Physics plexity of a system through a measure-theoretical of the Russian Academy of description of its trajectories. It has led to very Sciences “for his fundamen- important advances in the classification of dynami- tal contributions to dynamical cal systems. systems, ergodic theory, and Sinai has been at the forefront of ergodic theory. mathematical physics.” The He proved the first ergodicity theorems for scat- Photo courtesy of Princeton University Mathematics Department. Abel Prize recognizes contribu- tering billiards in the style of Boltzmann, work tions of extraordinary depth he continued with Bunimovich and Chernov. He Yakov Sinai and influence in the mathemat- constructed Markov partitions for systems defined ical sciences and has been awarded annually since by iterations of Anosov diffeomorphisms, which 2003. The prize carries a cash award of approxi- led to a series of outstanding works showing the mately US$1 million. Sinai received the Abel Prize power of symbolic dynamics to describe various at a ceremony in Oslo, Norway, on May 20, 2014. classes of mixing systems. With Ruelle and Bowen, Sinai discovered the Citation notion of SRB measures: a rather general and Ever since the time of Newton, differential equa- distinguished invariant measure for dissipative tions have been used by mathematicians, scientists, systems with chaotic behavior. -
At a Meeting of the Faculty of Arts and Sciences On
At a meeting of the FACULTY OF ARTS AND SCIENCES on February 12, 2008, the following tribute to the life and service of the late George Francis Carrier was spread upon the permanent records of the Faculty. GEORGE FRANCIS CARRIER BORN: May 4, 1918 DIED: March 8, 2002 George Francis Carrier, T. Jefferson Coolidge Professor of Applied Mathematics, Emeritus, at Harvard University, one of the world’s leading applied mathematicians, died of esophageal cancer in his 84th year, on March 8, 2002. He was born in Millinocket, Maine, on May 4, 1918. His father was a chemical engineer and manager of the Great Northern paper mill in Millinocket. As a teenager, George was a guide in his beloved Maine woods; he worked summer jobs at the mill without his father’s knowledge. Following in his father’s footsteps, he attended Cornell University, where he received a Masters of Engineering degree in 1939 and Ph.D. in 1944, working with Professor Norman Goodier. An accomplished clarinet and ocarina player, he organized a jazz band at Cornell and worked as a houseman at a local pool hall. In graduate school at Cornell, George contracted tuberculosis and spent a year in a sanitarium breathing fresh air and studying books on advanced mathematics. He then returned to graduate school where he taught courses in mechanical drawing and mechanisms and the first advanced course in applied mathematics for engineers at Cornell. Two students in the latter course, Julian Cole and Ivar Stackgold, said they first heard about asymptotics and perturbations analysis in that course and believed that experience had shaped their careers (they both became distinguished professors of applied mathematics). -
Professor Richard Courant (1888 – 1972)
Professor Richard Courant (1888 – 1972) From Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Richard_Courant Field: Mathematics Institutions: University of Göttingen University of Münster University of Cambridge New York University Alma mater: University of Göttingen Doctoral advisor: David Hilbert Doctoral students: William Feller Martin Kruskal Joseph Keller Kurt Friedrichs Hans Lewy Franz Rellich Known for: Courant number Courant minimax principle Courant–Friedrichs–Lewy condition Biography: Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there. As he found the courses not demanding enough, he continued his studies in Zürich and Göttingen. Courant eventually became David Hilbert's assistant in Göttingen and obtained his doctorate there in 1910. He had to fight in World War I, but he was wounded and dismissed from the military service shortly after enlisting. After the war, in 1919, he married Nerina (Nina) Runge, a daughter of the Göttingen professor for Applied Mathematics, Carl Runge. Richard continued his research in Göttingen, with a two-year period as professor in Münster. There he founded the Mathematical Institute, which he headed as director from 1928 until 1933. Courant left Germany in 1933, earlier than many of his colleagues. While he was classified as a Jew by the Nazis, his having served as a front-line soldier exempted him from losing his position for this particular reason at the time; however, his public membership in the social-democratic left was a reason for dismissal to which no such exemption applied.[1] After one year in Cambridge, Courant went to New York City where he became a professor at New York University in 1936. -
Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization
in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homog´en´eisationdans les ´equationsaux d´eriv´eespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Coll`egede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Coll`egede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. -
Arxiv:2105.10149V2 [Math.HO] 27 May 2021
Extended English version of the paper / Versión extendida en inglés del artículo 1 La Gaceta de la RSME, Vol. 23 (2020), Núm. 2, Págs. 243–261 Remembering Louis Nirenberg and his mathematics Juan Luis Vázquez, Real Academia de Ciencias, Spain Abstract. The article is dedicated to recalling the life and mathematics of Louis Nirenberg, a distinguished Canadian mathematician who recently died in New York, where he lived. An emblematic figure of analysis and partial differential equations in the last century, he was awarded the Abel Prize in 2015. From his watchtower at the Courant Institute in New York, he was for many years a global teacher and master. He was a good friend of Spain. arXiv:2105.10149v2 [math.HO] 27 May 2021 One of the wonders of mathematics is you go somewhere in the world and you meet other mathematicians, and it is like one big family. This large family is a wonderful joy.1 1. Introduction This article is dedicated to remembering the life and work of the prestigious Canadian mathematician Louis Nirenberg, born in Hamilton, Ontario, in 1925, who died in New York on January 26, 2020, at the age of 94. Professor for much of his life at the mythical Courant Institute of New York University, he was considered one of the best mathematical analysts of the 20th century, a specialist in the analysis of partial differential equations (PDEs for short). 1From an interview with Louis Nirenberg appeared in Notices of the AMS, 2002, [43] 2 Louis Nirenberg When the news of his death was received, it was a very sad moment for many mathematicians, but it was also the opportunity of reviewing an exemplary life and underlining some of its landmarks. -
Mathematics to the Rescue Retiring Presidential Address
morawetz.qxp 11/13/98 11:20 AM Page 9 Mathematics to the Rescue (Retiring Presidential Address) Cathleen Synge Morawetz should like to dedicate this lecture to my tial equation by a simple difference scheme and get- teacher, Kurt Otto Friedrichs. While many ting nonsense because the scheme is unstable. people helped me on my way professionally, The answers blow up. The white knights—Courant, Friedrichs played the central role, first as my Friedrichs, and Lewy (CFL)—would then step in thesis advisor and then by feeding me early and show them that mathematics could cure the Iresearch problems. He was my friend and col- problem. That was, however, not the way. But first, league for almost forty years. As is true among what is the CFL condition? It says [1] that for sta- friends, he was sometimes exasperated with me (I ble schemes the step size in time is limited by the was too disorderly), and sometimes he exasperated step size in space. The simplest example is the dif- me (he was just too orderly). However, I think we ference scheme for a solution of the wave equa- appreciated each other’s virtues, and I learned a tion great deal from him. utt uxx =0 Sometimes we had quite different points of − view. In his old age he agreed to be interviewed, on a grid that looks like the one in Figure 1. Sec- at Jack Schwartz’s request, for archival purposes. ond derivatives are replaced by second-order I was the interviewer. At one point I asked him differences with t = x. -
Louis Nirenberg
