EPADEL a Semisesquicentennial History, 1926-2000
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QUARTIC CM FIELDS 1. Background the Study of Complex Multiplication
QUARTIC CM FIELDS WENHAN WANG Abstract. In the article, we describe the basic properties, general and specific properties of CM degree 4 fields, as well as illustrating their connection to the study of genus 2 curves with CM. 1. Background The study of complex multiplication is closely related to the study of curves over finite fields and their Jacobian. Basically speaking, for the case of non-supersingular elliptic curves over finite fields, the endomorphism ring is ring-isomorphic to an order in an imaginary quadratic extension K of Q. The structure of imaginary extensions of Q has beenp thoroughly studied, and the ringsq of integers are simply generated by f1; Dg if D ≡ 1 mod 4, f D g ≡ or by 1; 4 if D 0 mod 4, where D is the discriminant of the field K. The theory of complex multiplication can be carried from elliptic curves to the (Jacobians) of genus 2 (hyperelliptic) curves. More explicitly, the Jacobian of any non-supersingular genus 2 (and hence, hyperelliptic) curve defined over a finite field has CM by an order in a degree 4, or quartic extension over Q, where the extension field K has to be totally imaginary. Description of the endomorphism ring of the Jacobian of a genus 2 curve over a finite field largely depends on the field K for which the curve has CM by. Many articles in the area of the study of genus two curves lead to the study of many properties of the field K. Hence the main goal of this article is, based on the knowledge of the author in the study of the genus 2 curves over finite fields, to give a survey of various, general or specific, properties of degree 4 CM fields. -
Local Time for Lattice Paths and the Associated Limit Laws
Local time for lattice paths and the associated limit laws Cyril Banderier Michael Wallner CNRS & Univ. Paris Nord, France LaBRI, Université de Bordeaux, France http://lipn.fr/~banderier/ http://dmg.tuwien.ac.at/mwallner/ https://orcid.org/0000-0003-0755-3022 https://orcid.org/0000-0001-8581-449X Abstract For generalized Dyck paths (i.e., directed lattice paths with any finite set of jumps), we analyse their local time at zero (i.e., the number of times the path is touching or crossing the abscissa). As we are in a discrete setting, the event we analyse here is “invisible” to the tools of Brownian motion theory. It is interesting that the key tool for analysing directed lattice paths, which is the kernel method, is not directly applicable here. Therefore, we in- troduce a variant of this kernel method to get the trivariate generating function (length, final altitude, local time): this leads to an expression involving symmetric and algebraic functions. We apply this analysis to different types of constrained lattice paths (mean- ders, excursions, bridges, . ). Then, we illustrate this approach on “bas- ketball walks” which are walks defined by the jumps −2, −1, 0, +1, +2. We use singularity analysis to prove that the limit laws for the local time are (depending on the drift and the type of walk) the geometric distribution, the negative binomial distribution, the Rayleigh distribution, or the half- normal distribution (a universal distribution up to now rarely encountered in analytic combinatorics). Keywords: Lattice paths, generating function, analytic combinatorics, singularity analysis, kernel method, gen- eralized Dyck paths, algebraic function, local time, half-normal distribution, Rayleigh distribution, negative binomial distribution 1 Introduction This article continues our series of investigations on the enumeration, generation, and asymptotics of directed lattice arXiv:1805.09065v1 [math.CO] 23 May 2018 paths [ABBG18a, ABBG18b, BF02, BW14, BW17, BG06, BKK+17, Wal16]. -
The Walk of Life Vol
THE WALK OF LIFE VOL. 03 EDITED BY AMIR A. ALIABADI The Walk of Life Biographical Essays in Science and Engineering Volume 3 Edited by Amir A. Aliabadi Authored by Zimeng Wan, Mamoon Syed, Yunxi Jin, Jamie Stone, Jacob Murphy, Greg Johnstone, Thomas Jackson, Michael MacGregor, Ketan Suresh, Taylr Cawte, Rebecca Beutel, Jocob Van Wassenaer, Ryan Fox, Nikolaos Veriotes, Matthew Tam, Victor Huong, Hashim Al-Hashmi, Sean Usher, Daquan Bar- row, Luc Carney, Kyle Friesen, Victoria Golebiowski, Jeffrey Horbatuk, Alex Nauta, Jacob Karl, Brett Clarke, Maria Bovtenko, Margaret Jasek, Allissa Bartlett, Morgen Menig-McDonald, Kate- lyn Sysiuk, Shauna Armstrong, Laura Bender, Hannah May, Elli Shanen, Alana Valle, Charlotte Stoesser, Jasmine