Department of Mathematics Annual Report Drexel University College of Arts & Sciences 2007 — 2008
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Matrix Theory
Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 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]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
State of the Art and Future Directions of Computational Mathematics And
STANFORD 50: State of the Art & Future Directions of Computational Mathematics & Numerical Computing A conference celebrating the 50th anniversary of George Forsythe’s arrival at Stanford and Gene Golub’s 75th birthday Incorporating the Eighth Bay Area Scientific Computing Day (BASCD) Stanford University March 29–31, 2007 Contents Welcome from the Organizers! 3 Conference Schedule 4 Thursday, March 29, 2007 ...................................... 4 Friday, March 30, 2007 ........................................ 6 Saturday, March 31, 2007 ...................................... 8 Abstracts 9 Posters 20 Graduate Student Posters ...................................... 20 Junior Scientist Posters ....................................... 21 Judges ................................................. 21 Participants 22 FOR SYTHEtation 23 Some of George and Alexandra Forsythe’s Books ......................... 23 Some of Gene Golub’s Books .................................... 24 GENEalogy 25 George Forsythe’s PhD Students .................................. 25 Some of Gene Golub’s Postdocs ................................... 25 Gene’s PhD Students ......................................... 26 Fondly Remembered NA Graduates and Faculty 27 Acknowledgments 28 Sponsors ................................................ 28 Session Chairs ............................................ 29 Organizers ............................................... 29 Welcome from the Organizers! Stanford 50: State of the Art and Future Directions of Computational Mathematics and Numerical -
THE BATEMAN–HORN CONJECTURE: HEURISTICS, HISTORY, and APPLICATIONS Contents 1. Introduction 2 2. Preliminaries 3 2.1. Asymptot
THE BATEMAN{HORN CONJECTURE: HEURISTICS, HISTORY, AND APPLICATIONS SOREN LAING ALETHEIA-ZOMLEFER, LENNY FUKSHANSKY, AND STEPHAN RAMON GARCIA Abstract. The Bateman{Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green{Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman{Horn conjecture, its applications, and its origins. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Asymptotic equivalence 3 2.2. Big-O and little-o notation 4 2.3. The logarithmic integral 4 2.4. Prime number theorem 5 3. A heuristic argument 5 3.1. A single polynomial 6 3.2. Effect of the degree. 7 3.3. A sanity check 7 3.4. Making a correction 8 3.5. More than one polynomial 9 3.6. The Bateman{Horn conjecture 10 4. Historical background 10 4.1. Predecessors of the conjecture 10 4.2. Bateman, Horn, and the ILLIAC 12 5. Why does the product converge? 18 5.1. Infinite products 18 5.2. Algebraic prerequisites 19 5.3. Analytic prerequisites 21 5.4. Convergence of the product 22 6. Single polynomials 24 6.1. Prime number theorem for arithmetic progressions 24 2010 Mathematics Subject Classification. 11N32, 11N05, 11N13. Key words and phrases. prime number, polynomial, Bateman{Horn conjecture, primes in arithmetic progressions, Landau's conjecture, twin prime conjecture, Ulam spiral. SRG supported by a David L. Hirsch III and Susan H. Hirsch Research Initiation Grant, the Institute for Pure and Applied Mathematics (IPAM) Quantitative Linear Algebra program, and NSF Grant DMS-1800123. -
Arxiv:1902.03719V4 [Math.CO] 1 Jul 2021 2.1
LORENTZIAN POLYNOMIALS PETTER BRAND¨ EN´ AND JUNE HUH ABSTRACT. We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge–Riemann rela- tions for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we pro- vide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete con- vex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove that the homogenized multi- variate Tutte polynomial of a matroid is Lorentzian whenever the parameter q satisfies 0 ă q ¤ 1. Consequences are proofs of the strongest Mason’s conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave se- quence. -
Arxiv:1807.08899V4 [Math.NT] 5 Apr 2019 5.2
THE BATEMAN{HORN CONJECTURE: HEURISTICS, HISTORY, AND APPLICATIONS SOREN LAING ALETHEIA-ZOMLEFER, LENNY FUKSHANSKY, AND STEPHAN RAMON GARCIA Abstract. The Bateman{Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green{Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman{Horn conjecture, its applications, and its origins. Contents 1. Introduction 2 2. Preliminaries 3 2.1. Asymptotic equivalence 3 2.2. Big-O and little-o notation 4 2.3. The logarithmic integral 4 2.4. Prime number theorem 5 3. A heuristic argument 5 3.1. A single polynomial 6 3.2. Effect of the degree. 7 3.3. A sanity check 7 3.4. Making a correction 8 3.5. More than one polynomial 9 3.6. The Bateman{Horn conjecture 10 4. Historical background 10 4.1. Predecessors of the conjecture 10 4.2. Bateman, Horn, and the ILLIAC 12 5. Why does the product converge? 18 5.1. Infinite products 18 arXiv:1807.08899v4 [math.NT] 5 Apr 2019 5.2. Algebraic prerequisites 19 5.3. Analytic prerequisites 21 5.4. Convergence of the product 22 6. Single polynomials 24 6.1. Prime number theorem for arithmetic progressions 24 2010 Mathematics Subject Classification. 11N32, 11N05, 11N13. Key words and phrases. prime number, polynomial, Bateman{Horn conjecture, primes in arithmetic progressions, Landau's conjecture, twin prime conjecture, Ulam spiral. SRG supported by a David L. Hirsch III and Susan H.