Surveys in Applied Mathematics. Essays Dedicated to S.M. Ulam

300 Pages·1976·15.1 MB·English

by  N. Metropolis, etc

Proceedings of the First Los Alamos
Symposium on Mathematics
in the Natural Sciences
Surveys in
Applied
Mathematics
Essays dedicated to
S.M. Ulam
EDITED BY
N. METROPOLIS
S. ORSZAG
G.-C. ROTA
ACADEMIC PRESS, INC. New York San Francisco London 1976
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1976, BY ACADEMIC PRESS, INC.
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Library of Congress Cataloging in Publication Data
Los Alamos Symposium on Mathematics in the Natural
Sciences, 1st, 1974
Proceedings of the first Los Alamos Symposium on
Mathematics in the Natural Sciences.
(Surveys in applied mathematics)
1. Mathematics-Congresses. 2. Science-Congresses.
I. Metropolis, Nicholas Constantine, Date II. Ors-
zag, S. III. Rota, Gian Carlo, Date IV. Series:
Surveys in applied mathematics (New York, 1976- )
QA1.L588 1974 510 76-43107
ISBN 0-12-492150-7
PRINTED IN THE UNITED STATES OF AMERICA
This volume
is affectionately dedicated to
STANISLAW M.U LAM
who
for many years
dedicated his efforts
to the well being of
The Los Alamos Scientific Laboratory
CONTRIBUTORS
KENNETH BACLAWSKI JAMES GLIMM
Department of Mathematics Courant Institute of Mathematical
Harvard University Sciences
Cambridge, Massachusetts 02138 New York University
New York, New York 10012
LIPMAN BERS
Department of Mathematics LOUIS N.HOWARD
Columbia University Massachusetts Institute of Technology
New York, New York 10027 Cambridge, Massachusetts 021 39
RAOUL BOTT M. KAC
Department of Mathematics The Rockerfel/er University
Harvard University New York, New York 10021
Cambridge, Massachusetts 02138
D.J. KLEITMAN
JAMES H. BRAMBLE Mathematics Department
Department of Mathematics Massachusetts Institute of Technology
Cornell University Cambridge, Massachusetts 02139
Ithaca, New York 14850
ROBERT H.KRAICHNAN
J. D. COLE Dub/in, New Hampshire 03444
Department of Mathematics
University of California, Los Angeles
PETER D. LAX
Los Angeles, California 90024
Courant Institute of Mathematical
Sciences
BRADLEY EFRON
251 Mercer Street
Department of Mathematics New York, New York 10012
Stanford University
Stanford, California 94305 C.C.LIN
Massachusetts Institute of Technology
F. FALTIN Cambridge, Massachusetts 02138
Department of Mathematics
Cornell University N.METROPOLIS
Ithaca, New York 14850
Los Alamos Scientific Laboratory
Los Alamos, New Mexico 87544
ix
CONTRIBUTORS
PIERRE VAN MOERBEKE G.-C. ROTA
University of Lou va in Massachusetts Institute of Technology
Louvain, Belgium Cambridge, Massachusetts 02139
J. MOSER FRANK SPITZER
Courant Institute of Mathematical Department of Mathematics
Sciences Cornell University
New York University Ithaca, New York 14850
New York, New York 10012
KENNETH G.WILSON
R. D. RICHTMYER Laboratory of Nuclear Studies
T-Division Cornell University
Los Alamos Scientific Laboratory Ithaca, New York 14850
Los Alamos, New Mexico 87544
B. ROSS
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
X
PREFACE
Since its recent formation, the U.S. Energy Research and Development Adminis-
tration has become involved in scientific programs whose complexity and impene-
trability exceeds those of the past. It was foreseen that these programs would
require mathematical sophistication well beyond that of even ten years ago.
The Los Alamos Scientific Laboratory decided to organize a workshop with two
objectives. It was observed that some of the ongoing projects were not making full
use of the latest mathematics and theoretical physics. Secondly, it was felt that a
small group of outstanding scientists should be invited to Los Alamos and exposed
to some of the applied mathematical aspects of important national problems.
This volume contains the collection of papers based upon most of the lectures
delivered at the workshop. Discussions with Los Alamos staff members were also
an integral part of the workshop. These consisted of panel discussions, wherein
staff members along with invited speakers and visiting staff members participated.
The staff made presentations of the mathematical aspects of problems and ques-
tions in the various areas of Laboratory interest; the invited speakers responded in
general or particular terms depending upon the extent of overlap of interest. The
reports of these discussions are not available for publication at present.
The Committee and all participants in the workshop thank Dr. Milton Rose for
making it possible and lending his support.
N. Metropolis, Chairman
Steven A. Orszag
Gian-Carlo Rota
xi
INTRODUCTION
C.C. Lin in "On the Role of Applied Mathematics" stresses the interaction be-
tween mathematical theory and the physical world. Lin argues the importance of
improved communications between mathematicians and other scientists and engi-
neers. We hope that the present volume helps to achieve this goal.
Raoul Bott in "On the Shape of a Curve" gives an introductory exposition of
topology and algebraic geometry. Bott discusses the "shape" of the solution set to
polynomial equations in several complex variables and explains the relationship of
these results to modern developments in algebraic number theory including the
famous Weil conjectures.
Lipman Bers in "Automorphic Forms for Schottky Groups" addresses the theory
of complex variables via group theory. He provides an introduction to the theory of
Kleinian groups and investigates their application to the deformation theory of
Riemann surfaces.
Bradley Efron in "Biased versus Unbiased Estimation" shows that deliberate
biasing of results can sometimes drastically improve statistical predictions. If the
