Univalent Functions. the Bieberbach Conjecture
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A Study on Concave Functions in Geometric Function Theory
TOHOKU UNIVERSITY Graduate School of Information Sciences A Study on Concave Functions in Geometric Function Theory Dissertation written by: Rintaro Ohno born 21. January 1984 in Tokyo (Japan) Sendai 2014 Division of Mathematics Prof. Toshiyuki Sugawa ii Supervisor: Prof. Toshiyuki Sugawa Examiners: Prof. Shigeru Sakaguchi Prof. Reika Fukuizumi Date of submission: 4. August 2014 Contents Introduction iv Notations vii 1 Basic Properties and Preliminaries1 1.1 Univalent Functions and Basic Principles......................1 1.2 Convex Functions....................................3 2 Concave Functions5 2.1 Characterizations for Concave Functions.......................6 2.2 Integral Representations for Concave Functions................... 12 3 Coefficients of Concave Functions - Known Results 18 3.1 Coefficients of the Taylor Series............................ 18 3.2 Coefficients of the Laurent Series........................... 24 3.3 Alternative Proof for the Residue........................... 33 4 Extension of Necessary and Sufficient Conditions for Concave Functions 35 4.1 Proofs of the Extended Formluas........................... 36 4.2 Application of the Extended Condition........................ 39 5 On a Coefficient Body of Concave Functions 42 5.1 Representation Formula for Cop and Lemmas.................... 44 5.2 Proof of the Theorems................................. 49 Acknowledgments 54 iii Introduction In the course of the last century, the field of geometric function theory presented many interesting and fascinating facts. Starting with the mapping theorem of Riemann, Bieberbach [5] gave a long-lasting conjecture in 1916, which attracted the attention of many mathematicians over the time. The conjecture concerned the class S of analytic and univalent functions in the unit disk 0 D = fz 2 C : jzj < 1g, normalized at the origin to have f(0) = f (0)−1 = 0, and stated that the P1 n Taylor coefficients an(f) of these functions f(z) = z + n=0 an(f)z 2 S satisfy the inequality jan(f)j ≤ n. -
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. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Arxiv:1803.02193V1 [Math.HO] 6 Mar 2018 AQE AR IT AZZK EE ENG IHI .KA G
KLEIN VS MEHRTENS: RESTORING THE REPUTATION OF A GREAT MODERN JACQUES BAIR, PIOTR BLASZCZYK, PETER HEINIG, MIKHAIL G. KATZ, JAN PETER SCHAFERMEYER,¨ AND DAVID SHERRY Abstract. Historian Herbert Mehrtens sought to portray the his- tory of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens’ conclusions have been picked up by both histo- rians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at G¨ottingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized anal- ysis as practiced by the Berlin school. Klein’s G¨ottingen lecture and other texts shed light on Klein’s modernism. Hilbert’s views on intuition are closer to Klein’s views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein’s credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens’ unsourced claim that Hilbert was interested in pro- duction rather than meaning appears to stem from Mehrtens’ marx- ist leanings. Mehrtens’ claim that [the future SS-Brigadef¨uhrer] “Theodor Vahlen . cited Klein’s racist distinctions within math- ematics, and sharpened them into open antisemitism” fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein’s position. arXiv:1803.02193v1 [math.HO] 6 Mar 2018 Contents 1. -
Lecture Notes on Multidimensional Complex Analysis
