Chapter 2. Vectors and Vector Spaces
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8. Gram-Schmidt Orthogonalization
Chapter 8 Gram-Schmidt Orthogonalization (September 8, 2010) _______________________________________________________________________ Jørgen Pedersen Gram (June 27, 1850 – April 29, 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares, and Investigations of the number of primes less than a given number. The process that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram ________________________________________________________________________ Erhard Schmidt (January 13, 1876 – December 6, 1959) was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia (now Estonia). His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations. Together with David Hilbert he made important contributions to functional analysis. http://en.wikipedia.org/wiki/File:Erhard_Schmidt.jpg ________________________________________________________________________ 8.1 Gram-Schmidt Procedure I Gram-Schmidt orthogonalization is a method that takes a non-orthogonal set of linearly independent function and literally constructs an orthogonal set over an arbitrary interval and with respect to an arbitrary weighting function. Here for convenience, all functions are assumed to be real. un(x) linearly independent non-orthogonal un-normalized functions Here we use the following notations. n un (x) x (n = 0, 1, 2, 3, …..). n (x) linearly independent orthogonal un-normalized functions n (x) linearly independent orthogonal normalized functions with b ( x) (x)w(x)dx i j i , j . -
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. -
Math 126 Lecture 4. Basic Facts in Representation Theory
Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius. -
Kailash C. Misra Rutgers University, New Brunswick
Nilos Kabasilas Heinri%h von )angenstein ;emetrios Ky2ones 4lissae"s J"2ae"s Universit> 2e /aris (1060) ,eorgios /lethon ,emistos Johannes von ,m"n2en !an"el Chrysoloras (10'5) UniversitIt *ien (1.56) ,"arino 2a ?erona $asilios $essarion ,eorg von /e"erba%h (1.5') !ystras (1.06) UniversitIt *ien (1..5) Johannes 8rgyro1o"los Johannes !Lller Regiomontan"s )"%a /a%ioli UniversitC 2i /a2ova (1...) UniversitIt )ei1Big : UniversitIt *ien (1.-6) ?ittorino 2a Aeltre !arsilio Ai%ino ;omeni%o !aria Novara 2a Aerrara Cristo3oro )an2ino UniversitC 2i /a2ova (1.16) UniversitC 2i AirenBe (1.6() UniversitC 2i AirenBe (1.'0) Theo2oros ,aBes 9gnibene (9mnibon"s )eoni%en"s) $onisoli 2a )onigo 8ngelo /oliBiano Constantino1le : UniversitC 2i !antova (1.00) UniversitC 2i !antova UniversitC 2i AirenBe (1.66) ;emetrios Chal%o%on2yles R"2ol3 8gri%ola S%i1ione Aortig"erra )eo 9"ters ,aetano 2a Thiene Sigismon2o /ol%astro Thomas C Kem1is Ja%ob ben Jehiel )oans !oses /ereB !ystras : 8%%a2emia Romana (1.-() UniversitC 2egli St"2i 2i Aerrara (1.6') UniversitC 2i AirenBe (1.90) Universit> CatholiK"e 2e )o"vain (1.'-) Jan"s )as%aris Ni%oletto ?ernia /ietro Ro%%abonella Jan Stan2on%& 8le7an2er Hegi"s Johann (Johannes Ka1nion) Re"%hlin AranNois ;"bois ,irolamo (Hieronym"s 8lean2er) 8lean2ro !aarten (!artin"s ;or1i"s) van ;or1 /elo1e !atthae"s 82rian"s Jean Taga"lt UniversitC 2i /a2ova (1.6() UniversitC 2i /a2ova UniversitC 2i /a2ova CollMge Sainte@$arbe : CollMge 2e !ontaig" (1.6.) (1.6.) UniversitIt $asel : Universit> 2e /oitiers (1.66) Universit> 2e /aris (1-16) UniversitC -
Zermelo in Z¨Urich
Zermelo in Zurich¨ ¤ Volker Peckhaus Institut fur¨ Philosophie der Universitat¨ Erlangen-Nurnberg¨ Bismarckstr. 1, D – 91054 Erlangen E-mail: [email protected] 1 Einleitung Meine sehr geehrten Damen und Herren! Unter dem eher lapidaren Titel Zerme- ” lo in Zurich“¨ mochte¨ ich eine Periode aus dem Leben des Logikers und Mengen- theoretikers Ernst Zermelo beleuchten, die zwar die Krone seiner verungluckten¨ akademischen Karriere darstellt, von der aber gleichwohl bisher kaum etwas be- kannt war. Was bekannt ist, und zwar wohl jedem, der sich schon einmal mit der Biographie Zermelos beschaftigt¨ hat, ist eine Anekdote, die Abraham A. Fraenkel in seiner Autobiographie Lebenskreise erzahlt.