DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter Selected Title s i n Thi s Serie s Volume 5 Emmanue l Hebe y Nonlinear analysi s o n manifolds : Sobole v space s an d inequalitie s 2000 3 Perc y Deif t Orthogonal polynomial s an d rando m matrices : A Riemann-Hilbert approac h 2000 2 Jala l Shata h an d Michae l Struw e Geometric wav e equation s 2000 1 Qin g Ha n an d Fanghu a Li n Elliptic partial differentia l equation s 2000 Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Production Editor Reeva Goldsmith Copy Editor Melissa Macasieb http://dx.doi.org/10.1090/cln/003 Percy Deif t Courant Institute of Mathematical Sciences 3 Orthogona l Polynomials and Random Matrices: A Riemann-Hilbert Approach Courant Institute of Mathematical Sciences New York University New York, New York American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primar y 30-XX , 33-XX , 60-XX , 15A90 , 26Cxx. Library o f Congres s Cataloging-in-Pubiicatio n Dat a Deift, Percy , 1945 - Orthogonal polynomial s an d rando m matrice s : a Riemann-Hiiber t approac h / Perc y Deift . p. cm . — (Couran t lectur e note s ; 3) Originally published : Ne w Yor k : Couran t Institut e o f Mathematica l Sciences , Ne w Yor k University, cl999 . Includes bibliographica l references . ISBN 0-8218-2695- 6 1. Orthogonal polynomials . 2 . Random matrices . I . Title. II . Series . QA404.5 .D37 200 0 515'.55—<lc21 00-06183 4 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , such a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the sourc e i s given . Republication, systemati c copying, or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission should be addressed to the Assistant to the Publisher, America n Mathematical Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionQams.org. © 199 8 held b y the author. Al l rights reserved . Printed i n the Unite d State s o f America . Reprinted b y the America n Mathematica l Society , 200 0 The America n Mathematica l Societ y retains al l right s except thos e granted t o the Unite d State s Government . @ Th e pape r use d i n this boo k i s acid-free an d fall s within the guideline s established t o ensure permanenc e and durability . Visit the AM S hom e page at URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 0 5 04 03 02 To Rebecca and Abby for your patience and support This page intentionally left blank Contents Preface be Chapter • 1. Riemann-Hilber t Problem s 1 1.1. What Is a Riemann-Hilbert Problem ? 1 1.2. Examples 4 Chapter • 2. Jacob i Operators 13 2.1. Jacobi Matrices 13 2.2. The Spectrum of Jacobi Matrices 23 2.3. The Toda Flow 25 2.4. Unbounded Jacobi Operators 26 2.5. Appendix: Suppor t of a Measure 35 Chapter • 3. Orthogona l Polynomials 37 3.1. Construction o f Orthogonal Polynomial s 37 3.2. A Riemann-Hilbert Proble m 43 3.3. Some Symmetry Consideration s 49 3.4. Zeros of Orthogonal Polynomials 52 Chapter • 4. Continue d Fractions 57 4.1. Continued Fraction Expansion of a Number 57 4.2. Measure Theory and Ergodic Theory 64 4.3. Application to Jacobi Operators 76 4.4. Remarks on the Continued Fraction Expansion o f a Number 85 Chapter • 5. Rando m Matrix Theory 89 5.1. Introduction 89 5.2. Unitary Ensembles 91 5.3. Spectral Variables for Hermitian Matrices 94 5.4. Distribution o f Eigenvalues 101 5.5. Distribution o f Spacings of Eigenvalues 113 5.6. Further Remarks on the Nearest-Neighbor Spacin g Distribution an d Universality 120 Chapter • 6. Equilibriu m Measures 129 6.1. Scaling 129 6.2. Existence of the Equilibrium Measure fi v 134 6.3. Convergence of kx* 145 vii Vlll CONTENTS 6.4. Convergence of ^3li(x\)dxi 149 6.5. Convergence of r\x* 159 6.6. Variational Problem for the Equilibrium Measure 167 6.7. Equilibrium Measure for V(x) — tx 2m 169 6.8. Appendix: The Transfinite Diameter and Fekete Sets 179 Chapter • 7. Asymptotic s for Orthogonal Polynomials 181 7.1. Riemann-Hilbert Problem: The Precise Sense 181 