Lecture notes on multidimensional complex analysis Harold P. Boas Draft of January 6, 2020 Contents 1 Introduction 1 1.1 A note on terminology ............................... 1 1.2 Power series .................................... 3 1.3 Integral representations .............................. 3 1.4 Partial differential equations ........................... 4 1.5 Geometry ...................................... 5 2 Power series 7 2.1 Domain of convergence .............................. 8 2.2 Characterization of domains of convergence ................... 9 2.3 Local properties of holomorphic functions .................... 16 2.4 The Hartogs phenomenon ............................. 18 2.5 Natural boundaries ................................ 20 2.6 Summary: domains of convergence ....................... 32 2.7 Separate holomorphicity implies joint holomorphicity ............. 32 3 Convexity 37 3.1 Real convexity ................................... 37 3.2 Convexity with respect to a class of functions .................. 38 3.2.1 Polynomial convexity ........................... 40 3.2.2 Linear and rational convexity ....................... 48 3.2.3 Holomorphic convexity .......................... 51 3.2.4 Pseudoconvexity .............................. 60 3.3 The Levi problem ................................. 71 3.3.1 The Levi form ............................... 72 3.3.2 Applications of the 휕 problem ...................... 76 3.3.3 Solution of the 휕-equation on smooth pseudoconvex domains ..... 81 4 Holomorphic mappings 86 4.1 Fatou–Bieberbach -
The Nazi Era: the Berlin Way of Politicizing Mathematics
The Nazi era: the Berlin way of politicizing mathematics Norbert Schappacher The first impact of the Nazi regime on mathematical life, occurring essentially be- tween 1933 and 1937. took the form of a wave of dismissals of Jewish or politically suspect civil servants. It affected, overall, about 30 per cent of all mathematicians holding positions at German universities. These dismissals had nothing to do with a systematic policy for science, rather they proceeded according to various laws and decrees which concerned all civil servants alike. The effect on individual institutes depended crucially on local circumstances see [6] , section 3.1, for details. Among the Berlin mathematicians, S0-year-old Richard von Mises was the first to emigrate. He went to Istanbul at the end of 1933. where he was joined by his assistant and future second wife Hilda Pollaczek-Geiringer who had been dismissed from her position in the summer of 1933. Formally, von N{ises resigned of his own free will he was exempt from the racial clause of the first Nazi law about civil servants because he had already been a civil servant before August 1914. Having seen what the "New Germany" of the Nazis was like, he preferred to anticipate later, stricter laws, which would indeed have cost him his job and citizenship in the autumn of 1935. The Jewish algebraist Issai Schur who, like von Mises. had been a civil servant already before World War I was temporarily put on Ieave in the summer of 1933, and even though this was revoked in October 1933 he would no longer teach the large classes which for years had been a must even for students who were not primarily into pure mathematics. -
In Memoriam Alexander M. Ostrowski (1893 Bis 1986)
In memoriam Alexander M. Ostrowski (1893 bis 1986) Autor(en): Jeltsch-Fricker, Rita Objekttyp: Article Zeitschrift: Elemente der Mathematik Band (Jahr): 43 (1988) Heft 2 PDF erstellt am: 04.10.2021 Persistenter Link: http://doi.org/10.5169/seals-40798 Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, www.library.ethz.ch http://www.e-periodica.ch ELEMENTE DER MATHEMATIK Revue de mathematiques elementaires - Rivista di matematica elementare Zeitschrift zur Pflege der Mathematik und zur Förderung des mathematisch-physikalischen Unterrichts El. Math. Vol. 43 Nr. 2 Seiten 33-64 Basel, März 1988 In memoriam Alexander M. Ostrowski (1893 bis 1986) vt+tttJä&r^ Bleistift-Zeichnung von Peter Birkhauser Am 20. -
Werner Fenchel, a Pioneer in Convexity Theory and a Migrant Scientist