¨ Ich mochte¨ Ihnen die Fußnote, in der er sich mit Zermelo beschaftigt,¨ vollstandig¨ zitieren (1967, 149, Fn. 55): Obgleich nicht hierher gehorig,¨ seien von diesem genialen und seltsamen Mathematiker, dessen Namen bis heute einen fast magischen Klang behalten hat, einige kaum bekannte Zuge¨ berichtet. Ein schlechter Lehrer, kam er in Gottingen¨ nicht vorwarts,¨ obgleich er 1904 einen drei Seiten langen Aufsatz — Beweis des Wohlordnungsgesetzes“ — publizierte, der die gesamte ma- ” thematische Welt in — zustimmende und ablehnende — Aufregung versetz- te; mit seinen Gegnern setzte er sich 1908 in einer Abhandlung auseinander, die an Sarkasmus nicht ihresgleichen in der mathematischen Literatur hat. 1910 wurde er endlich als Ordinarius an die kantonale Universitat¨ Zurich¨ be- rufen. Kurz vor dem Weltkrieg verbrachte er eine Nacht in den bayerischen Alpen und fullte¨ im Meldezettel des Hotels die Rubrik Staatsangehorigkeit“¨ ” mit den Worten aus: Gottseidank kein Schweizer.“ Das Ungluck¨ wollte, daß ” kurz danach der Leiter des Unterrichtsdepartments des Kantons Zurich¨ im gleichen Hotel wohnte und die Eintragung sah. -
(Emmy) Noether
Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether Gottingen¨ and Algebra Amalie (Emmy) Noether 1920-1933 Table of Contents German Nationalism: Exile Introduction from Germany Gottingen¨ and Hilbert and Einstein 1917-1920 References Larry Susanka October 3, 2019 Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References INTRODUCTION She was the eldest of four children. Little is known about her Emmy Noether (pronounced childhood beyond what can be inferred from her general NER-ter) was born to a situation. There are no anecdotes that reveal that she was prosperous Jewish family in “destined” to be a great mathematician. Girls at that time and the Bavarian university town place and social class simply did not study the sciences, and an of Erlangen on March 23, 1882. academic career was out of the question. She died unexpectedly in another university town, Bryn Mawr Pennsylvania, on April According to Wikipedia, 14, 1935 at the age of 53. As a girl, Noether was well liked. She did not stand out academically although she was known for being clever and The aim of this talk is to tell friendly. She was near-sighted and talked with a minor lisp part of the story of this during her childhood. remarkable woman and provide an outline of her ————— scientific legacy. ————— Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Amalie (Emmy) Noether TOC Intro Gottingen¨ 1917-1920 Gottingen¨ 1920-1933 Exile References Her father Max Noether, though, was a Professor of Mathematics at the University of Erlangen and studied what She went to the standard local we would call algebraic geometry. -
RM Calendar 2015
Rudi Mathematici x4–8228 x3+25585534 x2–34806653332 x+17895175197705=0 www.rudimathematici.com 1 G (1803) Guglielmo Libri Carucci dalla Sommaja RM132 (1878) Agner Krarup Erlang Rudi Mathematici (1894) Satyendranath Bose RM168 (1912) Boris Gnedenko 2 V (1822) Rudolf Julius Emmanuel Clausius (1905) Lev Genrichovich Shnirelman (1938) Anatoly Samoilenko 3 S (1917) Yuri Alexeievich Mitropolsky Gennaio 4 D (1643) Isaac Newton RM071 2 5 L (1723) Nicole-Reine Etable de Labrière Lepaute (1838) Marie Ennemond Camille Jordan Putnam 2000, A1 (1871) Federigo Enriques RM084 Sia A un numero reale positivo. Quali sono i possibili (1871) Gino Fano ∞ 6 M (1807) Jozeph Mitza Petzval valori di x , se x0, x1, … sono numeri positivi per cui ∑ i2 (1841) Rudolf Sturm =i 0 7 M (1871) Felix Edouard Justin Emile Borel ∞ ? (1907) Raymond Edward Alan Christopher Paley ∑ i A=x 8 G (1888) Richard Courant RM156 =i 0 (1924) Paul Moritz Cohn (1942) Stephen William Hawking Barzellette per élite 9 V (1864) Vladimir Adreievich Steklov È difficile fare giochi di parole con i cleptomani. (1915) Mollie Orshansky Prendono tutto alla lettera . 10 S (1875) Issai Schur (1905) Ruth Moufang 11 D (1545) Guidobaldo del Monte RM120 Titoli da un mondo matematico (1707) Vincenzo Riccati Con il passaggio ai nuovi test, le valutazioni crollano. (1734) Achille Pierre Dionis du Sejour Mondo matematico: Con il passaggio ai nuovi test, i 3 12 L (1906) Kurt August Hirsch risultati non sono più confrontabili. (1915) Herbert Ellis Robbins RM156 13 M (1864) Wilhelm Karl Werner Otto Fritz Franz Wien (1876) Luther Pfahler Eisenhart Alice rise: “È inutile che ci provi”, disse; “non si può (1876) Erhard Schmidt credere a una cosa impossibile”. -