7.2. Riemann-Hilbert Problem for Orthogonal Polynomials 189 7.3. Deformation o f a Riemann-Hilbert Proble m 191 7.4. Asymptotics o f Orthogonal Polynomials 201 7.5. Some Analytic Considerations o f Riemann-Hilbert Problem s 208 7.6. Construction o f the Parametrix 213 7.7. Asymptotics o f Orthogonal Polynomials on the Real Axis 230 Chaptei• 8. Universalit y 237 8.1. Universality 237 8.2. Asymptotics of Ps 251 Bibliography 259 Preface In the academic yea r 1996-1997 , I gave a course a t the Courant Institut e o n Riemann-Hilbert problems , orthogona l polynomials , an d rando m matri x theory . The lectures for the course were taken down and organized into note form by Ran- dall Pyke , Joh n Podesta , Jos e Ramirez , an d Wen-qin g Xu . Ove r th e las t year , Jinho Baik, Thomas Kriecherbauer, an d Ken McLaughlin have helped m e furthe r to bring these notes into their present form. Withou t their help, these notes would never hav e been published , an d I a m trul y thankfu l t o al l thes e peopl e fo r thei r efforts. I gave the course in 1996-1997 in an attempt to understand from a more rigor- ous mathematical point of view various results and formulae in Mehta's wonderfu l book Random Matrices [43]. At the same time, I was stimulated and challenged by a set of questions fro m Pete r Sarnak, wh o himself wa s trying to understand [43]. These notes are in many ways a response to his questions, and I deeply appreciat e his clear insights and ready help. The central question is the following: Wh y do very general ensembles o f ran- dom n x n matrices exhibit universal behavior as n —> oc ? My work and that of my collaborators Thomas Kriecherbauer, Ke n McLaugh- lin, Stephanos Venakides, and Xin Zhou on this question is reported in [15,16,17]. Apart from certain additional preparatory material , these notes are a pedagogic il- lustration o f the general methods an d results in [15,16,17], in a special case (see Sections 7 and 8) in which the technical difficulties ar e at a minimum. I thank my colleagues for allowing me to reproduce these results here. Pioneerin g mathemat - ical wor k o n universality fo r rando m matri x ensemble s wa s don e b y Pastu r an d Scherbina in [51], and Its and Bleher in [5]. In additio n t o the student s an d colleague s mentione d above , I would lik e t o thank Dais y Caldero n fo r he r skil l an d patienc e i n typin g th e final manuscript . Special thanks are also due to Melissa Macasieb for her expert copy-editing o f the text, and to Melissa and Reeva Goldsmith for their care in correcting the TgK file. The final figures were drawn by Daisy Calderon. Th e entire project o f preparin g the manuscript fo r publicatio n wa s overseen b y Paul Monsour, an d many , man y thanks are due to him for his great expertise and all his help. This work was supported in part by NSF Grant DMS-9500867. This page intentionally left blank This page intentionally left blank Bibliography [1] Abramowitz , M., an d Stegun, I. A., eds. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York, 1972. [2] Akhiezer , N. I. The classical moment problem and some related questions in analysis. Trans- lated by N. Kemmer. Hafner, Ne w York, 1965. [3] Beals , R., an d Coifman, R . Scatterin g an d inverse scatterin g fo r first order operators. Comm. PureAppl. Math. 37: 39-90, 1984 . [4] Beals , R. , Deift , P. , and Tomei , C . Direct and inverse scattering on the line. Mathematica l Surveys and Monographs, 28. American Mathematical Society, Providence, R.I., 1988. [5] Bleher , P. , an d Its , A . R . Semiclassica l asymptotic s o f orthogona l polynomials , Riemann - Hilbert problems, and universality in the matrix model. Ann. of Math. (2) 150: 185-266, 1999. [6] Clancey , K., and Gohberg, I. Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3, Birkhauser, Basel-Boston, 1981. [7] Coddington , E. A., and Levinson, N. Theory of ordinary differential equations.