Biasi, Keegan Cleghorn, Zofia Holland, Stephan Iskander, Michael Baldaro, Rosalee Calogero, Ye Eun Chai, and Samuel Descrochers 2018 c 2018 Amir A. Aliabadi Publications All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. ISBN: 978-1-7751916-1-2 Atmospheric Innovations Research (AIR) Laboratory, Envi- ronmental Engineering, School of Engineering, RICH 2515, University of Guelph, Guelph, ON N1G 2W1, Canada It should be distinctly understood that this [quantum mechanics] cannot be a deduction in the mathematical sense of the word, since the equations to be obtained form themselves the postulates of the theory. Although they are made highly plausible by the following considerations, their ultimate justification lies in the agreement of their predictions with experiment. —Werner Heisenberg Dedication Jahangir Ansari Preface The essays in this volume result from the Fall 2018 offering of the course Control of Atmospheric Particulates (ENGG*4810) in the Environmental Engineering Program, University of Guelph, Canada. -
The Kronecker-Weber Theorem
The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension of the rational numbers Q is con- tained in a cyclotomic extension. Recall that an abelian extension is a finite field extension K/Q such that the galois group Gal(K/Q) th is abelian, and a cyclotomic extension is an extension of the form Q(ζ), where ζ is an n root of unity. This paper consists of two proofs of the Kronecker-Weber theorem. The first is rather involved, but elementary, and uses the theory of higher ramification groups. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. An Elementary Proof Now we will present an elementary proof of the Kronecker-Weber theoerem, in the spirit of Hilbert’s original proof. The particular strategy used here is given as a series of exercises in Marcus [1]. Minkowski’s Theorem We first prove a classical result due to Minkowski. Theorem. (Minkowski) Any finite extension of Q has nonzero discriminant. In particular, such an extension is ramified at some prime p ∈ Z. Proof. Let K/Q be a finite extension of degree n, and let A = OK be its ring of integers. Consider the embedding: r s A −→ R ⊕ C x 7→ (σ1(x), ..., σr(x), τ1(x), ..., τs(x)) where the σi are the real embeddings of K and the τi are the complex embeddings, with one embedding chosen from each conjugate pair, so that n = r + 2s. -
Nominations for President
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society September 2013 Volume 60, Number 8 The Calculus Concept Inventory— Measurement of the Effect of Teaching Methodology in Mathematics page 1018 DML-CZ: The Experience of a Medium- Sized Digital Mathematics Library page 1028 Fingerprint Databases for Theorems page 1034 A History of the Arf-Kervaire Invariant Problem page 1040 About the cover: 63 years since ENIAC broke the ice (see page 1113) Solve the differential equation. Solve the differential equation. t ln t dr + r = 7tet dt t ln t dr + r = 7tet dt 7et + C r = 7et + C ln t ✓r = ln t ✓ WHO HAS THE #1 HOMEWORK SYSTEM FOR CALCULUS? THE ANSWER IS IN THE QUESTIONS. When it comes to online calculus, you need a solution that can grade the toughest open-ended questions. And for that there is one answer: WebAssign. WebAssign’s patent pending grading engine can recognize multiple correct answers to the same complex question. Competitive systems, on the other hand, are forced to use multiple choice answers because, well they have no choice. And speaking of choice, only WebAssign supports every major textbook from every major publisher. With new interactive tutorials and videos offered to every student, it’s not hard to see why WebAssign is the perfect answer to your online homework needs. It’s all part of the WebAssign commitment to excellence in education. Learn all about it now at webassign.net/math. 800.955.8275 webassign.net/math WA Calculus Question ad Notices.indd 1 11/29/12 1:06 PM Notices 1051 of the American Mathematical Society September 2013 Communications 1048 WHAT IS…the p-adic Mandelbrot Set? Joseph H. -
Herman Heine Goldstine
