Mathematical Genealogy of the Union College Department of Mathematics
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MA-302 Advanced Calculus 8
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore Aug-Dec 2002 MA-302 Advanced Calculus 8. Inverse function theorem Carl Gustav Jacob Jacobi† (1804-1851) The exercises 8.1 and 8.2 are only to get practice, their solutions need not be submitted. 8.1. Determine at which points a of the domain of definition G the following maps F have differentiable inverse and give the biggest possible open subset on which F define a diffeomorphism. a). F : R2 → R2 with F(x,y) := (x2 − y2, 2xy) . (In the complex case this is the function z → z2.) b). F : R2 → R2 with F(x,y) := (sin x cosh y,cos x sinh y). (In the complex case this is the function z → sin z.) c). F : R2 → R2 with F(x,y) := (x+ y,x2 + y2) . d). F : R3 → R3 with F(r,ϕ,h) := (r cos ϕ,rsin ϕ,h). (cylindrical coordinates) × 2 2 3 3 e). F : (R+) → R with F(x,y) := (x /y , y /x) . xy f). F : R2 → R2 with F(x,y) := (x2 + y2 ,e ) . 8.2. Show that the following maps F are locally invertible at the given point a, and find the Taylor-expansion F(a) of the inverse map at the point upto order 2. a). F : R3 → R3 with F(x,y,z) := − (x+y+z), xy+xz+yz,−xyz at a =(0, 1, 2) resp. a =(−1, 2, 1). ( Hint : The problem consists in approximating the three zeros of the monic polynomial of degree 3 whose coefficients are close to those of X(X−1)(X−2) resp. -
Der Vierfarbensatz. Geschichte, Topologische Grundlagen Und
Der Vierfarbensatz Geschichte, topologische Grundlagen und Beweisidee von Prof. Dr. RudolfjFritsch Universität München unter Mitarbeit von Gerda Fritsch, Gräfelfing Wissenschaftsverlag Mannheim • Leipzig - Wien • Zürich Mathematische Institut dar Universität München Die Deutsche Bibliothek - CIP-Einheitsaufnahme Fritsch, Rudolf: Der Vierfarbensatz: Geschichte, topologische Grundlagen und Beweisidee / von Rudolf Fritsch. Unter Mitarb. von Gerda Fritsch. - Mannheim; Leipzig; Wien; Zürich: BI-Wiss.-Verl., 1994 ISBN 3-411-15141-2 Gedruckt auf säurefreiem Papier mit neutralem pH-Wert (bibliotheksfest) Alle Rechte, auch die der Übersetzung in fremde Sprachen, vorbehalten. Kein Teil dieses Werkes darf ohne schriftliche Einwilligung des Verlages in irgendeiner Form (Fotokopie, Mikrofilm oder ein anderes Verfahren), auch nicht für Zwecke der Unterrichtsgestaltung, reproduziert oder unter Verwendung elektronischer Systeme verarbeitet, vervielfältigt oder verbreitet werden. © Bibliographisches Institut & F. A. Brockhaus AG, Mannheim 1994 Druck: RK Offsetdruck GmbH, Speyer Bindearbeit: Progressdruck GmbH, Speyer Printed in Germany ISBN 3-411-15141-2 Für Dorothee Veronika Bernhard Inhalt s Verzeichnis Statt eines Vorworts 3 Wie man dieses Buch lesen kann 6 1 Geschichte 7 2 (Topologische) Landkarten 45 2.1 Heuristische Vorüberlegungen 45 2.2 Grenzlinien 48 2.3 Formale Definition 61 Ecken 63 Länder 64 Zusammenhang von Landkarten 66 Reduktion auf Landkarten aus Streckenzügen 67 2.4 Grundlegende Beispiele 70 2.5 Landesgrenzen 75 2.6 Gemeinsame Grenzlinien -
Einstein's Physical Strategy, Energy Conservation, Symmetries, And
Einstein’s Physical Strategy, Energy Conservation, Symmetries, and Stability: “but Grossmann & I believed that the conservation laws were not satisfied” April 12, 2016 J. Brian Pitts Faculty of Philosophy, University of Cambridge [email protected] Abstract Recent work on the history of General Relativity by Renn, Sauer, Janssen et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein’s physical strategy and the particle physics deriva- tions compare? What energy-momentum complex(es) did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work relate to emerging knowledge (1911-14) of the canonical energy-momentum tensor and its translation-induced conservation? After initially using energy-momentum tensors hand-crafted from the gravitational field equa- ′ µ µ ν tions, Einstein used an identity from his assumed linear coordinate covariance x = Mν x to relate it to the canonical tensor. Usually he avoided using matter Euler-Lagrange equations and so was not well positioned to use or reinvent the Herglotz-Mie-Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor’s asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell’s theory (which avoids negative-energies but gets an asymmetric canonical tensor as a result) as on a scalar theory (the Newtonian limit). -