Load more

Recommended publications
  • CONTEMPORARY MATHEMATICS 458 Integrable Systems and Random Matrices in Honor of Percy Deift
    CONTEMPORARY MATHEMATICS 458 Integrable Systems and Random Matrices In Honor of Percy Deift Conference on Integrable Systems, Random Matrices, and Applications in Honor of Percy Deift' s 60th Birthday May 22-26, 2006 Courant Institute of Mathematical Sciences New York University, New York Jinho Baik Thomas Kriecherbauer Luen-Chau Li Kenneth D. T-R Mclaughlin Carlos Tomei Editors http://dx.doi.org/10.1090/conm/458 Integrable Systems and Random Matrices In honor of Percy Deift CoNTEMPORARY MATHEMATICS 458 Integrable Systems and Random Matrices In honor of Percy Deift Conference on Integrable Systems, Random Matrices, and Applications in Honor of Percy Deift' s 60th Birthday May 22-26,2006 Courant Institute of Mathematical Sciences New York University, New York Jinho Baik Thomas Kriecherbauer Luen-Chau Li Kenneth D. T-R Mclaughlin Carlos Tomei Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor George Andrews Abel Klein 2000 Mathematics Subject Classification. Primary 15A52, 35Q15, 35Q55, 35Q58, 37K15, 37K40, 42C05, 60K35, 60G60. Library of Congress Cataloging-in-Publication Data Integrable systems and random matrices : In honor of Percy Deift, a conference on integrable systems, random matrices, and applications in honor of Percy Deift's 60th birthday, May 22-26, 2006 / Jinho Baik ... [eta!.]. p. em. -(Contemporary mathematics, ISSN 0271-4132; v. 458) Includes bibliographical references. ISBN 978-0-8218-4240-9 (alk. paper) 1. Hamiltonian systems-Congresses. 2. Random matrices-Congresses. I. Baik, Jinho, 1973- 11. Deift, Percy, 1945- QA614 .83I662 2008 5151.39--dc22 2008007009 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given.
    [Show full text]
  • Contemporary Mathematics 122
    CONTEMPORARY MATHEMATICS 122 Inverse Scattering and Applications http://dx.doi.org/10.1090/conm/122 Contemporary Mathematics Standing orders are accepted for Contemporary Mathematics as well as other book series published by the American Mathematical Society. If you are interested in receiving purchasing information on each new volume in the Contemporary Mathematics senes as it is published, please write or call the American Mathematical Society. Customer Services American Mathematical Society Post Office Box 6248 Providence, Rhode Island 02940-6248 1-800-321-4AMS (321-4267) Titles in This Series Volume 1 Markov random fields and their 13 Algebraists' homage: Papers in applications, Ross Kindermann and ring theory and related topics, J. Laurie Snell S. A. Amitsur, D. J. Saltman, and 2 Proceedings of the conference on G. B. Seligman, Editors integration, topology, and geometry in 14 Lectures on Nielsen fixed point theory, linear spaces, William H. Graves, Editor Boju Jiang 3 The closed graph and P-closed 15 Advanced analytic number theory. graph properties in general topology, Part 1: Ramification theoretic methods, T. R. Hamlett and L. L. Herrington Carlos J. Moreno 4 Problems of elastic stability and 16 Complex representations of GL(2, K) for vibrations, Vadim Komkov, Editor finite fields K, llya Piatetski-Shapiro 5 Rational constructions of modules 17 Nonlinear partial differential equations, for simple Lie algebras, George B. Joel A. Smaller, Editor Seligman 18 Fixed points and nonexpansive 6 Umbral calculus and Hopf algebras, mappings, Robert C. Sine, Editor Robert Morris, Editor 19 Proceedings of the Northwestern 7 Complex contour integral homotopy theory conference, Haynes representation of cardinal spline R.