criterion is used that the statistical predictions should be very good most of the
time but sometimes rather poor, it is possible to introduce statistical functions
alternative to those that are best on the average. Efron investigates properties of
these biased estimators.
DJ. Kleitman in "Algorithms" addresses the classification of problems that can
and cannot be solved by efficient algorithms and illustrates the general results with
a number of specific results concerning multiplication of binary numbers, testing a
graph for planarity, and many other problems. This theory finds very useful appli-
cation in the design of algorithms for modern computers.
Kenneth Baclawski in "Whitney Numbers of Geometric Lattices" uses sheaf
theory to shed new light on the chromatic polynomials of graphs. He shows that
the coefficients of these polynomials are Betti numbers.
R.D. Richtmyer in "Continued Fraction Expansion of Algebraic Numbers" pre-
sents some computer tests of the conjecture that irrational algebraic numbers of
degree ^3 have continued fraction expansion coefficients that satisfy Khintchine's
law.
Frank Spitzer in "Random Time Evolution of Infinite Particle Systems" gives a
synopsis of the latest developments in the theory of behavior of infinite coupled
sets of dynamical equations. He states conditions for ergodic and nonergodic
behavior.
xiii
INTRODUCTION
Louis N. Howard in "Bifurcation in Reaction-Diffusion Problems" surveys classi-
cal bifurcation theorems. A bifurcation theorem gives conditions such that sta-
tionary and periodic solutions of differential equations be stable to small perturba-
tions. These theorems are important in such diverse fields as celestial mechanics,
hydrodynamics, solid mechanics, etc. Howard presents some new general results
and then applies the theory to the study of the Belousov-Zhabotinskii reaction, an
oscillatory chemical reaction that propagates as a wave.
J.D. Cole in "Singular Perturbation" gives a heuristic introduction to the modern
methods of perturbation theory. Cole gives representative applications of limit
process expansions (boundary layer problems) and multiple-scale problems. Cole
emphasizes the physical characteristics of the problems that are soluble by these
methods.
James H. Bramble in "A Survey of Some Finite Element Methods Proposed for
Treating the Dirichlet Problem" discusses several alternative methods for imposition
of boundary conditions in finite element approximations to elliptic boundary value
problems. Finite element methods are being actively applied to many kinds of
boundary value problems arising in solid mechanics and elsewhere.
James Glimm in "The Mathematics of Quantum Fields" gives a terse, clear intro-
duction to this intricate subject. The goal of quantum field theory is to obtain a
fundamental description of elementary particles by combining quantum mechanics
with special relativity in a se If-consistent way. Glimm emphasizes what is now
proved, what kinds of mathematics have been used to study quantum fields, and
what the status is of various unsolved problems.
Kenneth G. Wilson in "Renormalization Group Methods" gives an elementary
introduction to the renormalization group and its application to the solution of the
Kondo problem. In recent years, Wilson and his co-workers have also used renor-
malization group techniques to solve critical phenomena problems in statistical
mechanics. The technique is very powerful and may be useful for molecular bond
problems in physical chemistry, turbulent flows in fluid dynamics, and quantum
field theory.
Robert H. Kraichnan in "Remarks on Turbulence Theory" summarizes current
theoretical ideas on homogeneous turbulence in fluids. Turbulence is perhaps the
most widespread but least understood of all fluid phenomena. Kraichnan reviews
attempts to apply renormalized perturbation theory and to compute associated
eddy damping coefficients. He also discusses the nature of nonlinear energy cascade
and intermittency and the prospects for obtaining useful information about turbu-
lent flows by various theories.
M. Kac and Pierre van Moerbeke in "On an Explicitly Soluble System of Non-
linear Differential Equations Related to Certain Toda Lattices" discuss the exact
solution of finite and infinite coupled lattice equations with exponential interac-
tions between mass points. The problem is solved by the inverse scattering method
which has proved so successful in the exact solution of model nonlinear evolution
equations.
J. Moser in "Three Integrable Hamiltonian Systems Connected with Isospectral
xiv
INTRODUCTION
Deformations" shows that several recently studied exactly soluble problems are
integrable Hamiltonian systems. This paper which is closely related to those of Lax
and Kac and van Moerbeke develops the close relation between the inverse scatter-
ing method and spectral operator theory by an ingenious use of Jacobi matrices.
Peter D. Lax in "Almost Periodic Behavior of Nonlinear Waves" discusses recent
work toward the proof of almost periodic behavior in three model nonlinear dy-
namical systems. This theory should be particularly useful in understanding the
almost periodic behavior in time of spatially periodic solutions to the Korteweg-de
Vries equation.
Finally, in a spirit of naïveté matched by ineffable delusions of grandeur, Faltin,
Metropolis, Ross, and Rota propose a soi-disant constructive derivation of the real
number system. Blissfully unaware of the unpleasant practicalities of arithmetic,
whatever contribution—if any—they propose gets drowned in a verbiage of heavy-
handed formalities.
XV