On Resolving Singularities of Plane Curves Via a Theorem Attributed to Clebsch
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A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
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. -
Emmy Noether, Greatest Woman Mathematician Clark Kimberling
Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904. -
Notices of the American Mathematical Society ABCD Springer.Com
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook Collection of the American Mathematical Society August 2009 Volume 56, Number 7 Guido Castelnuovo and Francesco Severi: NEW NEW NEW Two Personalities, Two The objective of this textbook is the Blackjack is among the most popular This second edition of Alexander Soifer’s Letters construction, analysis, and interpreta- casino table games, one where astute How Does One Cut a Triangle? tion of mathematical models to help us choices of playing strategy can create demonstrates how different areas of page 800 understand the world we live in. an advantage for the player. Risk and mathematics can be juxtaposed in the Students and researchers interested in Reward analyzes the game in depth, solution of a given problem. The author mathematical modelling in math- pinpointing not just its optimal employs geometry, algebra, trigono- ematics, physics, engineering and the strategies but also its financial metry, linear algebra, and rings to The Dixmier–Douady applied sciences will find this text useful. performance, in terms of both expected develop a miniature model of cash flow and associated risk. mathematical research. Invariant for Dummies 2009. Approx. 480 p. (Texts in Applied Mathematics, Vol. 56) Hardcover 2009. Approx. 140 p. 23 illus. Hardcover 2nd ed. 2009. XXX, 174 p. 80 illus. Softcover page 809 ISBN 978-0-387-87749-5 7 $69.95 ISBN 978-1-4419-0252-8 7 $49.95 ISBN 978-0-387-74650-0 7 approx. $24.95 For access check with your librarian Waco Meeting page 879 A Primer on Scientific Data Mining in Agriculture Explorations in Monte Programming with Python A. -
Wolf Barth (1942--2016)
Wolf Barth (1942{2016) Thomas Bauer, Klaus Hulek, Slawomir Rams, Alessandra Sarti, Tomasz Szemberg 1. Life 1.1. Youth Wolf Paul Barth was born on 20th October 1942 in Wernigerode, a small town in the Harz mountains. This is where his family, originally living in Nuremberg, had sought refuge from the bombing raids. In 1943 the family moved to Simmelsdorf, which is close to Nuremberg, but less vulnerable to attacks. In 1945 Barth's brother Hannes was born. The family, the father was a town employee, then moved back to Nuremberg and Wolf Barth's home was close to the famous Nuremberg castle { his playgrounds were the ruins of Nuremberg. In 1961 Wolf Barth successfully completed the Hans Sachs Gymnasium in Nuremberg. He was an excellent student (except in sports), and he then chose to study mathemat- ics and physics at the nearby Universit¨atErlangen, officially called Friedrich-Alexander Universit¨atErlangen-N¨urnberg. 1.2. Studies and early academic career As it was the standard at the time, Wolf Barth enroled for the Staatsexamen, the German high school teacher's examination. At this time, Reinhold Remmert was a professor of mathematics in Erlangen and Wolf Barth soon felt attracted to this area of mathematics. So, when Remmert moved to G¨ottingenin 1963, Barth went with him. G¨ottingen had at that time started to recover to some extent after its losses in the Nazi period and Hans Grauert was one on the leading figures who attracted many talented mathematicians. And indeed, it was Grauert, who became Barth's second academic teacher and a figure who influenced his mathematical thinking greatly. -
Startling Simplicity
Startling simplicity P (6) iδui = δL − ( !) d P 1 @L 2 @L (1) k @L (k−1) − dx 1 (1) δui + 1 (2) δui + ··· + 1 (k) δui @ui @ui @ui ( !) d2 P 2 @L 3 @L (1) k @L (k−2) + dx2 2 (2) δui + 2 (3) δui + ··· + 2 (k) δui @ui @ui @ui k dk P k @L ··· + (−1) dxk k (k) δui : @ui On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . Direct consequence of additivity, chain rule, and integration by parts. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. In several independent variables those lines become a sum: @A @A Div(A) = 1 + ··· + n : @x1 @xn Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. Direct consequence of additivity, chain rule, and integration by parts. Noether’s Aj are not functions. They are differential operators. d The last three lines are a derivative dx . On its face, an equation Euler or Lagrange could have written. -
Amalie (Emmy) Noether (1882 - 1935)