Normat. Nordisk matematisk tidskrift 61:2–4, 133–152 Published in March 2019 Werner Fenchel, a pioneer in convexity theory and a migrant scientist Christer Oscar Kiselman Abstract. Werner Fenchel was a pioneer in introducing duality in convexity theory. He got his PhD in Berlin, was forced to leave Germany, moved to Denmark, and lived during almost two years in Sweden. In a private letter he explained his views on the development of duality in convexity theory and the many terms that have been used to describe it. The background for his move from Germany is sketched. 1. Introduction Werner Fenchel (1905–1988) was a pioneer in convexity theory and in particular the use of duality there. When asked about his views on the many terms used to express this duality he described in a private letter (1977) the whole development from Legendre and onwards, as well as his preferences concerning the choice of terms. The background for his leaving Germany and moving to Denmark and later to Sweden is sketched in Section 12. 2. Convex sets and convex functions In a vector space E over the field R of real numbers—think of E as the two- dimensional plane, identified with R2, or three-dimensional space, identified with R3—we define, given two points a and b, the segment with endpoints a and b: [a, b] = {(1 − t)a + tb; t ∈ R, 0 6 t 6 1}. A subset A of E is said to be convex if {a, b} ⊂ A implies that [a, b] ⊂ A. The modern theory of convex sets starts with the work of Hermann Minkowski (1864–1909); see his book (1910), a large part of which was published in 1896. -
Appendix a I
Appendix A I Appendix A Professional Institutions and Associations AVA: Aerodynamische Versuchsanstalt (see under --+ KWIS) DFG: Deutsche Forschungs-Gemeinschaft (previously --+ NG) German Scientific Research Association. Full title: Deutsche Gemeinschaft zur Erhaltung und Forderung der Forschung (German Association for the Support and Advancement of Sci entific Research). Successor organization to the --+ NG, which was renamed DFG unofficially since about 1929 and officially in 1937. During the terms of its presidents: J. --+ Stark (June 1934-36); R. --+ Mentzel (Nov. 1936-39) and A. --+ Esau (1939-45), the DFG also had a dom inant influence on the research policy of the --+ RFR. It was funded by government grants in the millions and smaller contributions by the --+ Stifterverband. Refs.: ~1entzel [1940]' Stark [1943]c, Zierold [1968], Nipperdey & Schmugge [1970]. DGtP: Deutsche Gesellschaft fiir technische Physik German Society of Technical Physics. Founded on June 6, 1919 by Georg Gehlhoff as an alternative to the --+ DPG with a total of 13 local associations and its own journal --+ Zeitschrift fUr technische Physik. Around 1924 the DGtP had approximately 3,000 members, thus somewhat more than the DPG, but membership fell by 1945 to around 1,500. Chairmen: G. Gehlhoff (1920-31); K. --+ Mey (1931-45). Refs.: Gehlhoff et al. [1920]' Ludwig [1974], Richter [1977], Peschel (Ed.) [1991]' chap. 1, Heinicke [1985]' p. 43, Hoffmann & Swinne [1994]. DPG: Deutsche Physikalische Gesellschaft German Physical Society. Founded in 1899 a national organization at to succeed the Berlin Physical Society, which dates back to 1845. The Society issued regular biweekly proceedings, reports (Berichte) on the same, as well as the journal: Fortschritte der Physik (since 1845). -
Loewner Theory for Quasiconformal Extensions: Old and New
Interdisciplinary Information Sciences J-STAGE Advance published June 7, 2019 #Graduate School of Information Sciences, Tohoku University ISSN 1340-9050 print/1347-6157 online DOI 10.4036/iis.2019.A.01 Loewner Theory for Quasiconformal Extensions: Old and New Ikkei HOTTAÃ Department of Applied Science, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichmu¨ller theory and classical and modern approaches based on Loewner theory. KEYWORDS: Loewner theory, quasiconformal mapping, evolution family, universal Teichmu¨ller space, univalent function 1. Introduction The Loewner differential equation is an ordinary/partial differential equation introduced by Lo¨wner in 1923 for the studyAdvance of conformal mappings in geometric function theory. Originally, a slit mapping, namely, a conformal map on the open unit disk onto the complex plane minus a Jordan arc whose one end point is 1, is dealt with. His work was later developed by Kufarev and Pommerenke to a conformal map whose image is a simply-connected domain on the complex plane. Lo¨wner’s theory has been successfully used to various problems in geometric function theory. In particular the Bieberbach conjecture, an extremal problem of the Taylor coefficients of the normalized univalent functions on the unit disk, was solved with the help of the Loewner’sView method. Recently, the Schramm–Loewner Evolution (SLE), a stochastic variant of the Loewner differential equation discovered by Oded Schramm in 2000, has attracted substantial attention in probability theory and conformal field theory. In this survey article we discuss one of applications of the Loewner differentialProofs equation, quasiconformal extensions of univalent functions. -