    [Show full text]
  • A Festschrift in Honor of Barry Simon's 60Th Birthday Quantum Field Theory, Statistical Mechanics, and Nonrelatmstic Quantum Systems
    http://dx.doi.org/10.1090/pspum/076.1 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Quantum Field Theory, Statistical Mechanics, and NonrelatMstic Quantum Systems Proceedings of Symposia in PURE MATHEMATICS Volume 76, Part 1 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Quantum Field Theory, Statistical Mechanics, and NonrelatMstic Quantum Systems A Conference on Spectral Theory and Mathematical Physics in Honor of Barry Simon's 60th Birthday March 27-31, 2006 California Institute of Technology Pasadena, California Fritz Gesztesy (Managing Editor) Percy Deift Cherie Galvez Peter Perry Wilhelm Schlag Editors 2000 Mathematics Subject Classification. Primary 35J10, 35P05, 47A55, 47A75, 47D08, 81Q15, 81T10, 81Uxx, 82B05, 82B10. Library of Congress Cataloging-in-Publication Data Spectral theory and mathematical physics : a festschrift in honor of Barry Simon's 60th birth• day : Quantum field theory, statistical mechanics, and nonrelativistic quantum systems / Fritz Gesztesy ... [et ah], editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 76, pt. 1) Includes bibliographical references. ISBN-13: 978-0-8218-4248-5 (alk. paper) (Part 1) ISBN-13: 978-0-8218-3783-2 (alk. paper) (Set) 1. Spectral theory (Mathematics)—Congresses. I. Simon, Barry, 1946- II. Gesztesy, Fritz, 1953- QC20.7.S646S64 2006 515/.7222—dc22 2006047073 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given.
    [Show full text]
  • 82Nd Josiah Willard Gibbs Lecture Monday, January 5, 2009 8:30 PM
    82nd Josiah Willard Gibbs Lecture Monday, January 5, 2009 8:30 PM JOSIAH WILLARD GIBBS 1839–1903 Photograph by Pictorial Mathematics Scripta Mathematica Yeshiva College, New York, 1942 American Mathematical Society Josiah Willard Gibbs Lecture Monday, January 5, 2009 8:30 PM Marriott Ballrooms 1 & 2 Marriott Wardman Park Washington, DC Integrable Systems: A Modern View Percy Deift Courant Institute of Mathematical Sciences New York University To commemorate the name of Professor Gibbs, the American Mathematical Society established an honorary lectureship in 1923 to be known as the Josiah Willard Gibbs Lectureship. The lectures are of a semipopular nature and are given by invitation. They are usually devoted to mathematics or its applications. It is hoped that these lectures will enable the public and the academic community to become aware of the contribu- tion that mathematics is making to present-day thinking and to modern civilization. Abstract The modern theory of integrable systems began with the solution of the Korteweg-de Vries equation by Gardner, Greene, Kruskal, and Miura in 1967. This led to the development of a variety of new mathematical tech- niques. Over time, and quite unexpectedly, these techniques have found applications in areas far beyond their dynamical origins. The applications include problems in algebraic geometry, numerical analysis, analytic num- ber theory, combinatorics and random matrix theory, among many others. In the lecture, the speaker will present some of these techniques and de- scribe some of their applications. 2 Percy Deift Percy Deift is a Professor of Mathematics at the Courant Institute, NYU. He obtained a master’s degree in chemical engineering from the University of Natal, Durban, South Africa, in 1970 and a master’s degree in physics from Rhodes University, Grahamstown, South Africa, in 1971.
