THE ISOPERIMETRIC INEQUALITY the Circle Is Uniquely Characterized
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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 -
A Quantitative Isoperimetric Inequality on the Sphere
A QUANTITATIVE ISOPERIMETRIC INEQUALITY ON THE SPHERE VERENA BOGELEIN,¨ FRANK DUZAAR, AND NICOLA FUSCO ABSTRACT. In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set E. 1. INTRODUCTION Recent years have seen an increasing interest in quantitative isoperimetric inequalities motivated by classical papers by Bernstein and Bonnesen [4, 6] and further developments in [14, 18, 17]. In the Euclidean case the optimal result (see [16, 13, 8]) states that if E is a set of finite measure in Rn then (1.1) δ(E) ≥ c(n)α(E)2 holds true. Here the Fraenkel asymmetry index is defined as jE∆Bj α(E) := min ; jBj where the minimum is taken among all balls B ⊂ Rn with jBj = jEj, and the isoperimetric gap is given by P (E) − P (B) δ(E) := : P (B) The stability estimate (1.1) has been generalized in several directions, for example to the case of the Gaussian isoperimetric inequality [7], to the Almgren higher codimension isoperimetric inequality [2, 5] and to several other isoperimetric problems [3, 1, 9]. In the present paper we address the stability of the isoperimetric inequality on the sphere by Schmidt [21] stating that if E ⊂ Sn is a measurable set having the same measure as a n geodesic ball B# ⊂ S for some radius # 2 (0; π), then P(E) ≥ P(B#); with equality if and only if E is a geodesic ball. Here, P(E) stands for the perimeter of E, that is P(E) = Hn−1(@E) if E is smooth and n ≥ 2 throughout the whole paper. -
Chapter 2. Vectors and Vector Spaces
2.2. Cartesian Coordinates and Geometrical Properties of Vectors 1 Chapter 2. Vectors and Vector Spaces Section 2.2. Cartesian Coordinates and Geometrical Properties of Vectors Note. There is a natural relationship between a point in Rn and a vector in n R . Both are represented by an n-tuple of real numbers, say (x1, x2, . , xn). In sophomore linear algebra, you probably had a notational way to distinguish vectors in Rn from points in Rn. For example, Fraleigh and Beauregard in Linear Algebra, n 3rd Edition (1995), denote the point x ∈ R as (x1, x2, . , xn) and the vector n ~x ∈ R as ~x = [x1, x2, . , xn]. Gentle makes no such notational convention so we will need to be careful about how we deal with points and vectors in Rn, when these topics are together. Of course, there is a difference between points in Rn and vectors in Rn (a common question on the departmental Linear Algebra Comprehensive Exams)!!! For example, vectors in Rn can be added, multiplied by scalars, they have a “direction” (an informal concept based on existence of an ordered basis and the component of the vector with respect to that ordered basis). But vectors don’t have any particular “position” in Rn and they can be translated from one position to another. Points in Rn do have a specific position given by the coordinates of the point. But you cannot add points, multiply them be scalars, and they have neither magnitude nor direction. So the properties which a vector in Rn has are not shared by a point in Rn and vice-versa. -
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. -
Classics in Mathematics Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg Gmbh Herbert Federer Was Born on July 23, 1920, in Vienna
Classics in Mathematics Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg GmbH Herbert Federer was born on July 23, 1920, in Vienna. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley. Affiliated to Brown University, Providence since 1945, he is now Professor Emeritus there. The major part of Professor Federer's scientific effort has been directed to the development of the subject of Geometric Measure Theory, with its roots and applications in classical geometry and analysis, yet in the functorial spirit of modern topology and algebra. His work includes more than thirty research papers published between 1943 and 1986, as well as this book. Herbert Federer Geometric Measure Theory Reprint of the 1969 Edition " Springer Herbert Federer (Professor Emeritus) Department of Mathematics Brown University Providence, RI 02912 USA Originally published as VoI. 153 of the Grundlehren der mathematischen Wissenschaften Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Federer, Herbert: Geometric measure theory / Herbert Federer. - Reprint of the 1969 ed. - Berlin; Heidelberg ; New York; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Grundlehren der mathematischen Wissenschaften ; VoI. 153) (Classies in mathematics) NE: 1. GT Mathematics Subject Classification (1991): 53C65, 46AXX ISBN 978-3-540-60656-7 ISBN 978-3-642-62010-2 (eBook) DOI 10.1007/978-3-642-62010-2 This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfIlm or in any other way, and storage in data banks. -
The Isoperimetric Inequality: Proofs by Convex and Differential Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 20 Issue 2 Article 4 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry Penelope Gehring Potsdam University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Part of the Analysis Commons, and the Geometry and Topology Commons Recommended Citation Gehring, Penelope (2019) "The Isoperimetric Inequality: Proofs by Convex and Differential Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 20 : Iss. 2 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol20/iss2/4 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry Cover Page Footnote I wish to express my sincere thanks to Professor Dr. Carla Cederbaum for her continuing guidance and support. I am grateful to Dr. Armando J. Cabrera Pacheco for all his suggestions to improve my writing style. And I am also thankful to Michael and Anja for correcting my written English. In addition, I am also gradeful to the referee for his positive and detailed report. This article is available in Rose-Hulman Undergraduate Mathematics Journal: https://scholar.rose-hulman.edu/rhumj/ vol20/iss2/4 Rose-Hulman Undergraduate Mathematics Journal VOLUME 20, ISSUE 2, 2019 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry By Penelope Gehring Abstract. The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of ele- mentary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn–Minkowski- Inequality. -
