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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. -
Algebraic Topology - Wikipedia, the Free Encyclopedia Page 1 of 5
Algebraic topology - Wikipedia, the free encyclopedia Page 1 of 5 Algebraic topology From Wikipedia, the free encyclopedia Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Contents 1 The method of algebraic invariants 2 Setting in category theory 3 Results on homology 4 Applications of algebraic topology 5 Notable algebraic topologists 6 Important theorems in algebraic topology 7 See also 8 Notes 9 References 10 Further reading The method of algebraic invariants An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex ). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. -
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. -
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]. -
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 -
Optimal Control for Automotive Powertrain Applications
Universitat Politecnica` de Valencia` Departamento de Maquinas´ y Motores Termicos´ Optimal Control for Automotive Powertrain Applications PhD Dissertation Presented by Alberto Reig Advised by Dr. Benjam´ınPla Valencia, September 2017 PhD Dissertation Optimal Control for Automotive Powertrain Applications Presented by Alberto Reig Advised by Dr. Benjam´ınPla Examining committee President: Dr. Jos´eGalindo Lucas Secretary: Dr. Octavio Armas Vergel Vocal: Dr. Marcello Canova Valencia, September 2017 To the geniuses of the past, who built our future; to those names in the shades, bringing our knowledge. They not only supported this work but also the world we live in. If you can solve it, it is an exercise; otherwise it's a research problem. | Richard Bellman e Resumen El Control Optimo´ (CO) es esencialmente un problema matem´aticode b´us- queda de extremos, consistente en la definici´onde un criterio a minimizar (o maximizar), restricciones que deben satisfacerse y condiciones de contorno que afectan al sistema. La teor´ıade CO ofrece m´etodos para derivar una trayectoria de control que minimiza (o maximiza) ese criterio. Esta Tesis trata la aplicaci´ondel CO en automoci´on,y especialmente en el motor de combusti´oninterna. Las herramientas necesarias son un m´etodo de optimizaci´ony una representaci´onmatem´aticade la planta motriz. Para ello, se realiza un an´alisiscuantitativo de las ventajas e inconvenientes de los tres m´etodos de optimizaci´onexistentes en la literatura: programaci´ondin´amica, principio m´ınimode Pontryagin y m´etodos directos. Se desarrollan y describen los algoritmos para implementar estos m´etodos as´ıcomo un modelo de planta motriz, validado experimentalmente, que incluye la din´amicalongitudinal del veh´ıculo,modelos para el motor el´ectricoy las bater´ıas,y un modelo de motor de combusti´onde valores medios. -
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. -
Agms20170007.Pdf
This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Le Donne, Enrico Title: A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries Year: 2017 Version: Please cite the original version: Le Donne, E. (2017). A Primer on Carnot Groups: Homogenous Groups, Carnot- Carathéodory Spaces, and Regularity of Their Isometries. Analysis and Geometry in Metric Spaces, 5(1), 116-137. https://doi.org/10.1515/agms-2017-0007 All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Anal. Geom. Metr. Spaces 2017; 5:116–137 Survey Paper Open Access Enrico Le Donne* A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries https://doi.org/10.1515/agms-2017-0007 Received November 18, 2016; revised September 7, 2017; accepted November 8, 2017 Abstract: Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. -
1 Background and History 1.1 Classifying Exotic Spheres the Kervaire-Milnor Classification of Exotic Spheres
A historical introduction to the Kervaire invariant problem ESHT boot camp April 4, 2016 Mike Hill University of Virginia Mike Hopkins 1.1 Harvard University Doug Ravenel University of Rochester Mike Hill, myself and Mike Hopkins Photo taken by Bill Browder February 11, 2010 1.2 1.3 1 Background and history 1.1 Classifying exotic spheres The Kervaire-Milnor classification of exotic spheres 1 About 50 years ago three papers appeared that revolutionized algebraic and differential topology. John Milnor’s On manifolds home- omorphic to the 7-sphere, 1956. He constructed the first “exotic spheres”, manifolds homeomorphic • but not diffeomorphic to the stan- dard S7. They were certain S3-bundles over S4. 1.4 The Kervaire-Milnor classification of exotic spheres (continued) • Michel Kervaire 1927-2007 Michel Kervaire’s A manifold which does not admit any differentiable structure, 1960. His manifold was 10-dimensional. I will say more about it later. 1.5 The Kervaire-Milnor classification of exotic spheres (continued) • Kervaire and Milnor’s Groups of homotopy spheres, I, 1963. They gave a complete classification of exotic spheres in dimensions ≥ 5, with two caveats: (i) Their answer was given in terms of the stable homotopy groups of spheres, which remain a mystery to this day. (ii) There was an ambiguous factor of two in dimensions congruent to 1 mod 4. The solution to that problem is the subject of this talk. 1.6 1.2 Pontryagin’s early work on homotopy groups of spheres Pontryagin’s early work on homotopy groups of spheres Back to the 1930s Lev Pontryagin 1908-1988 Pontryagin’s approach to continuous maps f : Sn+k ! Sk was • Assume f is smooth. -
Arxiv:1604.08579V1 [Math.MG] 28 Apr 2016 ..Caatrztoso I Groups Lie of Characterizations 5.1
A PRIMER ON CARNOT GROUPS: HOMOGENOUS GROUPS, CC SPACES, AND REGULARITY OF THEIR ISOMETRIES ENRICO LE DONNE Abstract. Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups. Contents 0. Prototypical examples 4 1. Stratifications 6 1.1. Definitions 6 1.2. Uniqueness of stratifications 10 2. Metric groups 11 2.1. Abelian groups 12 2.2. Heisenberg group 12 2.3. Carnot groups 12 2.4. Carnot-Carathéodory spaces 12 2.5. Continuity of homogeneous distances 13 3. Limits of Riemannian manifolds 13 arXiv:1604.08579v1 [math.MG] 28 Apr 2016 3.1. Tangents to CC-spaces: Mitchell Theorem 14 3.2. Asymptotic cones of nilmanifolds 14 4. Isometrically homogeneous geodesic manifolds 15 4.1. Homogeneous metric spaces 15 4.2. Berestovskii’s characterization 15 5. A metric characterization of Carnot groups 16 5.1. Characterizations of Lie groups 16 Date: April 29, 2016. 2010 Mathematics Subject Classification. 53C17, 43A80, 22E25, 22F30, 14M17. This essay is a survey written for a summer school on metric spaces. 1 2 ENRICO LE DONNE 5.2. A metric characterization of Carnot groups 17 6. Isometries of metric groups 17 6.1.