Current and Future Directions in Applied Mathematics

267 Pages·1997·12.785 MB·English

by  Roger Brockett (auth.), Mark Alber, Bei Hu, Joachim Rosenthal (eds.)

Current and Future
Directions
in Applied Mathematics
MarkAlber
BeiHu
Joachim Rosenthal
Editors
Springer-Science+Business Media, LLC
MarkAlber
BeiHu
Ioachim Rosenthal
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556-5683
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Printed on acid-free paper
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© 1997 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 1997
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ISBN 978-1-4612-7380-6 ISBN 978-1-4612-2012-1 (eBook)
DOI 10.1007/978-1-4612-2012-1
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987 6 543 2 1
Preface
Mark Alber, Bei Hu and Joachim Rosenthal ...................... vii
Part I
Some Remarks on Applied Mathematics
Roger Brockett .................................................... 1
Mathematics is a Profession
Christopher 1. Byrnes ............................................ 4
Comments on Applied Mathematics
Avner Friedman .................................................. 9
Towards an Applied Mathematics for Computer Science
Jeremy Gunawardena ............................................ 11
Infomercial for Applied Mathematics
Darryl Holm .................................................... 15
On Research in Mathematical Economics
M. Ali Khan .................................................... 21
Applied Mathematics in the Computer and Communications
Industry
Brian Marcus ................................................... 25
'frends in Applied Mathematics
Jerrold E. Marsden .............................................. 28
Applied Mathematics as an Interdisciplinary Subject
Clyde F. Martin ................................................. 31
vi Contents
Panel Discussion on Future Directions in Applied Mathe
matics
Laurence R. Taylor .............................................. 38
Part II
Feedback Stabilization of Relative Equilibria for Mechanical
Systems with Symmetry
A.M. Bloch, J.E. Marsden and G. Sanchez ....................... 43
Oscillatory Descent for Function Minimization
R. Brockett ...................................................... 65
On the Well-Posedness of the Rational Covariance Extension
Problem
C.l. Byrnes, H. J. Landau and A. Lindquist ...................... 83
Singular Limits in Fluid Mechanics
P. Constantin ................................................... 109
Singularities and Defects in Patterns Far from Threshold
N.M. Ercolani ................................................... 137
Mathematical Modeling and Simulation for Applications of
Fluid Flow in Porous Media
R.E. Ewing ..................................................... 161
On Loeb Measure Spaces and their Significance for N on
Cooperative Game Theory
M.A. Khan and Y. Sun ......................................... 183
Mechanical Systems with Symmetry, Variational Principles,
and Integration Algorithms
J.E. Marsden and J.M. Wendlandt .............................. 219
Preface
The applied sciences are faced with increasingly complex problems
which call for sophisticated mathematical models. Fast computers make
it possible to optimize strategic objectives and industry is applying mathe
matical models that aim at reducing production costs and increasing prof
itability. Investment and insurance companies rely on complicated stochas
tic models and the communication industry uses encoding and encryption
schemes which are based on algebraic geometry and number theory. Mod
ulational theory for semi-classical solutions of nonlinear equations has been
essential in recent developments of high bit-rate information transmission
and processing systems. Of course, these are only some of numerous fields
where applications of mathematical methods are crucial.
In order to tackle complex problems in the applied sciences there is an
increased demand for interdisciplinary research between mathematicians
and researchers working in engineering, the sciences and business. The
mathematical sciences are undergoing rapid changes and the boundaries
between the mathematical sciences and other disciplines are blurring. Si
multaneously, the job market for research mathematicians in academia has
been under stress for quite some time. In response to the changing envi
ronment, several mathematics departments in the US and in Europe have
started graduate programs in applied mathematics, industrial mathematics
and mathematical finance. At the present time, applied mathematics seems
to be both exciting and promising.
In April of 1996, the applied mathematics group in the Department
of Mathematics at the University of Notre Dame organized a Symposium
on Current and Future Directions in Applied Mathematics. The organizing
committee consisted of Mark Alber, Leonid Faybusovich, Bei Hu, Gerard
Misiolek, Joachim Rosenthal and Hong-Ming Yin. The symposium received
enthusiastic support from other members of the Department of Mathemat
ics.
The intention of the Symposium was to bring together experts in several
different areas of applied mathematics and to create the opportunity for
interactions, exchange of ideas, and discussions on the future of applied
mathematics.
Ten invited speakers delivered 50 minute plenary lectures about current
and future trends in their research field. The lectures were complemented by
a number of workshops which focused on specific research areas in applied
mathematics. The program of the workshops included 53 invited talks and
viii Preface
several discussions. A panel discussion was held on April 20, 1996, about
the role of applied mathematics in the next decade.
Altogether there were 120 participants with 15 international researchers
from Canada, France, Germany, Italy, Japan, Russia and the United King
dom. Researchers from the following Industrial and Government Laborato
ries actively participated in the Symposium: AT&T Research Lab, German
Telekom, Hewlett-Packard Research Lab, IBM Almaden Research Center,
Los Alamos National Lab and NASA Lewis Research Center.
The symposium featured an important educational component. Lec
tures by the invited speakers were prepared in such a way as to provide
students with a review of new results in applied mathematics and a list of
open problems. This was complemented by informal meetings with grad
uate students and a discussion which centered on future opportunities for
young researchers. One of the goals of this volume is to encourage young
people to enter the exciting field of applied mathematics.
The first part of the volume consists of reflections by several partici
pants of the Symposium on changes and important trends both in research
and education. In a second part the plenary speakers provide surveys on
their research fields, as well as new research results.
The Symposium was sponsored by the University of Notre Dame: Cen
ter for Applied Mathematics, College of Science, Department of Mathe
matics, with outside support from BRIMS Hewlett-Packard Research Lab
oratory, CNLS Los Alamos National Laboratory and the National Science
Foundation.
