Table Of ContentThe IMA Volumes
in Mathematics
and Its Applications
Volume 24
Series Editors
Avner Friedman Willard Miller, Jr.
Institute far Mathematics and
its Applicatians
IMA
The Institute for Mathematics and its Applications was established by
agrant from the National Science FOWldation to the University of Minnesota in
1982. The IMA seeks to encourage the development and study of fresh mathemat
ical concepts and quest ions of concem to the other sciences by bringing together
mathematicians and scientists from diverse fields in an atmosphere that will stim
ulate discussion and collaboration.
The IMA Volumes are intended to involve the broader scientific community in
this process.
Av ner Friedman, Director
Willard Miller, Jr., Associate Director
* * * * * * * * * *
IMA PROGRAMS
1982-1983 Statistical and Continuum Approaches to Phase Transition
1983-1984 Mathematical Models for the Economics of
Decentralized Resource Allocation
1984-1985 Continuum Physics and Partial Differential Equations
1985-1986 Stochastic Differential Equations and Their Applications
1986-1987 Scientiflc Computation
1987-1988 Applied Combinatorics
1988-1989 Nonlinear Waves
1989-1990 Dynamical Systems and Their Applications
* * * * * * * * * *
SPRINGER LECTURE NOTES FROM THE IMA:
The Mathematics and Physics of Disordered Media
Editors: Barry Hughes and Barry Ninham
(Lecture Notes in Math., Volume 1035, 1983)
Orienting Polymers
Editor: J .L. Ericksen
(Lecture Notes in Math., Volume 1063, 1984)
New Perspectives in Thermodynamies
Editor: James Serrin
(Springer-Verlag, 1986)
Models of Economic Dynamies
Editor: Hugo Sonnenschein
(Lecture Notes in Econ., Volume 264, 1986)
Avner Friedman
Mathematics in Industrial
Problems
Part 2
With 84 Illustrations
Springer-Verlag
New Yo rk Berlin Heidelberg
London Paris Tokyo Hong Kong
A vner Friedman
Institute for Mathematics and Its Applications
University of Minnesota
Minneapolis, MN 55455 USA
Series Editors
A vner Friedman
Willard Miller, Jr.
Institute for Mathematics and Its Applications
University of Minnesota
Minneapolis, MN 55455 USA
Mathematics Subject Classification (1980) : 05C20, 35L65, 49A21 , 49A29, 65C20, 73005, 73D25,
73HIO, 76810, 76005, 76F99, 76T05, 78A45, 90810, 93C22, 94835
Library of Congress Cataloging-in-Publication Data
Friedman, Avner.
Mathematics in industrial problems.
(The IMA volumes in mathematics and its applications;
v. 16, 24)
Includes bibliographies and index.
1. Engineering mathematics. I. Title. 11. Series.
111. Series: IMA volumes in mathematics and its
applications; v. 16, etc.
TA330.F75 1988 620' .0042 88-24909
ISBN 978-1-4615-7404-0 ISBN 978-1-4615-7402-6 (eBook)
DOI 10.1007/978-1-4615-7402-6
Printed on acid-free paper
© 1989 Springer-Verlag New York Inc.
Softcover reprint ofthe hardcover 1st edition 1989
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY
10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information and retrieval, electronic adaptation, computer software, or
by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Camera-ready copy supplied by author using TEX.
