DOCSLIB.ORG

OPSF, Random Matrices and Riemann-Hilbert Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
OPSF, Random Matrices and Riemann-Hilbert Problems OPSF, Random Matrices and Riemann-Hilbert problems Walter Van Assche School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, ICMAT 23{27 October, 2017 Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Plan of the course lecture 1: Orthogonal Polynomials and Random Matrices lecture 2: The Riemann-Hilbert problem for orthogonal polynomials lecture 3: Logarithmic potentials and equilibrium measures lecture 4: Asymptotics and universality Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Orthogonal polynomials We only use absolutely continuous measures: dµ(x) = w(x) dx The monic orthogonal polynomial Pn(x) for the weight w satisfies Z k Pn(x)x w(x) dx = 0; k = 0; 1;:::; n − 1; Z n 1 Pn(x)x w(x) dx = 2 6= 0: γn The Riemann-Hilbert1 problem for orthogonal polynomials gives a very different characterization in terms of a boundary value problem. It was first formulated by Fokas, Its and Kitaev in 1992 to investigate matrix models in 2D quantum gravity. 1Bernhard Riemann (1826{1866), David Hilbert (1862{1943) Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems A scalar Riemann-Hilbert problem First we deal with a scalar Riemann-Hilbert problem, which will be needed later. Find a function f : C ! C for which the following properties hold 1 f is analytic in C n R. 2 On R the boundary values f±(x) = lim!0+ f (x ± i) exist, and they are connected by f+(x) = f−(x) + w(x); x 2 R: 3 The asymptotic behavior is 1 f (z) = O( ); z ! 1: z Does such a function f exist? Is such a function unique? Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems A scalar Riemann-Hilbert problem Theorem (Sokhotsky-Plemelj) Suppose w is H¨oldercontinuous on R and integrable, then −1 Z 1 w(t) f (z) = dt; 2πi −∞ z − t and this is the only solution. −1 Z w(t) −1 Z x − t − i f (x + i) = dt = w(t) dt 2πi x + i − t 2πi (x − t)2 + 2 −1 Z w(t) −1 Z x − t + i f (x − i) = dt = w(t) dt 2πi x − i − t 2πi (x − t)2 + 2 Subtract to find 2i Z w(t) f+(x) − f−(x) = lim dt !0 2πi (x − t)2 + 2 1Julian Sokhotski (1842{1927), Josip Plemelj (1873{1967) Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems A scalar Riemann-Hilbert problem Change variable t = x + s to find Z w(t) 1 Z w(x + s) lim dt = lim ds !0 π (x − t)2 + 2 !0 π s2 + 1 1 Z 1 = w(x) ds = w(x): π 1 + s2 Hence the boundary conditions hold. The asymptotic behavior is also true and the function is analytic in C n R. Unicity: suppose g is another solution and consider f − g f − g is analytic in C n R, it has no jump on R, hence analytic everywhere in C. It is bounded, hence by Liouville's theorem it is a constant, and by the asymptotic behavior g(z) = f (z). Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Riemann-Hilbert problem for orthogonal polynomials 2×2 Find a matrix function Y : C ! C such that 1 Y is analytic in C n R. 2 On R the boundary values lim!0+ Y (x ± i) = Y±(x) exist, and they are related by 1 w(x) Y (x) = Y (x) ; x 2 : + − 0 1 R 3 Asymptotic behaviour 1 zn 0 Y (z) = + O( ) ; z ! 1: I z 0 z−n Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Riemann-Hilbert problem for orthogonal polynomials Suppose w is H¨oldercontinuous, then Z 0 1 Pn(x)w(x) 1 Pn(z) dx Y (z) = B 2πi x − z C @ Z P (x)w(x) A −2πiγ2 P (z) −γ2 n−1 dx n−1 n−1 n−1 x − z where Pn and Pn−1 are the monic orthogonal polynomials for w. Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Unicity of the solution Property For every solution of this RHP one has det Y = 1. Let Yb be a solution of the RHP, and consider YYb −1. −1 YYb is analytic in C n R. YYb −1 has no jump over the real line, hence YYb −1 is analytic in C. −1 1 −1 YYb = I + O( z ), hence by Liouville's theorem YYb = I, and thus Yb = Y . Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Recurrence relation Let Yn denote the solution with Pn in the upper left corner. −1 Consider R = Yn+1Yn , then 1 R is analytic in C n R. 2 R has no jump on the real line, hence R is analytic in C. 3 As z ! 