Herman Heine Goldstine Born September 13, 1913, Chicago, Ill.; Army representative to the ENIAC Project, who later worked with John von Neumann on the logical design of the JAS computer which became the prototype for many early computers-ILLIAC, JOHNNIAC, MANIAC author of The Computer from Pascal to von Neumann, one of the earliest textbooks on the history of computing. Education: BS, mathematics, University of Chicago, 1933; MS, mathematics, University of Chicago, 1934; PhD, mathematics, University of Chicago, 1936. Professional Experience: University of Chicago: research assistant, 1936-1937, instructor, 1937-1939; assistant professor, University of Michigan, 1939-1941; US Army, Ballistic Research Laboratory, Aberdeen, Md., 1941-1946; Institute for Advanced Study, Princeton University, 1946-1957; IBM: director, Mathematics Sciences Department, 1958-1965, IBM fellow, 1969. Honors and Awards: IEEE Computer Society Pioneer Award, 1980; National Medal of Science, 1985; member, Information Processing Hall of Fame, Infornart, Dallas, Texas, 1985. Herman H. Goldstine began his scientific career as a mathematician and had a life-long interest in the interaction of mathematical ideas and technology. He received his PhD in mathematics from the University of Chicago in 1936 and was an assistant professor at the University of Michigan when he entered the Army in 1941. After participating in the development of the first electronic computer (ENIAC), he left the Army in 1945, and from 1946 to 1957 he was a member of the Institute for Advanced Study (IAS), where he collaborated with John von Neumann in a series of scientific papers on subjects related to their work on the Institute computer. In 1958 he joined IBM Corporation as a member of the research planning staff. -
Volume 9, Number 4 N~NSLETTER July-August 1979 PRESIDENT' S REPORT Duluth Meeting. the 1979 AWM Summer Meeting Will Be Held at T
Volume 9, Number 4 N~NSLETTER July-August 1979 **************************************************************************************** PRESIDENT' S REPORT Duluth meeting. The 1979 AWM summer meeting will be held at the joint mathematics meetings at the University of Minnesota in Duluth. All of our scheduled events will be on Thursday, August 23. They are: a panel discussion at 4 p.m. in Bohannon 90 on "Math education: a feminist per- spective." Moderator: Judy Roltman, University of Kansas Panelists: Lenore Blum, Mills College Deborah Hughes Hallett, Harvard University Diane Resek, San Francisco State University We also hope to have materials to present from a re-entry program in the De Anza Community College District (California). a business meeting at 5 p.m. in Bohannon 90 (following the panel). Agenda: new by-laws Goal: to get a members' consensus on new by-laws to be voted on in the fall by mail ballot. a wine and cheese party at 8 p.m. in the lounge of the Kirby Student Center. There will also be an AWM table staffed throughout the meeting. Volunteers are needed to staff it - check with us when you come to the meetings. Come on by and visitl bring your friends too. Also in Duluth: the MAA's HedrIck Lectures will be presented by Mary Ellen Rudin of the Universlty of Wisconsin, Madison. By-laws. The by-laws will be written up by mld-July, and available from the AWM office. Suggestions and amendments are welcome. The procedure is that they will be presented at Duluth for amendment and non-blnding sense-of-the-meeting comments. -
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. -
The Vile, Dopy, Evil and Odious Game Players
The vile, dopy, evil and odious game players Aviezri S. Fraenkel¤ Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100, Israel February 7, 2010 Wil: We have worked and published jointly, for example on graph focality. Now that one of us is already an octogenarian and the other is only a decade away, let's have some fun; let's play a game. Gert: I'm game. Wil: In the Fall 2009 issue of the MSRI gazette Emissary, Elwyn Berlekamp and Joe Buhler proposed the following puzzle: \Nathan and Peter are playing a game. Nathan always goes ¯rst. The players take turns changing a positive integer to a smaller one and then passing the smaller number back to their op- ponent. On each move, a player may either subtract one from the integer or halve it, rounding down if necessary. Thus, from 28 the legal moves are to 27 or to 14; from 27, the legal moves are to 26 or to 13. The game ends when the integer reaches 0. The player who makes the last move wins. For example, if the starting integer is 15, a legal sequence of moves might be to 7, then 6, then 3, then 2, then 1, and then to 0. (In this sample game one of the players could have played better!) Assuming both Nathan and Peter play according to the best possible strategy, who will win if the starting integer is 1000? 2000?" Let's dub it the Mark game, since it's due to Mark Krusemeyer according to Berlekamp and Buhler. -
S, Soc 'At 'On For' Omen In, F Mat Cs
