CONTEMPORARY MATHEMATICS

275

Council for African American Researchers in the

Mathematical Sciences: Volume Ill

CAARMS Fifth Conference for African American Researchers

in the Mathematical Sciences June 22-25, 1999

University of Michigan, Ann Arbor, Michigan

Alfred G. Noel Earl Barnes

Sonya A. F. Stephens Editors

Selected Titles in This Series

275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for African American researchers in the mathematical sciences: Volume III, 2001

274 Ken~ichi Maruyama and John W. Rutter, Editors, Groups of homotopy self-equivalences and related topics, 2001

273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler, Editors, Abelian groups, rings and modules, 2001

272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic forms and their applications, 2000

271 J.P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy methods in algebraic topology, 2001

270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000

269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in dynamics, geometry and topology, 2001

268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000

267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000

266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000

265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000

264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000

263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors,

Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability

on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in

analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. Lopez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjmr Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999

(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/275

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CoNTEMPORARY MATHEMATICS

275

Council for African American Researchers in the

Mathematical Sciences: Volume Ill

CAARMS Fifth Conference for African American Researchers

in the Mathematical Sciences June 22-25, 1999

University of Michigan, Ann Arbor, Michigan

Alfred G. Noel Earl Barnes

Sonya A. F. Stephens Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains research and expository papers presented primarily at the CAARMS5 (5th Conference for African American Researchers in the Mathematical Sci-ences) meeting held June 22-25, 1999 at the University of Michigan in Ann Arbor, Michi-gan and supplemented by presentations made at the CAARMS3 meeting held June 17-20, 1997 at Morgan State University in Baltimore, Maryland. Financial support was provided by the National Security Agency, the University of Michigan, and Morgan State University.

2000 Mathematics Subject Classification. Primary 33F05, 47N40, 68R05, 68R10, 65F25, 11R37, 60K20, 60K25, 68M20, OOA30.

Figure 1 on page 119 of the article, "The response of the upper ocean to surface bouyancy forcing: A characteristic solution to wave propagation," M. Y. Stephens and Z. Liu, is reprinted with permission from Subduction of north Pacific temperature anom-alies: Observation and dynamics, Journal of Physical Oceanography, vol. 29, 1999, pp. 1056-1070. ©copyright 1999 by the American Meteorological Society.

Library of Congress Cataloging-in-Publication Data Conference for African American Researchers in the Mathematical Sciences (5th : 1999 : Ann Arbor, Mich.)

Council for African American Researchers in the Mathematical Sciences. Volume III: Fifth Conference for African American Researchers in the Mathematical Sciences, June 22-25, 1999, University of Michigan, Ann Arbor, Michigan/ Alfred G. Noel, Earl Barnes, Sonya A. F. Stephens, editors.

p. em. -(Contemporary mathematics, ISSN 0271-4132; 275) Includes bibliographical references. ISBN 0-8218-2141-5 (alk. paper) 1. Mathematics-Congresses. I. Noel, Alfred G., 1956- II. Barnes, Earl. III. Stephens,

Sonya A. F., 1970- IV. Council for African American Researchers in the Mathematical Sciences. V. Conference for African American Researchers in the Mathematical Sciences (3rd : 1997 : Bal-timore, Md.) VI. Title. VII. Contemporary mathematics (American Mathematical Society) v. 275.

QA1.C6253 1999 510-dc21 2001022099

© 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Information on copying and reprinting can be found in the back of this volume. Visit the AMS home page at URL: http: I /www. ams. org/

10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01

Contents

Preface ix

I. Research and Expository Papers 1

A lower bound for the chromatic number of a graph EARL R. BARNES 3

The fractional parts of ~ M. R. CURRIE AND E. H. GOINS 13

Ultrafilters and Ramsey theory DENNIS E. DAVENPORT 33

Artin's conjecture and elliptic curves EDRAY HERBER GOINS 39

Chaoticity results for "join the shortest queue" CARL GRAHAM 53

Maintenance of deteriorating machines with probabilistic monitoring and silent failures JULIE SIMMONS IVY AND STEPHEN M. POLLOCK 69

The application of numerical grid generation to problems in computational fluid dynamics BONITA V. SAUNDERS 95

The elementary residual method DESMOND STEPHENS AND GARY HOWELL 107

The response of the upper ocean to surface buoyancy forcing: A characteristic solution to wave propagation MONICA Y. STEPHENS AND ZHENGYU LIU 117

An overview of integro-differential equations and variational Lyapunov method SONYA A. F. STEPHENS AND V. LAKSHMIKANTHAM

II. Papers on Philosophy of Mathematics

The art of mathematics JERRY P. KING

vii

129

137

139

viii CONTENTS

Mathematics is four dimensional WILLIAM A. MASSEY 147

III. Tutorials 159

Compact! A tutorial SCOTT W. WILLIAMS 161

Preface

The Council for African American Researchers in the Mathematical Sciences (CAARMS) is a group dedicated to organizing an annual conference that show-cases the current research primarily, but not exclusively, of African Americans in the mathematical sciences, which includes mathematics, operations research, sta-tistics, and computer science. Held annually since 1995, the Conference for African American Researchers in the Mathematical Sciences (also called CAARMS) has had significant numbers of African American researchers presenting their work in hour long technical presentations and African American graduate students present-ing their current work in an organized poster session. These events create a forum for mentoring and networking, where attendees can meet an audience of African American researchers and graduate students interested in such fields. The members of the council work on other issues such as strategies for increasing the number of doctoral recipients from underrepresented minority groups.