1 z + O(1) O(1) 1 R(z) = + O( ) O(1) 0 z Hence Liouville's theorem gives that z − b c R(z) = n n ; dn 0 Conclusion z − bn cn Yn+1(z) = Yn(z) dn 0 2 Pn+1(z) = (z − bn)Pn(z) − 2πiγn−1cnPn−1(z) Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Liouville-Ostrogradski formula det Y (z) = 1 gives Z P (x)w(x) Z P (x)w(x) γ2 P (z) n−1 dx − P (z) n dx = 1 n−1 n z − x n−1 z − x Z p (x)w(x) In terms of p (z) = γ P (z) and q (z) = n dx n n n n z − x an pn(z)qn−1(z) − pn−1(z)qn(z) = 1: This is known as the Liouville-Ostrogradski formula: Wronskian [Casorati determinant] of two solutions of the recurrence relation. Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Christoffel-Darboux kernel Recall n−1 X Pn(x)Pn−1(y) − Pn−1(x)Pn(y) K (x; y) = γ2P (x)P (y) = γ2 : n k k k n−1 x − y k=0 Pn(x) ∗ −1 ∗ ∗ Yn(x) = 2 ; Yn (y) = 2 −2πiγn−1Pn−1(x) ∗ 2πiγn−1Pn−1(y) Pn(y) 1 0 1 Y −1(y)Y (x) = 2πiγ2 P (x)P (y)−P (x)P (y) n n 0 n−1 n n−1 n−1 n Property 1 1 K (x; y) = 0 1 Y −1(y)Y (x) : n 2πi(x − y) n n 0 Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Motivation for doing asymptotics In random matrix theory one wants to investigate the asymptotic distribution of the eigenvalues for large matrices (n ! 1). Recall that Z ρ1(x) dx = E(Nn(A)); Nn(A) = number of eigenvalues in A: A ρ1(x) = Kn(x; x)w(x): Hence Z 1 Nn(A) lim Kn(x; x)w(x) dx = E lim : n!1 n A n!1 n But more is true Z Z Nn(A) 1 lim = lim Kn(x; x)w(x) dx = v(x) dx n!1 n n!1 n A A and v is called the asymptotic density of the eigenvalues. Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Motivation for doing asymptotics One needs to know the asymptotic behavior of Kn(x; x) as n ! 1. But this only gives the global distribution of the eigenvalues. One really wants to understand the local behavior of the eigenvalues: fix a point in the spectrum, how do the eigenvalues behave near that point? In particular one wants to understand the spacing between eigenvalues. This immediately gives rise to investigating gap probabilities pA(0) = P(no eigenvalues in A): Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Gap probabilities Recall that for disjoint sets k Z Z Z Y ··· ρk (x1;:::; xk ) dx1 ::: dxk = E Nn(Aj ) A1 A2 Ak j=1 k ρk (x1;:::; xk ) = det Kn(xi ; xj ) : i;j=1 If all the sets are the same then k−1 Z Y ρk (x1;:::; xk ) dx1 ::: dxk = E (Nn(A) − j) k A j=0 is the average number of points of the n-point process for which k components are in A. Z 1 1 Nn(A) X j ρk (x1;:::; xk ) dx1 ::: dxk = E = pA(j); k! k k k A j=k where pA(j) = P(j eigenvalues in A). Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Gap probabilities Consider the infinite series 1 k Z 1 1 X (−1) X k X j ρk (x1;:::; xk ) dx1 ::: dxk = (−1) pA(j) k! k k k=0 A k=0 j=k interchange sums 1 j X X j = p (j) (−1)k A k j=0 k=0 Now use j X j (−1)k = δ k 0;j k=0 to find 1 X (−1)k Z ρk (x1;:::; xk ) dx1 ::: dxk = pA(0) = P(Nn(A) = 0): k! k k=0 A Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Gap probabilities 1 X (−1)k Z ρk (x1;:::; xk ) dx1 ::: dxk = pA(0) = P(Nn(A) = 0): k! k k=0 A The sum is the known as the Fredholm2 determinant of the operator Kn;A 1 X (−1)k Z k det(1 − Kn;A) = det Kn(xi ; xj ) dx1 ::: dxk k! k i;j=1 k=0 A with Z (Kn;Af )(x) = Kn(x; y)f (y) dy: A 2Erik Ivar Fredholm, Sweden (1866{1927) Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems Gap probabilities For local analysis around a point x∗ in the spectrum, we need 1 ∗ u ∗ v lim Kn x + ; x + = K(u; v) n!1 nc nγ nγ and the gap probablity will be the Fredholm determinant of KA with Z (KAf )(u) = K(u; v)f (v) dv: A If x∗ is in the bulk of the spectrum, then we have the sine kernel sin π(u − v) K(u; v) = : π(u − v) If x∗ is at the end of the spectrum, then we have the Airy kernel Ai(u)Ai0(v) − Ai0(u)Ai(v) Z 1 K(u; v) = = Ai(u+s)Ai(v+s) ds: u − v 0 Walter Van Assche OPSF, Random Matrices and Riemann-Hilbert problems References P.