Carl Gustav Jacob Jacobi
CARL GUSTAV JACOB JACOBI Along with Norwegian Niels Abel, German mathematician Carl Gustav Jacob Jacobi (December 10, 1804 – February 18, 1851) was one of the founders of the theory of elliptic functions, which he described in his 1829 work Fundamenta Nova Theoriae Functionum Ellipticarum (New Foundations of the Theory of Elliptic Functions). His achievements in this area drew praise from Joseph-Louis Lagrange, who had spent some 40 years studying elliptic integrals. Jacobi also investigated number theory, mathematical analysis, geometry and differential equations. His work with determinants, in particular the Hamilton-Jacobi theory, a technique of solving a system of partial differential equations by transforming coordinates, is important in the presentation of dynamics and quantum mechanics. V.I. Arnold wrote of the Hamilton-Jacobi method, “… this is the most powerful method known for exact integration.” Born in Potsdam, Jacobi was the son of a prosperous Jewish banker. His older brother Moritz Hermann Jacobi was a well-known physicist and engineer. The latter worked as a leading researcher at the Academy of Sciences in St. Petersburg. During their lifetimes Moritz was the better known of the two for his work on the practical applications of electricity and especially for his discovery in 1838 of galvanoplastics. Also called electrotyping, it is a process something like electroplating for making duplicate plates of relief, or letterpress, printing. Carl was constantly mistaken for his brother or even worse congratulated for having such a distinguished and accomplished brother. To one such compliment he responded with annoyance, “I am not his brother, he is mine.” Carl Jacobi demonstrated great talent for both languages and mathematics from an early age. -
Emil Artin in America
MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 321–330 S 0273-0979(2012)01398-8 Article electronically published on December 18, 2012 CREATING A LIFE: EMIL ARTIN IN AMERICA DELLA DUMBAUGH AND JOACHIM SCHWERMER 1. Introduction In January 1933, Adolf Hitler and the Nazi party assumed control of Germany. On 7 April of that year the Nazis created the notion of “non-Aryan descent”.1 “It was only a question of time”, Richard Brauer would later describe it, “until [Emil] Artin, with his feeling for individual freedom, his sense of justice, his abhorrence of physical violence would leave Germany” [5, p. 28]. By the time Hitler issued the edict on 26 January 1937, which removed any employee married to a Jew from their position as of 1 July 1937,2 Artin had already begun to make plans to leave Germany. Artin had married his former student, Natalie Jasny, in 1929, and, since she had at least one Jewish grandparent, the Nazis classified her as Jewish. On 1 October 1937, Artin and his family arrived in America [19, p. 80]. The surprising combination of a Roman Catholic university and a celebrated American mathematician known for his gnarly personality played a critical role in Artin’s emigration to America. Solomon Lefschetz had just served as AMS president from 1935–1936 when Artin came to his attention: “A few days ago I returned from a meeting of the American Mathematical Society where as President, I was particularly well placed to know what was going on”, Lefschetz wrote to the president of Notre Dame on 12 January 1937, exactly two weeks prior to the announcement of the Hitler edict that would influence Artin directly. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Academic Genealogy of the Oakland University Department Of