NBS-INA-The Institute for Numerical Analysis
t PUBUCATiONS fl^ United States Department of Commerce I I^^^V" I ^1 I National Institute of Standards and Tectinology NAT L INST. OF STAND 4 TECH R.I.C. A111D3 733115 NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R, Hestenes and John Todd -QC- 100 .U57 #730 1991 C.2 i I NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R. Hestenes John Todd Department of Mathematics Department of Mathematics University of California California Institute of Technology at Los Angeles Pasadena, CA 91109 Los Angeles, CA 90078 Sponsored in part by: The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, DC 20036 August 1991 U.S. Department of Commerce Robert A. Mosbacher, Secretary National Institute of Standards and Technology John W. Lyons, Director National Institute of Standards U.S. Government Printing Office For sale by the Superintendent and Technology Washington: 1991 of Documents Special Publication 730 U.S. Government Printing Office Natl. Inst. Stand. Technol. Washington, DC 20402 Spec. Publ. 730 182 pages (Aug. 1991) CODEN: NSPUE2 ABSTRACT This is a history of the Institute for Numerical Analysis (INA) with special emphasis on its research program during the period 1947 to 1956. The Institute for Numerical Analysis was located on the campus of the University of California, Los Angeles. It was a section of the National Applied Mathematics Laboratories, which formed the Applied Mathematics Division of the National Bureau of Standards (now the National Institute of Standards and Technology), under the U.S. -
Mathematiker in Der NS-Zeit
Mathematiker in der NS-Zeit • Thomas Huckle • Technische Universität München • Fakultät für Informatik • Wissenschaftliches Rechnen Augsburg, 21.10.2019 Überblick (1) Mathematik in Deutschland vor 1933 (2) Verfolgung 1933-1945 (3) Schicksale jüdische Mathematiker 1933-1945 (3) Mathematiker im Krieg (1) Mathematik in Deutschland vor 1933 Göttingen als Zentrum von Mathematik und Physik mit Felix Klein, David Hilbert, und den Physikern um Max Born und Ludwig Prandtl. Entwicklung der Grundlagen der - Mathematik (Hilbert Probleme) - allgemeinen Relativitätstheorie (H. Minkowski, D. Hilbert) - Quantenmechanik (Max Born) - Aerodynamik (Ludwig Prandtl) Wikipedia „Hilbert’s Göttingen is still remembered as a Camelot for mathematics and physics. All roads led to Göttingen from 1900 to 1933, just as the mathematical community there had designated a point on the town square ´the origin of the coordinates`.” Yandell, The Honors Class Ernst Hellinger Alfred Haar Hans Minkowski David Hilbert http://news.lib.uchicago.edu/blog/2012/09/21/transcending-tradition-exhibit-opens-oct-4/ Mathematik in Deutschland vor 1933 Andere Schwerpunkte neben Göttingen: - Berlin (Schur, Bieberbach, v.Neumann, ..) - Bonn (Toeplitz, Hausdorff,..) - München (Pringsheim, Perron, Caratheodory) Jüdische Mathematiker waren in herausragenden Stellungen und spielten eine wichtige Rolle in der Gesellschaft: Pringsheim, Hausdorff, … Es fand aber auch schon vor 1933 eine Auswanderung nach USA statt: J. von Neumann, T. v. Kármán, bzw. Emil J. Gumbel. (2) Verfolgung jüdischer Mathematiker - Antijüdische Gesetzgebung - Emigration und Verfolgung - Nationalsozialistische Mathematik 1933 Erste Schritte der Nazis: - Neues Nazi-Gesetz „Wiederherstellung des Berufsbeamtentum“, 7.April: Entlassung aller jüdischen Beamten mit Ausnahme von Wikipedia - WWI Soldaten, - Personen, die seit 1914 angestellt waren, - Personen, die Vater oder Sohn in WWI verloren hatten.