Load more

Recommended publications
  • Comparative Literature in Slovenia
    CLCWeb: Comparative Literature and Culture ISSN 1481-4374 Purdue University Press ©Purdue University Volume 2 (2000) Issue 4 Article 11 Comparative Literature in Slovenia Kristof Jacek Kozak University of Alberta Follow this and additional works at: https://docs.lib.purdue.edu/clcweb Part of the Comparative Literature Commons, and the Critical and Cultural Studies Commons Dedicated to the dissemination of scholarly and professional information, Purdue University Press selects, develops, and distributes quality resources in several key subject areas for which its parent university is famous, including business, technology, health, veterinary medicine, and other selected disciplines in the humanities and sciences. CLCWeb: Comparative Literature and Culture, the peer-reviewed, full-text, and open-access learned journal in the humanities and social sciences, publishes new scholarship following tenets of the discipline of comparative literature and the field of cultural studies designated as "comparative cultural studies." Publications in the journal are indexed in the Annual Bibliography of English Language and Literature (Chadwyck-Healey), the Arts and Humanities Citation Index (Thomson Reuters ISI), the Humanities Index (Wilson), Humanities International Complete (EBSCO), the International Bibliography of the Modern Language Association of America, and Scopus (Elsevier). The journal is affiliated with the Purdue University Press monograph series of Books in Comparative Cultural Studies. Contact: <[email protected]> Recommended Citation Kozak, Kristof Jacek. "Comparative Literature in Slovenia." CLCWeb: Comparative Literature and Culture 2.4 (2000): <https://doi.org/10.7771/1481-4374.1094> This text has been double-blind peer reviewed by 2+1 experts in the field. The above text, published by Purdue University Press ©Purdue University, has been downloaded 2344 times as of 11/ 07/19.
    [Show full text]
  • EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]
    CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner
    [Show full text]
  • Implementation of Sustainable Mobility in Education
    University of Ljubljana, Faculty of Arts, Department of Geography GeograFF 23 Implementation of Sustainable Mobility in Education Tatjana Resnik Planinc, Matej Ogrin, Mojca Ilc Klun, Kristina Glojek Ljubljana 2018 Geograff23_FINAL.indd 1 10.5.2017 10:39:08 GeograFF 23 Implementation of Sustainable Mobility in Education Authors/avtorji: Tatjana Resnik Planinc, Matej Ogrin, Mojca Ilc Klun, Kristina Glojek Editor/urednica: Katja Vintar Mally Reviewers/recenzentki: Ana Vovk Korže, Mimi Urbanc Translators/prevajalca: James Cosier, Ana Mihor Published by/založila: Ljubljana University Press, Faculty of Arts/Znanstvena založba Filozofske fakultete Univerze v Ljubljani Issued by/izdal: Department of Geography/Oddelek za geografijo For the publisher/odgovorna oseba: zanjo Roman Kuhar, dean of the Faculty of Arts/ dekan Filozofske fakultete Layout/Prelom: Aleš Cimprič DOI: 10.4312/9789610600145 First edition, Digital edition/prva izdaja, elektronska izdaja Publication is free of charge./Publikacija je brezplačna. Delo je ponujeno pod licenco Creative Commons Attribution-ShareAlike 4.0 International License (priznanje avtorstva, deljenje pod istimi pogoji). Kataložni zapis o publikaciji (CIP) pripravili v Narodni in univerzitetni knjižnici v Ljubljani COBISS.SI-ID=293545984 ISBN 978-961-06-0013-8 (epub) ISBN 978-961-06-0014-5 (pdf) Geograff23_FINAL.indd 2 10.5.2017 10:39:08 Implementation of Sustainable Mobility in Education GeograFF 23 Geograff23_FINAL.indd 3 10.5.2017 10:39:09 Geograff23_FINAL.indd 4 10.5.2017 10:39:09 GeograFF 23 Contents 1 Introductory