    [Show full text]
  • Combinatorics and Random Matrix Theory
    GRADUATE STUDIES IN MATHEMATICS 172 Combinatorics and Random Matrix Theory Jinho Baik Percy Deift 8SY½G7YMHER American Mathematical Society Combinatorics and Random Matrix Theory https://doi.org/10.1090//gsm/172 GRADUATE STUDIES IN MATHEMATICS 172 Combinatorics and Random Matrix Theory Jinho Baik Percy Deift To Y½c 7Yidan %merican MathematicaP 7ociety Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 05A15, 15B52, 33E17, 35Q15, 41A60, 47B35, 52C20, 60B20, 60K35, 82C23. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-172 Library of Congress Cataloging-in-Publication Data Names: Baik, Jinho, 1973- | Deift, Percy, 1945- | Suidan, Toufic Mubadda, 1975- Title: Combinatorics and random matrix theory / Jinho Baik, Percy Deift, Toufic Suidan. Description: Providence, Rhode Island : American Mathematical Society, 2016. | Series: Graduate studies in mathematics ; volume 172 | Includes bibliographical references and index. Identifiers: LCCN 2015051274 | ISBN 9780821848418 (alk. paper) Subjects: LCSH: Random matrices. | Combinatorial analysis. | AMS: Combinatorics – Enumera- tive combinatorics – Exact enumeration problems, generating functions. msc | Linear and mul- tilinear algebra: matrix theory – Special matrices – Random matrices. msc | Special functions (33-XX deals with the properties of functions as functions) – Other special functions – Painlev´e- type functions. msc | Partial differential equations
    [Show full text]
  • Arxiv:2011.12335V2 [Math-Ph] 5 May 2021 References 155
    TWELVE TALES IN MATHEMATICAL PHYSICS: AN EXPANDED HEINEMAN PRIZE LECTURE BARRY SIMON1;2 Abstract. This is an extended version of my 2018 Heineman prize lecture describing the work for which I got the prize. The citation is very broad, so this describes virtually all my work prior to 1995 and some afterwards. It discusses work in non-relativistic quantum mechanics, constructive quantum field theory and statis- tical mechanics. Contents 0. Introduction 2 1. Summability of Divergent Eigenvalue Perturbation Series 2 2. Complex Scaling Theory of Resonances 12 3. Statistical Mechanical Methods in EQFT 19 4. Thomas{Fermi Theory 25 5. Infrared Bounds and Continuous Symmetry Breaking 32 6. N{Body quantum mechanics 45 7. Magnetic Fields in NRQM 80 8. Quasi-classical and Non{quasi-classical limits 86 9. Almost Periodic and Ergodic Schr¨odinger Operators 103 10. Topological Methods in Condensed Matter Physics 122 11. Anderson Localization: The Simon-Wolff Criterion 133 12. Generic Singular Continuous Spectrum 144 13. Further Remarks 152 arXiv:2011.12335v2 [math-ph] 5 May 2021 References 155 Date: May 7, 2021. 2010 Mathematics Subject Classification. Primary: 81Q10, 81T08, 82B24; Sec- ondary: 47A55, 81Q15, 81Q20, 81Q70, 81T25, 81U24. Key words and phrases. Simon, Schr¨odingeroperators, quantum mechanics, quantum field theory, statistical mechanics. 1 Departments of Mathematics and Physics, Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: [email protected]. 2 Research supported in part by NSF grants DMS-1265592 and DMS-1665526 and in part by Israeli BSF Grant No. 2014337. 1 2 B. SIMON 0. Introduction The citation for my 2018 Dannie Heineman prize for Mathematical Physics reads: for his fundamental contributions to the mathematical physics of quantum mechanics, quantum field theory, and statistical mechanics, including spectral theory, phase transitions, and geometric phases, and his many books and monographs that have deeply influenced generations of researchers.
    [Show full text]
  • Mathematics People
    NEWS Mathematics People equations, quadratic forms and elliptic curves. His work Munshi Awarded ICTP-IMU also makes clear that he is far from done, and that we Ramanujan Prize should expect to see many more interesting results from him in the future.” Ritabrata Munshi of the Indian Munshi received his PhD from Princeton University in Statistical Institute and the Tata In- 2006 under the direction of Andrew Wiles. His honors in- stitute of Fundamental Research has clude the 2015 Shanti Swarup Bhatnagar Prize for Science been awarded the 2018 Ramanujan and Technology in mathematical sciences and the 2017 Prize for Young Mathematicians Infosys Prize in Mathematical Sciences. from Developing Countries for his The Ramanujan Prize is awarded annually to a young outstanding work in number theory. researcher from a developing country. The prize carries The prize is awarded by the Abdus a cash award of US$15,000, and the recipient is invited to Salam International Centre for Theo- deliver a lecture at ICTP. retical Physics (ICTP), the Interna- Ritabrata Munshi tional Mathematical Union (IMU), —From an ICTP announcement and the Department of Science and Technology of the Government of India. The prize citation reads: “Ritabrata Munshi has made profound contributions to analytic number theory, in par- Sisto Receives 2018 Duszenko ticular to the study of analytic properties of L-functions and automorphic forms. L-functions were defined in Award great generality by Robert Langlands, and while much is Alessandro Sisto of ETH Zürich known about them from the representation theoretic and has been named the recipient of the arithmetic geometry points of view, their deeper analytic 2018 Duszenko Award for his sig- properties are largely unknown.