Rudi Mathematici
Rudi Mathematici x4-8188x3+25139294x2-34301407052x+17549638999785=0 Rudi Mathematici January 53 1 S (1803) Guglielmo LIBRI Carucci dalla Sommaja Putnam 1999 - A1 (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE Find polynomials f(x), g(x), and h(x) _, if they exist, (1912) Boris GNEDENKO such that for all x 2 S (1822) Rudolf Julius Emmanuel CLAUSIUS f (x) − g(x) + h(x) = (1905) Lev Genrichovich SHNIRELMAN (1938) Anatoly SAMOILENKO −1 if x < −1 1 3 M (1917) Yuri Alexeievich MITROPOLSHY 4 T (1643) Isaac NEWTON = 3x + 2 if −1 ≤ x ≤ 0 5 W (1838) Marie Ennemond Camille JORDAN − + > (1871) Federigo ENRIQUES 2x 2 if x 0 (1871) Gino FANO (1807) Jozeph Mitza PETZVAL 6 T Publish or Perish (1841) Rudolf STURM "Gustatory responses of pigs to various natural (1871) Felix Edouard Justin Emile BOREL 7 F (1907) Raymond Edward Alan Christopher PALEY and artificial compounds known to be sweet in (1888) Richard COURANT man," D. Glaser, M. Wanner, J.M. Tinti, and 8 S (1924) Paul Moritz COHN C. Nofre, Food Chemistry, vol. 68, no. 4, (1942) Stephen William HAWKING January 10, 2000, pp. 375-85. (1864) Vladimir Adreievich STELKOV 9 S Murphy's Laws of Math 2 10 M (1875) Issai SCHUR (1905) Ruth MOUFANG When you solve a problem, it always helps to (1545) Guidobaldo DEL MONTE 11 T know the answer. (1707) Vincenzo RICCATI (1734) Achille Pierre Dionis DU SEJOUR The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of (1906) Kurt August HIRSCH 12 W mathematics. -
THE ISOPERIMETRIC INEQUALITY the Circle Is Uniquely Characterized
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 6, November 1978 THE ISOPERIMETRIC INEQUALITY BY ROBERT OSSERMAN1 The circle is uniquely characterized by the property that among all simple closed plane curves of given length L, the circle of circumference L encloses maximum area. This property is most succintly expressed in the isoperimetric inequality 2 L > 4<irA9 (1) where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. The purpose of this paper is to recount some of the most interesting of the many sharpened forms, generalizations, and applications of this inequality, with emphasis on recent contributions. Earlier work is summarized in the book of Hadwiger [1], Other general references, varying from very elementary to quite technical are Kazarinoff [1], Pólya [2, Chapter X], Porter [1], and the books of Blaschke listed in the bibliography. Most books on convexity also contain a discussion of the isoperimetric inequality from that perspective. One aspect of the subject is given by Burago [1]. Others may be found in a recent paper of the author [4] on Bonnesen inequalities and in the book of Santaló [4] on integral geometry and geometric probability. An important note: we shall not go into the area of so-called "isoperimetric problems". Those are simply variational problems with constraints, whose name derives from the fact that inequality (1) corresponds to the first example of such a problem: maximize the area of a domain under the constraint that the length of its boundary be fixed. -
Mind and Nature: Selected Writings on Philosophy, Mathematics, And