    [Show full text]
  • The Pentagon
    THE PENTAGON Volume XXV Spring, 1966 Number~2 CONTENTS Page National Officers 68 Computer Application to Symmetric Double Integration by Hypercubes By )erry L. Lewis 69 Conic Sections with Circles as Focal Points By Thomas M. Potts 78 Concerning Functional Conjugates By Alan R. Grissom 86 Incorporation of Some Mathematical Ideas through Application to An Electrical Circuit By Jerry R. Ridenhour and William B. Chauncey 90 Factoring a Polynomial of the Fourth Degree By R. S. Luthar L06 The Problem Corner 109 Installation of New Chapters 115 The Book Shelf 116 The Mathematical Scrapbook 125 Kappa Mu Epsilon News 128 Eflual Jfomk GMltmustt 3n ifflraortam Carl V. Fronabarger, Past President Members of Kappa Mu Epsilon have been saddened by the knowledge that Dr. Loyal F. Ollmann, National President of Kappa Mu Epsilon, passed from this life on April 8, 1966. Surviving him are his wife, Nila M. (Schwartz) Ollmann, and three children: Naida Jane, Mary Joan, and Loyal Taylor. Loyal F. Ollmann was born on August 28, 1905. He received an A.B. from Ripon College, 1926; a M.S. from the University of Wisconsin, 1928; and a M.A. and a Ph.D. from the University of Michigan in 1938 and 1939, respectively. His professional teaching and administrative experiences in cluded serving as: Assistant Instructor of Physics at the University of Wisconsin, 1926-27; Professor of Physics and Mathematics, Elmhurst College, 1929-36; part-time Instructor in Mathematics, University of Michigan, 1936-39; Instructor of Mathematics, Texas Technological College, 1939-40; Assistant Professor of Mathematics, College of Wooster, 1940-41; and he was associated with Hofstra University from 1941 until the time of his death, first as Associate Professor and then as Head of the Mathematics Department; he was Chairman of the Division of Natural Sciences, Mathematics and Engineering, 1957-61.
    [Show full text]
  • Mathematics in the Austrian-Hungarian Empire
    Mathematics in the Austrian-Hungarian Empire Christa Binder The appointment policy in the Austrian-Hungarian Empire In: Martina Bečvářová (author); Christa Binder (author): Mathematics in the Austrian-Hungarian Empire. Proceedings of a Symposium held in Budapest on August 1, 2009 during the XXIII ICHST. (English). Praha: Matfyzpress, 2010. pp. 43–54. Persistent URL: http://dml.cz/dmlcz/400817 Terms of use: © Bečvářová, Martina © Binder, Christa Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz THE APPOINTMENT POLICY IN THE AUSTRIAN- -HUNGARIAN EMPIRE CHRISTA BINDER Abstract: Starting from a very low level in the mid oft the 19th century the teaching and research in mathematics reached world wide fame in the Austrian-Hungarian Empire before World War One. How this was complished is shown with three examples of careers of famous mathematicians. 1 Introduction This symposium is dedicated to the development of mathematics in the Austro- Hungarian monarchy in the time from 1850 to 1914. At the beginning of this period, in the middle of the 19th century the level of teaching and researching mathematics was very low – with a few exceptions – due to the influence of the jesuits in former centuries, and due to the reclusive period in the first half of the 19th century. But even in this time many efforts were taken to establish a higher education.