Obituary Louis Nirenberg (1925–2020) Mathematician who transformed the study of partial differential equations. fter the Second World War, PDE in Novosibirsk, which helped to redefine mathe matics in the United States the relationship between the Soviet Union flourished owing to a convergence of and the United States. There, he forged close interests. Mathematicians had shown friendships in an environment he compared to their worth to military and indus- a voyage at sea. A later geopolitically signifi- Atry patrons, who underwrote far-reaching cant trip took him to China toward the end of empires of theories and people, including the the Cultural Revolution. After being assigned consummate problem-solver Louis Nirenberg. a PhD thesis in Italian as the subject for a term One of the world’s most cited and paper during his graduate studies, he devel- productive mathematicians, Nirenberg was oped a lifelong affinity for Italy. also among the most collaborative. His work Nirenberg was known for using methods in continued to make waves until he was well their most fruitful generality. “I have made a into his eighties, and reshaped how mathe- living off the maximum principle,” he quipped, maticians understand and study dynamical referring to a fundamental technique for systems, from cells to markets. Winning the establishing inequalities in PDE. He demon- 2015 Abel Prize (shared with John Nash, made strated its versatile potential to researchers in NEW YORK UNIVERSITY ARCHIVES NEW YORK famous by the 2001 film A Beautiful Mind) was many fields. As a young man, he had worried just a bookend to a fêted career. -
2004 Steele Prizes
2004 Steele Prizes The 2004 Leroy P. Steele Prizes were awarded at the The 2004 Steele Prizes were awarded to JOHN W. 110th Annual Meeting of the AMS in Phoenix in MILNOR for Mathematical Exposition, to LAWRENCE C. January 2004. EVANS and NICOLAI V. KRYLOV for a Seminal Contri- The Steele Prizes were established in 1970 in honor bution to Research, and to CATHLEEN SYNGE MORAWETZ of George David Birkhoff, William Fogg Osgood, and for Lifetime Achievement. The text that follows William Caspar Graustein. Osgood was president of presents, for each awardee, the selection commit- the AMS during 1905–06, and Birkhoff served in that tee’s citation, a brief biographical sketch, and the capacity during 1925–26. The prizes are endowed awardee’s response upon receiving the prize. under the terms of a bequest from Leroy P. Steele. Up to three prizes are awarded each year in the follow- Mathematical Exposition: John W. Milnor ing categories: (1) Lifetime Achievement: for the Citation cumulative influence of the total mathematical work The Leroy P. Steele Prize for Mathematical Exposi- of the recipient, high level of research over a period tion is awarded to John W. Milnor in recognition of time, particular influence on the development of of a lifetime of expository contributions ranging a field, and influence on mathematics through Ph.D. across a wide spectrum of disciplines including students; (2) Mathematical Exposition: for a book topology, symmetric bilinear forms, characteristic or substantial survey or expository-research paper; classes, Morse theory, game theory, algebraic K- (3) Seminal Contribution to Research (limited for theory, iterated rational maps…and the list goes on. -
Mathematics Newsletter Volume 21. No4, March 2012
MATHEMATICS NEWSLETTER EDITORIAL BOARD S. Ponnusamy (Chief Editor) Department of Mathematics Indian Institute of Technology Madras Chennai - 600 036, Tamilnadu, India Phone : +91-44-2257 4615 (office) +91-44-2257 6615, 2257 0298 (home) [email protected] http://mat.iitm.ac.in/home/samy/public_html/index.html S. D. Adhikari G. K. Srinivasan Harish-Chandra Research Institute Department of Mathematics, (Former Mehta Research Institute ) Indian Institute of Technology Chhatnag Road, Jhusi Bombay Allahabad 211 019, India Powai, Mumbai 400076, India [email protected] [email protected] C. S. Aravinda B. Sury, TIFR Centre for Applicable Mathematics Stat-Math Unit, Sharadanagar, Indian Statistical Institute, Chikkabommasandra 8th Mile Mysore Road, Post Bag No. 6503 Bangalore 560059, India. Bangalore - 560 065 [email protected], [email protected] [email protected] M. Krishna G. P. Youvaraj The Institute of Mathematical Sciences Ramanujan Institute CIT Campus, Taramani for Advanced Study in Mathematics Chennai-600 113, India University of Madras, Chepauk, [email protected] Chennai-600 005, India [email protected] Stefan Banach (1892–1945) R. Anantharaman SUNY/College, Old Westbury, NY 11568 E-mail: rajan−[email protected] To the memory of Jong P. Lee Abstract. Stefan Banach ranks quite high among the founders and developers of Functional Analysis. We give a brief summary of his life, work and methods. Introduction (equivalent of middle/high school) there. Even as a student Stefan revealed his talent in mathematics. He passed the high Stefan Banach and his school in Poland were (among) the school in 1910 but not with high honors [M].