    [Show full text]
  • Spectral Theory and Mathematical Physics
    http://dx.doi.org/10.1090/pspum/076.2 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Ergodic Schrodinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory Proceedings of Symposia in PURE MATHEMATICS Volume 76, Part 2 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Ergodic Schrodinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory A Conference on Spectral Theory and Mathematical Physics in Honor of Barry Simon's 60th Birthday March 27-31, 2006 California Institute of Technology Pasadena, California Fritz Gesztesy (Managing Editor) Percy Deift Cherie Galvez Peter Perry Wilhelm Schlag Editors ^(fl^^fm American Mathematical Society IVflf1f1f|S//l Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 34A55, 34F05, 34L05, 34L40, 37H15, 42C05, 47A10, 47B36, 60H25, 81Q10. Library of Congress Cataloging-in-Publication Data Spectral theory and mathematical physics : a festschrift in honor of Barry Simon's 60th birthday : Ergodic Schrodinger operators, singular spectrum, orthogonal polynomials, and inverse spectral theory / Fritz Gesztesy... [et al.], editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 76, pt. 2) Includes bibliographical references. ISBN-13: 978-0-8218-4249-2 (alk. paper) (Part 2) ISBN-13: 978-0-8218-3783-2 (alk. paper) (Set) 1. Spectral theory (Mathematics)—Congresses. I. Simon, Barry, 1946— II. Gesztesy, Fritz, 1953- QC20.7.S646S64 2006 515/.7222—dc22 2006047073 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given.
    [Show full text]
  • The Toda Rarefaction Problem
    The Toda Rarefaction Problem PERCY DEIFT Courant Institute SPYRIDON KAMVISSIS Universite' de Paris XI11 Ecole Normale Superieure THOMAS KRIECHERBAUER Universitat Augsburg AND XIN ZHOU Duke University Abstract In the Toda shock problem (see [7], (111, [S], and also 131) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas. We make use of the Riemann-Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 21al of the driving particle is sufficiently large (la1 > l), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops. 0 1996 John Wiley & Sons, Inc. 1. Introduction The Toda rarefaction problem concerns the long-time behavior of the solution of the following initial-boundary-value problem, where xu(?)is prescribed by xdt) = 2at, for all t, xA0) = 0, for n 2 1, yJ0) = 0, for n 2 1, and the constant a < 0. Communications on Pure and Applied Mathematics, Vol. XLIX, 3543 (1996) 0 1996 John Wiley & Sons, Inc. CCC 001 0-3640/96/01003549 36 P.