    [Show full text]
  • Analysis As a Life: to the 80Th Birthday of Professor H. Begehr
    Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. More information about this series at http://www.springer.com/series/4961 Sergei Rogosin • Ahmet Okay Çelebi Editors Analysis as a Life Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday Editors Sergei Rogosin Ahmet Okay Çelebi Department of Economics Mathematics Department Belarusian State University Yeditepe University Minsk, Belarus Ata¸sehir, Istanbul, Turkey ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-030-02649-3 ISBN 978-3-030-02650-9 (eBook) https://doi.org/10.1007/978-3-030-02650-9 Library of Congress Control Number: 2018967001 © Springer Nature Switzerland AG 2019 This work is subject to copyright.
    [Show full text]
  • On Spectral Properties of Single Layer Potentials Seyed Zoalroshd University of South Florida, [email protected]
    University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 6-28-2016 On Spectral Properties of Single Layer Potentials Seyed Zoalroshd University of South Florida, [email protected] Follow this and additional works at: http://scholarcommons.usf.edu/etd Part of the Mathematics Commons Scholar Commons Citation Zoalroshd, Seyed, "On Spectral Properties of Single Layer Potentials" (2016). Graduate Theses and Dissertations. http://scholarcommons.usf.edu/etd/6445 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. On Spectral Properties of Single Layer Potentials by Seyed M. Zoalroshd A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: D. Khavinson, Ph.D. Catherine A. B´en´eteau,Ph.D. Arthur Danielyan, Ph.D. Sherwin Kouchekian, Ph.D. David Rabson, Ph.D. Date of Approval: March 7, 2016 Keywords: Eigenfunctions, Eigenvalues, Isoperimetric inequality, Plemelj-Sokhotski theorem, Single layer operator, Schatten ideals, Singular numbers. Copyright c 2016, Seyed M. Zoalroshd Dedication To the memory of my beloved mother. Acknowledgments I would like to express my sincere gratitude to my advisor Prof. Dmitry Khavinson for his continuous support, patience and encouragement. Professor Khavinson profoundly changed my perspective on working in the field of mathematics. Besides my advisor, I would like to thank the rest of my thesis committee: Prof.
    [Show full text]
  • Maths Challenges News
    United Kingdom Mathematics Trust Maths Challenges News Issue 40 September 2012 In this UKMT Maths Challenges issue Challenge your pupils and test their problem solving skills by entering your students in the 2012/13 UKMT Maths Challenges. The questions are accessible but are designed to stretch pupils beyond the school curriculum and make them think. 2 Diary dates The Challenges are straightforward to administer in schools, and the papers are sent back to UKMT for marking. Feedback is quick, as results are now emailed (and posted) for 2012/13 and you will receive detailed analysis of your school‘s performance and how it compares to the nation. Extended solutions, including new extension material, are available to download from the UKMT website after the event. Volunteer for UKMT Students are recognised for their performance, with every school not only receiving a certificate to present to their best performer overall, but also Fundraising certificates are now awarded to the best student in each year group. The top 40% of students nationwide receive a gold, silver or bronze certificate (the top 60% for the Senior Challenge). 3 IMO 2012 - High scoring pupils are invited to participate in follow-on competitions. These awards provide an JMC certificate winners from Furze Platt Senior School, Berks The Best excellent opportunity to highlight mathematics in Europe within your school and celebrate your students‘ mathematical success in the local press. Can you and your pupils do these questions from the 2011/12 Challenges? Mentoring Senior Maths Challenge Intermediate Maths Challenge Junior Maths Challenge Schemes A triangle has two edges Alex Erlich and Paneth Farcas Tommy Thomas's tankard of length 5.
    [Show full text]
  • 2018-06-108.Pdf
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions.
    [Show full text]
  • 242 Reviews HM3 Basic Philosophical Language Was, As Was That of His
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector 242 Reviews HM3 basic philosophical language was, as was that of his readers. This explains why he defines new terms such as the various measures of vis centripeta, but does not define other terms, such as actio, reactio, status, or mutatio, that were part of the common philoso- phical language of the time. (As for vis insita and vis impressa, terms that had a long history before Newton, the special senses he gives them cannot be fully understood without reference to their previously accepted meanings. (For example, in reading Definition 111 (Materiae vis insita est potentia resistendi...) one is re- minded of Goclenius’ article on “Vis” (in the general sense): Vis Insita est, vel Violentia. Insita, ut naturalis potestas.) Three short examples will illustrate the point. In Eli and ~~a Newton deleted the opening line of Lex III, Actioni contrariam semper & aequalem esse reactionem, leaving only the sentence Corporum duorum actiones in se mutuo semper esse aequales & in par- tes contrarias dirigi. Whatever the full interpretation of this deletion may be, it seems at least to imply that the term reactio (though not the idea) is not indispensable for a statement of the Law. Of course this is the case, since reactio means simply actio mutua, as the explanatory paragraph accompanying Lex III suggests, and as Micraelius [1653, Actio] explicitly states: Actio mutua est, qua agens repatitur propter contactum corporeum, & dicitur reactio .,. Goclenius’ article on Reactio [Lexicon Philosophicurn 16131 makes the same point in a more developed way: Reactio est retributa seu reciprocata patientis actio quaedam, qua resistit agenti, (renititur adversus agens) & id commutat dum ab eo commutatur.