Although the current proceedings are considered to be "Volume III" of this series, Volumes I and II were published under different titles. Volume I is called "African Americans in Mathematics" and contains proceedings from the second CAARMS meeting, called CAARMS2, held June 25-28, 1996 at the Center for Discrete Mathematics and Computer Science (DIMACS) in Piscataway, New Jer-sey. It is also Volume 34 of the Series in Discrete Mathematics and Theoretical Computer Science. Volume II is called "African Americans in Mathematics II" and contains the proceedings from the CAARMS4 meeting, held June 16-19, 1998 at Rice University, in Houston, Texas. It is Volume 252 of the Contemporary Mathe-matics Series of the American Mathematical Society. Henceforth, the current title of Volume III, "Council for African American Researchers in the Mathematical Sci-ences" will be the permanent title for the series. This volume includes research and expository papers presented at both the CAARMS3 meeting held June 17-20, 1997 at Morgan State University in Baltimore, Maryland and at the CAARMS5 meeting held June 22-25 1999 at the University of Michigan in Ann Arbor, Michigan. The work covers a wide range of mathematics and every reader will find something of interest. Some papers are comprehensive surveys of research topics for which no text book exists. Other papers develop mathematics to obtain a deeper theoretical insight into mathematics or to solve problems in manufacturing and oceanography. Finally, some papers discuss new ways to view the totality of mathematics and move be•;')nd the "pure" versus "applied" paradigm. The general style is rather expository which should help the reader.

Those meriting special thanks include the organizers of the CAARMS5 meet-ing: William Massey from Bell Laboratories of Lucent Technologies, Leon Woodson from Morgan State University, and our University of Michigan hosts John Birge,

ix

X PREFACE

currently the Dean of Engineering at Northwestern University, and Robert Meg-ginson of the University of Michigan. William Massey is one of the co-founders of CAARMS and has been one of the main organizers of every CAARMS meeting. John Birge and Robert Megginson respectively represent the joint participation of the University of Michigan's Operations and Industrial Engineering department and the Mathematics department with the CAARMS5 meeting. We also thank the National Security Agency, the University of Michigan, and Morgan State University for their financial support.

Finally we express our deep gratitude to all the anonymous referees and to Christine Thivierge of the American Mathematical Society for their advice and comments.

Alfred G. Noel University of Massachusetts Boston

Earl Barnes Georgia Institute of Technology

Sonya A. F. Stephens Florida A & M University

December 2000

Copying and reprinting. Material in this book may be reproduced by any means for educational 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 acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams. org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

Selected Titles in This Series (Continued from the front of this publication)

251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000

250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer

Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum

affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and Dav!d Royal Larson, Editors, The functional and

harmonic analysis of wavelets and frames, 1999 246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in

dynamics, 1999 245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias L6pez, and Joseph Lipman, Studies in

duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and

education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric

aspects of partial differential equations, 1999 241 Piotr Pragacz, Michal Szurek, and Jaroslaw Wisniewski, Editors, Algebraic

geometry: Hirzebruch 70, 1999 240 Angel Carocca, Victor Gonzalez-Aguilera, and Rubi E. Rodriguez, Editors,

Complex geometry of groups, 1999 239 Jean-Pierre Meyer, Jack Morava, and W. Stephen Wilson, Editors, Homotopy

invariant algebraic structures, 1999 238 Gui-Qiang Chen and Emmanuele DiBenedetto, Editors, Nonlinear partial

differential equations, 1999 237 Thomas Branson, Editor, Spectral problems in geometry and arithmetic, 1999 236 Bruce C. Berndt and Fritz Gesztesy, Editors, Continued fractions: From analytic

number theory to constructive approximation, 1999 235 Walter A. Carnielli and !tala M. L. D'Ottaviano, Editors, Advances in

contemporary logic and computer science, 1999 234 Theodore P. Hill and Christian Houdre, Editors, Advances in stochastic

inequalities, 1999 233 Hanna Nencka, Editor, Low dimensional topology, 1999 232 Krzysztof Jarosz, Editor, Function spaces, 1999 231 Michael Farber, Wolfgang Liick, and Shmuel Weinberger, Editors, Tel Aviv

topology conference: Rothenberg Festschrift, 1999 230 Ezra Getzler and Mikhail Kapranov, Editors, Higher category theory, 1998 229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the

representation theory of finite dimensional algebras, 1998 228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang,

Editors, Operator algebras and operator theory, 1999 227 John McCleary, Editor, Higher homotopy structures in topology and mathematical

physics, 1999 226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation:

Applications to geometry and optimization, 1999

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.

Council for African American Researchers in the Mathematical Sciences: Volume III

Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors

This volume presents research and expository papers presented at the third and fifth meet-ings of the Council for African American Researchers in the Mathematical Sciences (CAARMS). The CAARMS is a group dedicated to organizing an annual conference that showcases the current research primarily, but not exclusively, of African Americans in the mathematical sciences, including mathematics, operations research, statistics, and computer science. Held annually since 1995, significant numbers of researchers have presented their current work in hour-long technical presentations, and graduate students have presented their work in organized poster sessions. The events create an ideal forum for mentoring and networking where attendees can meet researchers and graduate students interested in the same fields. For volumes based on previous CAARMS proceedings, see African Americans in Mathematics II (Volume 252 in the AMS series, Contemporary Mathematics) , and African Americans in Mathematics (Volume 34 in the AMS series, DIMACS).

ISBN 0-8218-2141-5

9 780821 821411 CONM/275