    [Show full text]
  • Acknowledgements List of Sponsors
    Acknowledgements The ICM 2006 is held under the auspices of the International Mathematical Union and the sponsorship of the following public and academic bodies and private companies and foundations. List of Sponsors Public Bodies Ministerio de Educaci´ony Ciencia Ministerio de Asuntos Exteriores y de Cooperaci´on Agencia Espa˜nolade Cooperaci´onInternacional Ministerio de Cultura Biblioteca Nacional Direcci´onGeneral de Comunicaci´ony Cooperaci´onCultural Sociedad Estatal de Conmemoraciones Culturales Comunidad de Madrid Consejer´ıade Educaci´on Instituto Madrile˜node Desarrollo Ayuntamiento de Madrid Madrid-Convention Bureau Concejal´ıa de las Artes Centro Cultural Conde Duque Consejo Superior de Investigaciones Cient´ıficas Correos Fundaci´onCarolina Fundaci´onEspa˜nola para la Ciencia y la Tecnolog´ıa Academic Bodies Universidad Aut´onomade Madrid Universitat de Barcelona Universitat Aut`onoma de Barcelona Universidad de Castilla-La Mancha Universidad Complutense de Madrid Universidad Nacional de Educaci´ona Distancia Universidad de Sevilla Association for Women in Mathematics Canadian Mathematical Society Real Sociedad Matem´atica Espa˜nola Royal Dutch Mathematical Society Sociedad de Estad´ıstica e Investigaci´onOperativa Sociedad Espa˜nolade Matem´atica Aplicada Societat Catalana de Matem`atiques Soci´et´eMath´ematique de France Irish Mathematical Society Mathematical Society of Japan National Council of Teachers of Mathematics, USA Sociedad Espa˜nolade Investigaci´onen Educaci´onMatem´atica Federaci´onEspa˜nola de Sociedades de Profesores de Matem´aticas Sociedad Espa˜nolade Historia de las Ciencias y de las T´ecnicas Private Companies and Foundations Enterasys Fundaci´onPedro Barri´ede la Maza Fundaci´onRam´onAreces Fundaci´onVodafone Grupo SM, Editorial SM ONCE Springer The King Juan Carlos I of Spain Center of N.
    [Show full text]
  • IAMP News Bulletin October 2018
    IAMP News Bulletin October 2018 International Association of Mathematical Physics Contents International Association of Mathematical Physics News Bulletin, October 2018 Contents Prizes from the IAMP3 In Noether’s Words: Invariant Variational Problems5 Noether’s landmark paper one hundred years later 10 Noether’s Cosmology 28 Treasurer’s report 48 News from the IAMP Executive Committee 52 Contact Coordinates for this Issue 55 Bulletin Editor Editorial Board Evans Harrell Rafael Benguria, Virginie Bonnaillie-Noel,¨ Yasuyuki Kawahigashi, Manfred Salmhofer, Robert Sims Contacts. http://www.iamp.org and e-mail: [email protected] Cover picture: One Hundred Years of Noether’s Theorem The views expressed in this IAMP News Bulletin are those of the authors and do not necessarily represent those of the IAMP Executive Committee, Editor or Editorial Board. Any complete or partial performance or reproduction made without the consent of the author or of his successors in title or assigns shall be unlawful. All reproduction rights are henceforth reserved, and mention of the IAMP News Bulletin is Mobligatory in the reference. (Art.L.122-4 of the Code of Intellectual Property). Φ ISSN 2304-7348 News Bulletin (International Association of Mathematical Physics) 2 ∩IAMP News Bulletin, October 2018 Prizes from the IAMP Prizes from the IAMP On July 23, at the 2018 International Congress in Montreal, Canada, the International Associ- ation of Mathematical Physics (IAMP) awarded the 2018 Henri Poincare´ Prizes for mathemat- ical physics to Michael Aizenman, Princeton
    [Show full text]
  • 1 Elliptic Partial Differential Equations Second Edition
    COURANT 1 QING HAN LECTURE FANGHUA LIN NOTES Elliptic Partial Differential Equations Second Edition American Mathematical Society Courant Institute of Mathematical Sciences Elliptic Partial Differential Equations Second Edition http://dx.doi.org/10.1090/cln/001 Qing Han University of Notre Dame Fanghua Lin Courant Institute of Mathematical Sciences 1 Elliptic Partial Differential Equations Second Edition Courant Institute of Mathematical Sciences New York University New York, New York American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 35–01, 35Jxx. For additional information and updates on this book, visit www.ams.org/bookpages/cln-1 Library of Congress Cataloging-in-Publication Data Han, Qing. Elliptic partial differential equations / Qing Han, Fanghua Lin. — 2nd ed. p. cm. — (Courant lecture notes ; v. 1) Includes bibliographical references. ISBN 978-0-8218-5313-9 (alk. paper) 1. Differential equations, Elliptic. I. Lin, Fanghua. II. Title. QA377.H3182 2011 515.3533—dc22 2010051489 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected].
    [Show full text]