    [Show full text]
  • Differential Galois Groups Over Laurent Series Fields
    DIFFERENTIAL GALOIS GROUPS OVER LAURENT SERIES FIELDS DAVID HARBATER, JULIA HARTMANN, AND ANNETTE MAIER Abstract. In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of char- acteristic zero. More precisely, we show that any linear algebraic group (i.e. affine group scheme of finite type) over such a Laurent series field does occur as the differential Galois group of a linear differential equation with coefficients in any such function field (of one or several variables). Introduction Differential Galois theory studies linear homogeneous differential equations by means of their symmetry groups, the differential Galois groups. Such a group acts on the solution space of the equation under consideration, and this furnishes it with the structure of a linear algebraic group over the field of constants of the differential field. In analogy to the inverse problem in ordinary Galois theory, an answer to the question of which linear algebraic groups occur as differential Galois groups over a given differential field provides information about the field and its extensions. Classically, differential Galois theory was mostly concerned with one variable function fields over the complex numbers. In that case, the inverse problem is related to the Riemann- Hilbert problem (Hilbert’s 21st problem) about monodromy groups of differential equations. In fact, Tretkoff and Tretkoff ([TT79]) showed that every linear algebraic group over C occurs as the differential Galois group of some equation over C(x), as a consequence of Plemelj’s solution to the (modified) Riemann-Hilbert problem ([Ple08]; see also [AB94]).
    [Show full text]
  • HOMOTECIA No. 2-12 Febrero 2014
    HOMOTECIA Nº 2 – Año 12 Lunes, 3 de Febrero de 2014 1 ¿En cuáles disciplinas donde se forman docentes a nivel universitario, se deben o no incluir en sus respectivos pensum una fuerte carga curricular en matemática? No nos estamos refiriendo a la inclusión de una gran cantidad de asignaturas de contenido matemático sino aun siendo pocas estas, su exigencia sea de un rigor muy significativo. Esta interrogante nos la hacemos como producto de la reflexión sobre las directrices que, hasta los momentos, nos han indicado pautan el diseño de un Currículo por Competencias a implantar y desarrollar lo más pronto posible en nuestra Facultad de Ciencias de la Educación de la Universidad de Carabobo. Para los técnicos curriculares que asesoran este trabajo, pareciera que las matemáticas sólo son necesarias e importantes para los estudiantes que se forman como docentes de matemática, física y química. Siempre hemos apoyado el cambio curricular a un diseño basado en competencias puesto que lo que se pretende es “… aumentar, de manera exponencial, el intercambio de nuevos conocimientos en cantidad, calidad y complejidad, propiciándose con ello nuevos abordajes desde el punto de vista inter y transdisciplinario en el tratamiento y resolución de situaciones problemáticas… (para dar respuesta a) nuevos retos de la educación, con la integración de los saberes, actitudes y el hacer de variadas disciplinas relacionadas con MAX BORN una determinada situación problemática…” (Durant y Naveda, 2012, p. 9: “Transformación curricular por competencias (1882 – 1970) en la educación universitaria bajo el enfoque ecosistémico formativo”). El insigne educador Federico Froebel afirmó Nació el 11 de diciembre de 1882 en Breslau, Alemania, en la “Sin las matemáticas o, por lo menos, sin el conocimiento fundamental del cálculo que se apropia del conocimiento de actualidad llamada Wroclaw y ubicada en Polonia; y murió el 5 de la forma y de la magnitud como condiciones necesarias, la educación del hombre es una obra incompleta.
    [Show full text]