Mind and Nature 8 This page intentionally left blank MIND AND NATURE Selected Writings on Philosophy, Mathematics, and Physics HERMANN WEYL Edited and with an introduction by Peter Pesic 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 Weyl, Hermann, 1885–1955. Mind and nature : selected writings on philosophy, mathematics, and physics / Hermann Weyl; edited and with an introduction by Peter Pesic. p. cm. Includes bibliographical references and index. ISBN 978-0-691-13545-8 (cloth : alk. paper) 1. Mathematics—Philosophy. 2. Physics—Philosophy. I. Pesic, Peter. II. Title. QA8.4.W49 2009 510.1—dc22 2008049964 British Library Cataloging-in-Publication Data is available This book has been composed in Adobe Garamond Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10987654321 Contents 8 Introduction 1 1921 1 Electricity and Gravitation 20 1922 2 Two Letters by Einstein and Weyl on a Metaphysical Question 25 1927 3 Time Relations in the Cosmos, Proper Time, Lived Time, and Metaphysical Time 29 1932 4 The Open World: Three Lectures on the Metaphysical Implications of Science 34 1934 5 Mind and Nature 83 1946 6 Address at the Princeton Bicentennial Conference 162 ca. 1949 7 Man and the Foundations of Science 175 1954 8 The Unity of Knowledge 194 1955 9 Insight and Reflection 204 Notes 223 References 241 Acknowledgments 253 Index 255 This page intentionally left blank Mind and Nature 8 This page intentionally left blank Introduction 8 “It’s a crying shame that Weyl is leaving Zurich. -
Torsten Ekedahl Landau and Schur Bodil Branner Kavli IPMU, Tokyo P
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Obituaries History Interview Centres Mikael Passare, Torsten Ekedahl Landau and Schur Bodil Branner Kavli IPMU, Tokyo p. 12 p. 31 p. 37 p. 47 June 2012 Issue 84 ISSN 1027-488X S E European M M Mathematical E S Society Journals published by the ISSN print 1661-7207 Editor-in-Chief: ISSN online 1661-7215 Rostislav Grigorchuk (Texas A&M University, College Station, USA; Steklov Institute of Mathematics, Moscow, 2012. Vol. 6. 4 issues Russia) Approx. 800 pages. 17.0 x 24.0 cm Aims and Scope Price of subscription: Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or 258¤ online only group actions as well as articles in other areas of mathematics in which groups or group actions are used 298 ¤ print+online as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields. ISSN print 1463-9963 Editors-in-Chief: ISSN online 1463-9971 José Francisco Rodrigues (Universidade de Lisboa, Portugal) 2012. Vol. 14. 4 issues Charles M. Elliott (University of Warwick, Coventry, UK) Approx. 500 pages. 17.0 x 24.0 cm Robert V. Kohn (New York University, USA) Price of subscription: Aims and Scope 390 ¤ online only Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of 430 ¤ print+online interfaces and free boundary problems in all areas where such phenomena are pertinent. -
Hilbert & Schmidt Aneb O Jednom Mezníku V Historii Matematiky
Pokroky matematiky, fyziky a astronomie Albrecht Pietsch Hilbert & Schmidt aneb O jednom mezníku v historii matematiky Pokroky matematiky, fyziky a astronomie, Vol. 39 (1994), No. 2, 65--94 Persistent URL: http://dml.cz/dmlcz/138061 Terms of use: © Jednota českých matematiků a fyziků, 1994 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Hilbert & Schmidt aneb O jednom mezníku v historii matematiky Albrecht Pietsch, Jena Úvod překladatele V roce 1984 začalo nakladatelství Teubner v Lipsku vydávat řadu Teubner-Archiv zur Mathematik, v níž čtenáře seznamovalo s pracemi významných německých ma tematiků, působících v druhé polovině devatenáctého a na počátku dvacátého století. Šlo především o přetisky (faksimile) či přepisy důležitých článků nebo (univerzitních) přednášek, doprovázených zasvěcenými komentáři. Jedenáctý svazek uvedené řady vyšel v roce 1989. Nesl název Integrální rovnice a rovnice o nekonečně mnoha neznámých a obsahoval některé z prací, které v letech 1904 až 1910 publikovali David Hilbert a jeho žák Erhard Schmidt. Svazek uspořádal Albrecht Pietsch, který také napsal závěrečný doslov. Informace o dalších svazcích uvedené řady najde čtenář na 3. straně obálky tohoto čísla. D. Hilberta a E. Schmidta jistě není třeba našim čtenářům představovat. Práce, které v uvedené periodě vytvořili, představují nepochybně mezník v historii lineární funkcionální analýzy. Zde chceme čtenáře seznámit s Pietschovým doslovem, který výrazně přesahuje rámec toho, co bychom od „pouhého doslovua očekávali.