Basilios Bessarion Mystras 1436 Guarino da Verona Johannes Argyropoulos 1408 Università di Padova 1444 Academic Genealogy of the Oakland University Vittorino da Feltre Marsilio Ficino Cristoforo Landino Università di Padova 1416 Università di Firenze 1462 Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo Angelo Poliziano Florens Florentius Radwyn Radewyns Geert Gerardus Magnus Groote Università di Mantova 1433 Università di Mantova Università di Firenze 1477 Constantinople 1433 DepartmentThe Mathematics Genealogy Project of is a serviceMathematics of North Dakota State University and and the American Statistics Mathematical Society. Demetrios Chalcocondyles http://www.mathgenealogy.org/ Heinrich von Langenstein Gaetano da Thiene Sigismondo Polcastro Leo Outers Moses Perez Scipione Fortiguerra Rudolf Agricola Thomas von Kempen à Kempis Jacob ben Jehiel Loans Accademia Romana 1452 Université de Paris 1363, 1375 Université Catholique de Louvain 1485 Università di Firenze 1493 Università degli Studi di Ferrara 1478 Mystras 1452 Jan Standonck Johann (Johannes Kapnion) Reuchlin Johannes von Gmunden Nicoletto Vernia Pietro Roccabonella Pelope Maarten (Martinus Dorpius) van Dorp Jean Tagault François Dubois Janus Lascaris Girolamo (Hieronymus Aleander) Aleandro Matthaeus Adrianus Alexander Hegius Johannes Stöffler Collège Sainte-Barbe 1474 Universität Basel 1477 Universität Wien 1406 Università di Padova Università di Padova Université Catholique de Louvain 1504, 1515 Université de Paris 1516 Università di Padova 1472 Università -
Mathematical Genealogy of the Wellesley College Department Of
Nilos Kabasilas Mathematical Genealogy of the Wellesley College Department of Mathematics Elissaeus Judaeus Demetrios Kydones The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society. http://www.genealogy.math.ndsu.nodak.edu/ Georgios Plethon Gemistos Manuel Chrysoloras 1380, 1393 Basilios Bessarion 1436 Mystras Johannes Argyropoulos Guarino da Verona 1444 Università di Padova 1408 Cristoforo Landino Marsilio Ficino Vittorino da Feltre 1462 Università di Firenze 1416 Università di Padova Angelo Poliziano Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo 1477 Università di Firenze 1433 Constantinople / Università di Mantova Università di Mantova Leo Outers Moses Perez Scipione Fortiguerra Demetrios Chalcocondyles Jacob ben Jehiel Loans Thomas à Kempis Rudolf Agricola Alessandro Sermoneta Gaetano da Thiene Heinrich von Langenstein 1485 Université Catholique de Louvain 1493 Università di Firenze 1452 Mystras / Accademia Romana 1478 Università degli Studi di Ferrara 1363, 1375 Université de Paris Maarten (Martinus Dorpius) van Dorp Girolamo (Hieronymus Aleander) Aleandro François Dubois Jean Tagault Janus Lascaris Matthaeus Adrianus Pelope Johann (Johannes Kapnion) Reuchlin Jan Standonck Alexander Hegius Pietro Roccabonella Nicoletto Vernia Johannes von Gmunden 1504, 1515 Université Catholique de Louvain 1499, 1508 Università di Padova 1516 Université de Paris 1472 Università di Padova 1477, 1481 Universität Basel / Université de Poitiers 1474, 1490 Collège Sainte-Barbe -
A History of Mathematics in America Before 1900.Pdf
THE BOOK WAS DRENCHED 00 S< OU_1 60514 > CD CO THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Publication Committee GILBERT AMES BLISS DAVID RAYMOND CURTISS AUBREY JOHN KEMPNER HERBERT ELLSWORTH SLAUGHT CARUS MATHEMATICAL MONOGRAPHS are an expression of THEthe desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemination of mathe- matical knowledge by making accessible at nominal cost a series of expository presenta- tions of the best thoughts and keenest re- searches in pure and applied mathematics. The publication of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The expositions of mathematical subjects which the monographs will contain are to be set forth in a manner comprehensible not only to teach- ers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and trea- tises. The scope of this series includes also historical and biographical monographs. The Carus Mathematical Monographs NUMBER FIVE A HISTORY OF MATHEMATICS IN AMERICA BEFORE 1900 By DAVID EUGENE SMITH Professor Emeritus of Mathematics Teacliers College, Columbia University and JEKUTHIEL GINSBURG Professor of Mathematics in Yeshiva College New York and Editor of "Scripta Mathematica" Published by THE MATHEMATICAL ASSOCIATION OF AMERICA with the cooperation of THE OPEN COURT PUBLISHING COMPANY CHICAGO, ILLINOIS THE OPEN COURT COMPANY Copyright 1934 by THE MATHEMATICAL ASSOCIATION OF AMKRICA Published March, 1934 Composed, Printed and Bound by tClfe QlolUgUt* $Jrr George Banta Publishing Company Menasha, Wisconsin, U. -