Amalie (Emmy) Noether (1882 - 1935) Mairi Sakellariadou King’s College London Emmy Noether was born in Erlangen, Germany on March 23, 1882 She was named Amalie, but always called "Emmy" 2 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family The father: Max Noether (1844 Mannheim– 1921 Erlangen) From a Jewish family of wealthy wholesale hardware dealers. At 14, Max contracted polio and was afflicted by its effects for the rest of his life. Through self-study, he learned advanced mathematics and entered the University of Heidelberg in 1865. He moved to the University of Erlagen in 1888. While there, he helped to found the field of algebraic geometry. In 1880 he married Ida Amalia Kaufmann, the daughter of another wealthy Jewish merchant family. Two years later they had their first child, named Amalia (“ Emmy “) after her mother. 3 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family The mother: Ida Amalia Kaufmann (1852 Koln – 1951 Erlagen) From a wealthy Jewish merchant family. Ida had a brother who was a professor at the University of Berlin. In 1880 she married Max Noether ; they had four children. Ida was a skilled pianist. 4 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family . Emmy 1882 – 1935 Professor of Mathematics in Erlagen, Gottingen, and Bryn Mawr (USA) . Alfred 1883 – 1918 Chemist . Fritz 1884 – 1937 Professor of Mathematics in Breslavia (Germany) and in Tomsk (Russia) . Gustav Robert 1889 -- 1928 5 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family a bit before the first world war 6 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) Emmy's childhood was unexceptional, going to school, learning domestic skills, and taking piano lessons. -
Borsuk-Ulam Theorem and Hilbert's Nullstellensatz
Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Borsuk-Ulam Theorem and Hilbert’s Dilip P. Patil Nullstellensatz § 1 Borsuk-Ulam — and vice versa Theorem §2 Borsuk’s Nullstellensatz and its Dilip P. Patil Equivalents §3 2-Fields Department of Mathematics §4 Dimension Indian Institute of Science, Bangalore and Multiplicity §5 Projective 1 Nullstellensatz Bhaskaracharya Pratishthana , Pune §6 Analogs of July 22, 2018 HNS to 2fields 1On the occasion of Birthday of Late Professor Shreeram S. Abhyankar (1930 – 2012), Founder Director, Bhaskaracharya Pratishthana, Pune. Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) Founder Director Bhaskaracharya Pratishthana, Pune. Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Founder Director Bhaskaracharya Pratishthana, Pune. Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) and Multiplicity §5 Projective Nullstellensatz §6 Analogs of HNS to 2fields Dilip P. Patil Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Late Professor Shreeram Shankar Abhayankar Borsuk-Ulam Theorem and Hilbert’s Nullstellensatz Dilip P. Patil § 1 Borsuk-Ulam Theorem §2 Borsuk’s Nullstellensatz and its Equivalents §3 2-Fields §4 Dimension Shreeram Shankar Abhayankar (22 July, 1930 – 02 Nov, 2012) and Founder Director Multiplicity §5 Projective Bhaskaracharya Pratishthana, Pune. -
Notices of the American Mathematical Society ABCD Springer.Com
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook Collection of the American Mathematical Society September 2009 Volume 56, Number 8 Bôcher, Osgood, ND and the Ascendance NEW NEW 2 EDITION EDITION of American Mathematics forms bridges between From the reviews of the first edition The theory of elliptic curves is Mathematics at knowledge, tradition, and contemporary 7 Chorin and Hald provide excellent distinguished by the diversity of the Harvard life. The continuous development and explanations with considerable insight methods used in its study. This book growth of its many branches permeates and deep mathematical understanding. treats the arithmetic theory of elliptic page 916 all aspects of applied science and 7 SIAM Review curves in its modern formulation, technology, and so has a vital impact on through the use of basic algebraic our society. The book will focus on these 2nd ed. 2009. X, 162 p. 7 illus. number theory and algebraic geometry. aspects and will benefit from the (Surveys and Tutorials in the Applied contribution of world-famous scientists. Mathematical Sciences) Softcover 2nd ed. 2009. XVIII, 514 p. 14 illus. Invariant Theory ISBN 978-1-4419-1001-1 (Graduate Texts in Mathematics, 2009. XI, 263 p. (Modeling, Simulation & 7 Approx. $39.95 Volume 106) Hardcover of Tensor Product Applications, Volume 3) Hardcover ISBN 978-0-387-09493-9 7 $59.95 Decompositions of ISBN 978-88-470-1121-2 7 $59.95 U(N) and Generalized Casimir Operators For access check with your librarian page 931 Stochastic Partial Linear Optimization Mathematica in Action Riverside Meeting Differential Equations The Simplex Workbook The Power of Visualization A Modeling, White Noise Approach G. -
The Bicentennial Tribute to American Mathematics 1776–1976
