Table Of ContentA Celebration of Mathematical Modeling
A Celebration of
Mathematical Modeling
The Joseph B. Keller Anniversary Volume
Edited by
DAN GIVOLI
Department of Aerospace Engineering,
Technion – Israel Institute of Technology, Haifa, Israel
MARCUS J. GROTE
Department of Mathematics,
University of Basel, Basel, Switzerland
and
GEORGE C. PAPANICOLAOU
Department of Mathematics,
Stanford University, Stanford, California, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6526-1 ISBN 978-94-017-0427-4 (eBook)
DOI 10.1007/978-94-017-0427-4
Printed on acid-free paper
All Rights Reserved
©2004 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2004
Softcover reprint of the hardcover 1st edition 2004
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permission from the Publisher, with the
exception of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
For Joe Keller,
colleague, teacher and friend,
on the occasion of his
eightieth birthday
Table of Contents
Preface ix
Publications of Joseph B. Keller xiii
Monte Carlo Simulation for American Options 1
R. Caflisch and S. Chaudhary
Some Problems in Electromagnetics 17
M. Cheney
Pattern Formation in a Suspension of Swimming
Microorganisms: Nonlinear Aspects 33
R. Childress and E. Spiegel
Non-Reflecting Boundaries: High-Order Treatment 53
D. Givoli
Nonreflecting Boundary Conditions for Time Dependent
Waves 73
M. Grote
Weak Shock Reflection 93
J. Hunter and A.M. Tesdall
Bifurcation Theory, Symmetry Breaking and
Homogenization in Continuum Mechanics Descriptions of DNA 113
J. Maddocks
On Flames as Discontinuity Surfaces in Gasdynamic
Flows 137
B.J. Matkowsky
Contact Lines 161
M. Miksis
Eikonal Functions: Old and New 181
J. Rubinstein and G. Wolansky
vii
viii
Physical Insight, Mathematical Modeling and
Asymptotics 199
L. Ting
Waves, Bubbles and Jets 221
J.-M. Vanden-Broeck
Index 239
Preface
ThisvolumecelebratestheeightiethbirthdayofJosephB.Keller. The
authors who contributed to this volume belong to what can be called
the “Keller school of applied mathematics.” They are former students,
postdoctoral fellows and visiting scientists who have collaborated with
Joe (some of them still do) during his long career. They all look at Joe
as their ultimate (role) model.
JoeKeller’sdistinguishedcareerhasbeendividedbetweentheCourant
Institute of Mathematical Sciences at New York University, where he
received all his degrees (his PhD adviser being the great R. Courant
himself) and served as a professor for 30 years, and Stanford University,
where he has been since 1978. The appended photos highlight some
scenes from the old days.
Those who know Joe Keller’s work have been always amazed by its
diversity and breadth. It is considered a well-known truth that there
is not a single important area in applied mathematics or physics which
Keller did not contribute to. This can be appreciated, for example, by
glancing through his list of publication included in this volume. Appro-
priately, the papers in this book, written with Joe’s inspiration, cover
a variety of application areas; together they span the broad subject of
mathematical modeling. The models discussed in the book describe the
behavior of various systems such as those related to finance, waves, mi-
croorganisms, shocks, DNA, flames, contact, optics, fluids, bubbles and
jets. Joe’s activity includes many more areas, which unfortunately are
not represented here.
Joe Keller is undoubtly one of the greatest applied mathematicians of
the 20th century, and has won almost any possible prize. His consider-
able abilities have been well summarized by Bernard Matkowsky in the
June 1996 issue of SIAM News: “He has the ability to see clearly the
essential elements of a physical phenomenon in a particular discipline,
and then to pose the simplest possible mathematical model to describe
the phenomenon. Keller then uses (and more often than not invents)
sophisticated mathematical methods to solve the problem. These meth-
ods can be used for problems other than the one for which they were
developed, and thus become available to mathematicians, scientists, and
engineers worldwide. Finally he extracts the physical implications of his
mathematical results, so that his work is appreciated not only by math-
ematicians, but also by people in the discipline in which the problem
originated.”
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One major field of interest of Joe is wave propagation. His famous
invention of the Geometrical Theory of Diffraction (GTD) for solving
problems of high-frequency waves started during World War II while he
was working on sonar problems for the Columbia University Division of
War Research. Since then GTD has become an indispensable tool for
waveproblemsinavarietyofapplicationareassuchaselectromagnetics,
acoustics, non-destructive testing of structures, earthquake engineering
and oil exploration. Joe’s contributions include many other important
inventions and methods that will not be mentioned here.
A talk by Joe at a seminar or a conference is an event that should
never be missed. Here is just one example out of many. In a special
lecture which he gave just before Christmas 1987 in his department, Joe
started by saying that during Christmas time many people bring home
toy models (like model airplanes, etc.), and so he would like to give his
audience as a Christmas present a bunch of mathematical models. The
talk itself was superb as always. Some of his “models” were serious but
amusing (like a model for ordering teams in a tournament and a model
for the way a worm advances using a one-dimensional wave equation),
while others were simply jokes. One model concerned a proof that if one
flips a perfect coin one has a 50% chance of getting Head. Joe showed
this by performing the entire calculation of the nonlinear dynamics of
the coin, from the initial flip to the landing on the floor!
For many years Joe received constant funding from federal agencies,
including ONR, AFOSR and the Army. This in itself is not that unique.
Whatisuniqueisthathiscontractwiththeseagenciessimplysaid“Prof.
J.B. Kellerwill study various problems in applied mathematics.” That’s
it! Probably Joe is one of the few researchers in the world who have
received this kind of freedom from the funding agencies, and rightly so.
It is a great honor for the editors and authors of this book to be
associated with this publication. We present this volume to Joe with
infiniteappreciationandaffection,andwishhimmanymorehappyyears
of mathematical modeling.
Dan Givoli, Haifa
Marcus Grote, Basel
George Papanicolaou, Stanford
October 2003
Biographical Sketch of Joseph B. Keller
Joseph B. Keller was born in Paterson, New Jersey in 1923 and ed-
ucated at New York University (Ph.D. 1948). He remained there as a
Professor of Mathematics in the Courant Institute of Mathematical Sci-
ences until 1979. Then he moved to Stanford University where he was
Professor of Mathematics and Mechanical Engineering until 1993, when
he became Professor Emeritus.
His research concerns the use of mathematics to solve problems of
science and engineering. For example, he developed the Geometrical
Theory of Diffraction to describe the propagation of waves. It is widely
used to analyze radar reflection from objects, to calculate elastic wave
scattering from flaws in solids, to study acoustic wave propagation in
the ocean, etc. Another example is his formulation of the EBK method
of quantization to determine energy levels of atoms and molecules in
quantum mechanics and to solve characteristic value problems in other
fields. Other problems he has worked on are described in his list of
publications.
Professor Keller is a member of the National Academy of Science, a
Foreign Member of the Royal Society of London, and he was Honorary
Professor of Mathematical Sciences at the University of Cambridge. He
is a recipient of the Wolf Prize (1997), the Frederick E. Nemmers Prize
(1996),theNASAwardinAppliedMathematicsandNumericalAnalysis
(1995), the National Medal of Science (1988), the Timoshenko Medal
(1984), the Eringen Medal (1981), the von Karman Prize (1979) and he
was the Gibbs Lecturer of the American Mathematical Society (1977),
and the von Neumann Lecturer of SIAM (1983).
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