I. Mcmullen's Decomposition and the Steiner Formula the Steiner Formula
I. McMullen’s decomposition and the Steiner formula The Steiner formula describes how the volume of a convex body behaves when the convex body is enlarged in every direction by a given radius r. In particular, the dependence on r is polynomial. It was first described by Jakob Steiner [5] in 1840 for dimensions 2 and 3. The Steiner formula can be deduced easily from a much more general result on the homogeneous decomposition of translation invariant valuations, i.e. maps that behave similar to volume and surface area. The existence of such a decomposition was conjectured by McMullen and then independently proved by McMullen [1], by Meier [2] and by Spiegel [4]. Remarks: • Spiegel’s proof is probably the easiest to understand and uses mostly discrete geometry, see also Schneider’s book [3]. • Since this topic will introduce mixed volumes, it should be one of the first in the seminar. References: [1] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. [2] Christoph Meier, Multilinearität bei Polyederaddition, Arch. Math. (Basel) 29 (1977), no. 2, 210–217 (German). [3] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclo- pedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. [4] Wolfgang Spiegel, Ein Beitrag über additive, translationsinvariante, stetige Eikörperfunktionale, Geometriae Dedicata 7 (1978), no. 1, 9–19 (German). [5] Jakob Steiner, Über parallele Flächen, Bericht über die zur Bekanntmachung geeigneten Ver- handlungen der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1840), 114– 118, available at http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/ schriften/anzeige/index_html?band=08-verh/1840&seite:int=114 (German). -
Arxiv:1711.01585V1 [Math.MG] 5 Nov 2017
SUB-FINSLER HEISENBERG PERIMETER MEASURES AYLA P. SANCHEZ´ Abstract. This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Caratheodory´ metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface area in R3. Using this result, I study two perimeter measures that arise from the study of Finsler normed planes, and provide evidence that Pansu’s conjecture regarding the isoperimetric problem in the sub-Riemannian case appears to hold in the more general sub-Finsler case. This is contrary to the relationship between Finsler and Riemannian isoperimetrices. In particular, I show that for any CC-metric there exist a class of surfaces with Legendrian foliation by CC-geodesics generalizing Pansu’s bubble set, but that even in their natural metric using either perimeter measure, the computed examples of such surfaces have lower isoperimetric ratio than the Pansu bubble set, which has Legendrian foliation by sub-Riemannian geodesics. 1. Introduction 1.1. The isoperimetric problem. In this paper I study the Heisenberg group, which I will consider under exponential 3 1 coordinates H(R) (R ; ∗) , where (x1; y1; z1) ∗ (x2; y2; z2) = (x1 + x2; y1; +y2; z1 + z2 − 2 (x1y2 − x2y1). It has a natural notion of volume as Haar measure, which can be taken to be Lebesgue measure λ. I equip H(R) with a Carnot- Caratheodory´ metric dQ that arises from a norm in the plane jj · jjQ and is in general sub-Finsler. -
Inequalities That Imply the Isoperimetric Inequality
March 4, 2002 Inequalities that Imply the Isoperimetric Inequality Andrejs Treibergs University of Utah Abstract. The isoperimetric inequality says that the area of any region in the plane bounded by a curve of a fixed length can never exceed the area of a circle whose boundary has that length. Moreover, if some region has the same length and area as some circle, then it must be the circle. There are dozens of proofs. We give several arguments which depend on more primitive geometric and analytic inequalities. \The cicrle is the most simple, and the most perfect figure. " Poculus. Commentary on the first book of Euclid's Elements. \Lo cerchio `e perfettissima figura." Dante. In these notes, I will present a few of my favorite proofs of the isoperimetric inequality. It is amusing and very instructive to see that many different ideas can be used to establish the same statement. I will concentrate on proofs based on more primitive inequalities. Several important proofs are omitted (using the calculus of variations, integral geometry or Steiner symmetrization), due to the fact that they have already been discussed by others at previous occasions [B], [BL], [S]. Among all closed curve of length L in the plane, how large can the enclosed area be? Since the regions are enclosed by the curves with the same perimeter, we are asking for the largest area amongst isoperiometric regions. Which curve (or curves) encloses the largest possible area? We could ask the dual question: Among all regions in the plane with prescribed area A, at least how long should the perimiter be? Since the regions all have the same area, we are asking for the smallest length amongst isopiphenic regions.