THE BICENTENNIAL TRIBUTE TO AMERICAN MATHEMATICS Donald J. Albers R. H. McDowell Menlo College Washington University Garrett Birkhof! S. H. Moolgavkar Harvard University Indiana University J. H. Ewing Shelba Jean Morman Indiana University University of Houston, Victoria Center Judith V. Grabiner California State College, C. V. Newsom Dominguez Hills Vice President, Radio Corporation of America (Retired) W. H. Gustafson Indiana University Mina S. Rees President Emeritus, Graduate School P. R. Haimos and University Center, CUNY Indiana University Fred S. Roberts R. W. Hamming Rutgers University Bell Telephone Laboratories R. A. Rosenbaum I. N. Herstein Wesleyan University University of Chicago S. K. Stein Peter J. Hilton University of California, Davis Battelle Memorial Institute and Case Western Reserve University Dirk J. Struik Massachusetts Institute of Technology Morris Kline (Retired) Brooklyn College (CUNY) and New York University Dalton Tarwater Texas Tech University R. D. Larsson Schenectady County Community College W. H. Wheeler Indiana University Peter D. Lax New York University, Courant Institute A. B. Willcox Executive Director, Mathematical Peter A. Lindstrom Association of America Genesee Community College W. P. Ziemer Indiana University The BICENTENNIAL TRIBUTE to AMERICAN MATHEMATICS 1776 1976 Dalton Tarwater, Editor Papers presented at the Fifty-ninth Annual Meeting of the Mathematical Association of America commemorating the nation's bicentennial Published and distributed by The Mathematical Association of America © 1977 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 77-14706 ISBN 0-88385-424-4 Printed in the United States of America Current printing (Last Digit): 10 987654321 PREFACE It was decided in 1973 that the Mathematical Association of America would commemorate the American Bicentennial at the San Antonio meet- ing of the Association in January, 1976, by stressing the history of Ameri- can mathematics. -
Basic Algebra
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore May-July 2003 Basic Algebra 3. Modules 1) — Submodules ;— Ideals Max Noether† Emmy Amalie Noether†† (1844-1921) (1882-1935) 3.1. (Modular Law)Forsubmodules U,W,N of an A–module V with U ⊆ W, prove that: W ∩ (U + N) = U + (W ∩ N). 3.2. Let U,W,U,W be submodules of an A–module V with U ∩ W = U ∩ W . Prove that U is the intersection of U + (W ∩ U ) and U + (W ∩ W ). 3.3. LetA be a ring and let Vi ,i∈ I, be an infinite family of non-zero A–modules. Prove that W := i∈I Vi is not a finite A–module. 3.4. For every natural number m ≥ 1, give a minimal generating system for the Z–module Z consisting of m elements. 3.5. Let K be a field and let A be a subring of K such that every element of K can be expressed as a quotient a/b with a,b ∈ A, b = 0. (i.e. K is the quotient field of A). If K is a finite A–module, then prove that A = K. In particular, Q is not a finite Z–module. ( Hint : Suppose 2 K = Ax1 +···+Axn and b ∈ A, b = 0, with bxi ∈ A for i = 1,...,n. Now, try to express 1/b as a linear combination of xi .) 3.6. Let K be a field and let V be a K–vector space. Suppose that V1,...,Vn be distinct K- subspaces of V .IfK has at least n elements (in particular, if K is infinite), then V1 ∪···∪Vn = V . -
To “Applied” Mathematics: Tracing an Important Dimension of Mathematics and Its History
Mathematisches Forschungsinstitut Oberwolfach Report No. 12/2013 DOI: 10.4171/OWR/2013/12 From “Mixed” to “Applied” Mathematics: Tracing an important dimension of mathematics and its history Organised by Moritz Epple, Frankfurt Tinne Hoff Kjeldsen, Roskilde Reinhard Siegmund-Schultze, Kristiansand 3 March – 9 March 2013 Abstract. The workshop investigated historical variations of the ways in which historically boundaries were drawn between ‘pure’ mathematics on the one hand and ‘mixed’ or ‘applied’ mathematics on the other from about 1500 until today. It brought together historians and philosophers of mathematics as well as several mathematicians working on applications. Emphasis was laid upon the clarification of the relation between the historical use and the historiographical usefulness and philosophical soundness of the various cate- gories. Mathematics Subject Classification (2010): 01A40 to 01A99. Introduction by the Organisers For a long time, research in the history of mathematics has mostly focused on developments in pure mathematics. In recent years, however, some significant research has changed this situation. The conference brought together historians of mathematics actively involved in this reorientation, in order to take stock of what has been achieved, and to identify historical problems yet to be solved. In having done so, the organizers hope to have advanced the general understanding of the relations of mathematics with neighbouring scientific fields, and with technological, economic and social practices involving mathematical methods, in the period since ca. 1500. A fundamental historical insight provided a starting point for planning this meeting: There is no, and there has never been, a once and for all fixed notion of ‘applied’ mathematics.