Special thanks go to Laurence R. Taylor, Chair of the Department of
Mathematics at the time of the symposium and Hafiz Atassi, Director of
the Center for Applied Mathematics for their advice and support. We also
want to thank Gregory Luther for the help with organizing the symposium.
Secretarial and organizational work of Fern Martin, Patti Strauch and Rita
Vanderbosch is gratefully acknowledged.
Mark Alber, Bei Hu, Joachim Rosenthal
University of Notre Dame, Notre Dame, Indiana
November, 1996
Part I
Some Remarks on Applied Mathematics *
Roger Brockett t
Encourage students to focus on fundamentals. In mathematics this
means algebra, analysis and geometry, but it is also desirable for an ap
plied mathematician to have a broad education in science and engineering,
including the perspectives found in computer science. It is impossible to
predict what kinds of mathematics will be especially useful for the problems
yet to be encountered. Being prepared to read about the work of others and
having a first hand knowledge of many good examples is about the best one
can do. One of the very positive side effects associated with the increasing
number of scientifically and mathematically literate people in the world is
that there are now readable books and survey papers covering a vast ar
ray of scientific work. It is essential that students who aspire to be model
builders and problem solvers should have the tools necessary to make use
of this resource. This point is well illustrated by recent developments in
mathematical physics involving the use of some of the very latest ideas in
geometry to provide a language suitable for unifying field theories.
When it comes time to pick an area to work in, there are many choices
and each individual will have his/her favorite. Among the exciting chal
lenges that compete for the attention of fledgling applied mathematicians
wanting to work in control theory, the ones that interest me the most are
those that seem to have the potential to extend the applicability of math
ematics into new domains. I include in this category the various attempts
now being made to model the mechanisms animals use to control their
bodies, attempts being made to model the processes of perception (e.g.
mathematical approaches to image understanding) and the emerging mod
els of learning based on the melding of inductive and deductive reasoning in
a probabilistic setting. One of the stumbling blocks in these areas seems to
be the lack of tools for merging the analysis of signals in one two and three
dimensions and the analysis of the "tokens" which emerge when the data
carried by the signal is compressed for abstract processing and/or storage.
The process of reducing an image into a set of objects has, in most cases,
"Received December, 1996
tDivision of Engineering and Applied Sciences, Harvard University, Cambridge, MA
02183
M. Alber et al. (eds.), Current and Future Directions in Applied Mathematics
© Birkhäuser Boston 1997
2 Roger Brockett
a robustness about it, not unlike the invariance's that allow one to reason
about manifolds using homotopy theory. Likewise, methods for specify
ing motion in a partially known world seem to rely on a blending of quite
abstract, high level, ideas together with feedback rules that are sensitive
to specific details. Again one senses a need for theories that encompass
both the discrete and the continuous. There has been interest in this type
of information processing for a long time but I have the sense that only
recently have researchers been willing to acknowledge the depth of ques
tions and begun to focus on modes of analysis that are up to the task.
It may be noted that because these problems are difficult, and the course
uncharted, dealing with them requires conviction and an effective strategy.
The same is true in many areas of applied mathematics. After a problem
is cast in mathematical terms it often happens that the person with the
greatest mastery of the set of techniques will contribute the most. But in
that formative period, in which progress is often slow and unpredictable,
it is useful to spend time thinking about various ways one might define
problems in the general area, rather than delving deeply into a particular
question whose significance might be quite limited.
What should colleges and universities do about applied mathematics?
Some years ago I had a chance conversation with a mathematics graduate
student who expressed the view that because he had taken a course on
the theory of differential equations he could do any engineering problem
involving differential equations and he did not see why he would need to
study engineering to be as useful to a prospective employer as an engineer.
More recently I had a note from a mathematics major saying that she was
graduating in a few months, had not taken any applied mathematics, but
would like to talk to me because she was looking for a job in applied math
ematics. It seems she, too, had been led to believe that training in applied
mathematics was optional even for those who want to do it for a living.
However, as we all know, change is everywhere and institutions, be they
beholden to stockholders or be they nonprofits devoted to higher learn
ing, are constantly engaged in a balancing act, trying to be true to their
traditional values while redefining themselves to maintain relevance. One
frequently reads that the world is becoming more dependent on applications
of mathematics all the time. Isn't it obvious, then, that as a result of the
redefinition there will emerge more and larger applied mathematics depart
ments? I am afraid the answer is no, it is not obvious to everyone. There
is a competing idea; one can let mathematics enter the various disciplines,
as needed, rather than attempting to put it all "under one roof'. Although
Some Remarks on Applied Mathematics 3
the debate centering around the competition for resources between various
disciplines is, in the present climate, especially difficult, there seems to be
some agreement that universities are wise to invest in the here and now
as well as the there and possibly never. Applied sciences are a good way
to do this and applied mathematics especially so. As an academic unit,
applied mathematics requires fewer resources than, say, an engineering de
partment, while also serving the purpose of showing that knowledge has
practical value as well as admirable beauty. Whether applied mathemat
ics is done in mathematics departments, applied mathematics departments,
departments of biology, engineering, physics or chemistry, it seems that stu
dents need more rather than less of the clarity and precision it brings to the
table. In some frameworks it may be best to add mathematical thinking
in a department of biology whereas in others it will make sense to add a
mathematical biologist to an applied mathematics group. There need not
be a universal solution. However, we do need mechanisms that will en
courage the development of specialists and insure that future generations
of doctors, lawyers, engineers and business people are adequately trained
in mathematics and its application to practical problems.