s,soc 'at 'on for' omen in, f mat cs Office Address: Box "178, Wellesley College, Wellesley, Massachusetts 02 ~8~ Telephone: 617-235-0320 Ext. 2643 Volume 18, Number 2 NEWSLETTER March-April 1988 PRESIDENT'S REPORT Dr. Marguerite Lehr, Professor Emerita of Mathematics at Bryn Mawr College, died on December 14, 1987. I first met Miss Lehr when I was interviewing for a position at Bryn Mawr. She enthusiastically attended my talk and most of our colloquia for the next few years; some of you may remember her from the panel discussion at the Emmy Noether Symposium. Several years ago, Pat Kenschaft wrote an article about Miss Lehr that appeared in this Newsletter. Pat tells me that Marguerite liked the article very much, and so we are reprinting it in this issue, as a tribute to an extraordinary woman. Con~atulations and welcome to our new officers, President-Elect Jill Mesirov, Treasurer Jenny Baglivo, and Members-at-Large Rebekka Struik and Carol Wood. Also, many thanks to outgoing officers Vivienne Mayes and Evelyn Silvia (Members-at-Large) and Treasurer Lyrmell Stern. Lynnell has devoted considerable time to AWM, and we are most grateful for the expert job she has done. Atlanta. After the tranquility of Salt Lake City, the Atlanta meetings seemed somewhat hectic. AWM activities were very exciting, with large audiences attending all of our events. Karen Uhlenbeck (University of Texas at Austin) presented this year's Emmy Noether Lecture, "Moment Maps in Stable Bundles: Where Analysis, Algebra, and Topology Meet." The AWM panel, "Is the Climate for Women in Mathematics Changing?," was moderated by Judith Roitman, with panelists Mary Ellen Rudin, Louise Hay, Karen Uhlenbeck, and Nancy Stanton. -
Matical Society Was Held at Columbia University on Friday and Saturday, April 25-26, 1947
THE APRIL MEETING IN NEW YORK The four hundred twenty-fourth meeting of the American Mathe matical Society was held at Columbia University on Friday and Saturday, April 25-26, 1947. The attendance was over 300, includ ing the following 300 members of the Society: C. R. Adams, C. F. Adler, R. P. Agnew, E.J. Akutowicz, Leonidas Alaoglu, T. W. Anderson, C. B. Allendoerfer, R. G. Archibald, L. A. Aroian, M. C. Ayer, R. A. Bari, Joshua Barlaz, P. T. Bateman, G. E. Bates, M. F. Becker, E. G. Begle, Richard Bellman, Stefan Bergman, Arthur Bernstein, Felix Bernstein, Lipman Bers, D. H. Blackwell, Gertrude Blanch, J. H. Blau, R. P. Boas, H. W. Bode, G. L. Bolton, Samuel Borofsky, J. M. Boyer, A. D. Bradley, H. W. Brinkmann, Paul Brock, A. B. Brown, G. W. Brown, R. H. Brown, E. F. Buck, R. C. Buck, L. H. Bunyan, R. S. Burington, J. C. Burkill, Herbert Busemann, K. A. Bush, Hobart Bushey, J. H. Bushey, K. E. Butcher, Albert Cahn, S. S. Cairns, W. R. Callahan, H. H. Campaigne, K. Chandrasekharan, J. O. Chellevold, Herman Chernoff, Claude Chevalley, Ed mund Churchill, J. A. Clarkson, M. D. Clement, R. M. Cohn, I. S. Cohen, Nancy Cole, T. F. Cope, Richard Courant, M. J. Cox, F. G. Critchlow, H. B. Curry, J. H. Curtiss, M. D. Darkow, A. S. Day, S. P. Diliberto, J. L. Doob, C. H. Dowker, Y. N. Dowker, William H. Durf ee, Churchill Eisenhart, Benjamin Epstein, Ky Fan, J.M.Feld, William Feller, F. G. Fender, A. D. -
Curriculum Vita
CURRICULUM VITA NAME: Mourad E.H. Ismail AFFILIATION: University of Central Florida and King Saud University and ADDRESS: Department of Mathematics University of Central Florida 4000 Central Florida Blvd. P.O. Box 161364 Orlando, FL 32816-1364, USA. Telephones: Cell (407) 770-9959. POSITION: Research Professor at U of Central Florida PERSONAL DATA: Born April 27, 1944, in Cairo, Egypt. Male. Married. Canadian and Egyptian citizen, Permanent resident in the United States. EDUCATION: Ph.D. 1974 (Alberta), M.Sc. 1969 (Alberta), B.Sc. 1964 (Cairo). MATHEMATICAL GENEALOGY: Source: The Mathematics Genealogy Project, http://www.genealogy.ams.org • Yours truly, Ph. D. 1974, Advisor: Waleed Al-Salam. • Waleed Al-Salam, Ph. D. 1958, Advisor: Leonard Carlitz. • Leonard Carlitz, Ph. D. 1930, Advisor: Howard Mitchell. • Howard Mitchell, Ph. D. 1910, Advisor: Oswald Veblen. • Oswald Veblen, Ph. D. 1903, Advisor: Eliakim Hastings Moore. • Eliakim Hastings Moore, Ph. D. 1885, Advisor: H. A. Newton. • H. A. Newton, B. S. Yale 1850, Advisor: Michel Chasles. • Michel Chasles, Ph. D. 1814, Advisor: Simeon Poisson. • Simeon Poisson, Advisor: Joseph Lagrange. • Joseph Lagrange, Ph. D. Advisor: Leonhard Euler. 1 RESEARCH INTERESTS: Approximation theory, asymptotics, combinatorics, integral transforms and operational calculus, math- ematical physics, orthogonal polynomials and special functions. POSITIONS HELD: 2010{2012 Chair Professor, City University of Hong Kong 2003-2012 and 2013-present Professor, University of Central Florida 2008{2016 Distinguished Research