Hamiltonian System and Dissipative System
Hamiltonian system and Dissipative system: Motivating example: (energy and DE) y00 + qy = 0 system: y0 = v, v0 = −qy y00 + py0 + qy = 0 system: y0 = v, v0 = −qy − pv 1 1 Energy function: E(y, v) = v2 + qy2 2 2 1 where v2 is the kinetic energy (note that m = 1) 2 1 and qy2 is the potential energy (example: q = g) 2 1 y00 + qy = 0 d d 1 1 E(y(t), v(t)) = v2(t) + qy2(t) = v(t)v0(t) + qy(t)y0(t) dt dt 2 2 = v(t)(−qy(t)) + qy(t) · v(t) = 0 (energy is conserved) y00 + py0 + qy = 0 d d 1 1 E(y(t), v(t)) = v2(t) + qy2(t) = v(t)v0(t) + qy(t)y0(t) dt dt 2 2 = v(t)(−qy(t) − pv(t)) + qy(t) · v(t) = −p[v(t)]2 ≤ 0 (energy is dissipated) 2 Definition: dx dy = f(x, y), = g(x, y). dt dt If there is a function H(x, y) such that for each solution orbit d (x(t), y(t)), we have H(x(t), y(t)) = 0, then the system is a dt Hamiltonian system, and H(x, y) is called conserved quantity. (or energy function, Hamiltonian) If there is a function H(x, y) such that for each solution orbit d (x(t), y(t)), we have H(x(t), y(t)) ≤ 0, then the system is a dt dissipative system, and H(x, y) is called Lyapunov function. (or energy function) 3 Example: If a satellite is circling around the earth, it is a Hamil- tonian system; but if it drops to the earth, it is a dissipative system. -
Some Comments on Multiple Discovery in Mathematics
Journal of Humanistic Mathematics Volume 7 | Issue 1 January 2017 Some Comments on Multiple Discovery in Mathematics Robin W. Whitty Queen Mary University of London Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the History of Science, Technology, and Medicine Commons, and the Other Mathematics Commons Recommended Citation Whitty, R. W. "Some Comments on Multiple Discovery in Mathematics," Journal of Humanistic Mathematics, Volume 7 Issue 1 (January 2017), pages 172-188. DOI: 10.5642/jhummath.201701.14 . Available at: https://scholarship.claremont.edu/jhm/vol7/iss1/14 ©2017 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. Some Comments on Multiple Discovery in Mathematics1 Robin M. Whitty Queen Mary University of London [email protected] Synopsis Among perhaps many things common to Kuratowski's Theorem in graph theory, Reidemeister's Theorem in topology, and Cook's Theorem in theoretical com- puter science is this: all belong to the phenomenon of simultaneous discovery in mathematics. We are interested to know whether this phenomenon, and its close cousin repeated discovery, give rise to meaningful questions regarding causes, trends, categories, etc. -
Mathematics in the Austrian-Hungarian Empire
Mathematics in the Austrian-Hungarian Empire Christa Binder The appointment policy in the Austrian-Hungarian Empire In: Martina Bečvářová (author); Christa Binder (author): Mathematics in the Austrian-Hungarian Empire. Proceedings of a Symposium held in Budapest on August 1, 2009 during the XXIII ICHST. (English). Praha: Matfyzpress, 2010. pp. 43–54. Persistent URL: http://dml.cz/dmlcz/400817 Terms of use: © Bečvářová, Martina © Binder, Christa Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz THE APPOINTMENT POLICY IN THE AUSTRIAN- -HUNGARIAN EMPIRE CHRISTA BINDER Abstract: Starting from a very low level in the mid oft the 19th century the teaching and research in mathematics reached world wide fame in the Austrian-Hungarian Empire before World War One. How this was complished is shown with three examples of careers of famous mathematicians. 1 Introduction This symposium is dedicated to the development of mathematics in the Austro- Hungarian monarchy in the time from 1850 to 1914. At the beginning of this period, in the middle of the 19th century the level of teaching and researching mathematics was very low – with a few exceptions – due to the influence of the jesuits in former centuries, and due to the reclusive period in the first half of the 19th century. But even in this time many efforts were taken to establish a higher education.