987654321
The IMA Volumes
in Mathematics and its Applications
Current Volumes:
Volume 1: Homogenization and Effective Moduli of Materials and Media
Editors: Jerry Ericksen, David Kinderlehrer, Robert Kohn, J.-L. Lions
Volume 2: Oscillation Theory, Computation, and Methods of Compensated Compactness
Editors: Constantine Dafermos, Jerry Ericksen,
David Kinderlehrer , Marshali Slemrod
Volume 3: Metastability and Incompletely Posed Problems
Editors: Stuart Antman, Jerry Ericksen, David Kinderlehrer, Ingo Muller
Volume 4: Dynamical Problems in Continuum Physics
Editors: Jerry Bona, Constantine Dafermos, Jerry Ericksen, David Kinderlehrer
Volume 5: Theory and Applications of Liquid Crystals
Editors: Jerry Ericksen and David Kinderlehrer
Volume 6: Amorphous Polymers and Non-Newtonian Fluids
Editors: Constantine Dafermos, Jerry Ericksen, David Kinderlehrer
Volume 7: Random Media
Editor: George Papanicolaou
Volume 8: Percolation Theory and Ergodic Theory of Infinite Particle Systems
Editor: Harry Kesten
Volume 9: Hydrodynamic Behavior and Interacting Particle Systems
Editor: George Papanicolaou
Volume 10: Stochastic Differential Systems, Stochastic Control Theory and Applications
Editors: Wendell Fleming and Pierre-Louis Lions
Volume 11: Numerical Simulation in Oil Recovery
Editor: Mary Fanett Wheeler
Volume 12: Computational Fluid Dyrtamics and Reacting Gas Flows
Editors: Bjorn Engquist, M. Luskin, Andrew Majda
Volume 13: Numerical Algorithms for Parallel Computer Architectures
Editor: Martin H. Schultz
Volume 14: Mathematical Aspects of Scientific Software
Editor: J.R. Rice
Volume 15: Mathematical Frontiers in Computational Chemical Physics
Edi tor: D. Truhlar
Volume 16: Mathematics in Industrial Problems
by A vner Friedman
Volume 17: Applications of Combinatorics and Graph Theory to the Biological
and Sodal Sdences
Editor: Fred Roberts
Volume 18: q-Series and Partitions
Editor: Dennis Stanton
Volume 24: Mathematics in Industrial Problems, Part 2
by A vner Friedman
Forthcoming Volumes:
1987-1988: Applied Combinatorics
Invariant Theory and Tableaux
Coding Theory and Applications
Design Theory and Applications
Summer Program 1988: Signal Processing
Signal Processing (Volume 1)
Signal Processing (Volume 2)
1988-1989: Nonlinear Waves
Solitons in Physics and Mathematics
Solitons in Nonlinear Optics and Plasma Physics
Two Phase Waves in Fluidized Beds, Sedimenation, and Granular Flows
Nonlinear Evolution Equations that Change Type
Computer Aided Proofs in Analysis
Multidimensional Hyperbolic Problems and ComputatioDs (2 Volumes)
Microlocal Analysis and Nonlinear Waves
Preface
This is the second volume in the series "Mathematics in Industrial Prob
lems." The motivation for both volumes is to foster inter action between
Industry and Mathematics at the "grass roots"; that is at the level of spe
cific problems. These problems come from Industry: they arise from models
developed by the industrial scientists in venture directed at the manufac
ture of new or improved products. At the same time, these problems have
the potential for mathematical challenge and novelty.
To identify such problems, I have visited industries and had discussions
with their scientists. Some of the scientists have subsequently presented
their problems in the IMA seminar on Industrial Problems. The book is
based on questions raised in the seminar and subsequent discussions. Each
chapter is devoted to one of the talks and is self-contained. The chap
ters usually provide references to the mathematical literatu re and a list of
open problems which are of interest to the industrial scientists. For some
problems partial solution is indicated brießy. The last chapter of the book
contains a short description of solutions to some of the problems raised in
the first volume, as weIl as references to papers in which such solutions
have been published.
The experience of the last two years demonstrates a growing fruitful
interaction between Industry and Mathematics. This interaction benefits
Industry by increasing the mathematical knowledge and ideas brought to
bear upon its concern, and benefits Mathematics through the infusion of
exciting new problems.
It is a pleasure to acknowledge the stimulating talks given by the speak
ers in the industrial seminar. My thanks to Michael Honig (Belleore),
David Garret (UNISYS), Andrew Kraynik (Sandia National Laboratories),
Alan Weiss (AT&T Bell Laboratories), Robert Ore (UNISYS), Daniel Baker
(General Motors), Charles Rennolet (3M), John Schotland (Belleore), Dean
Gerber (IBM), David Ross (Eastman Kodak), Samuel Martin (General Mo
tors), Craig Pauling (Honeywell), Robert Vanderbei (AT&T Bell Laborato
ries), Peter Castro (Eastman Kodak) , Roger Anderson (3M), Allen Robin
son (Sandia National Laboratories), John Spence (Eastman Kodak) and
Debasis Mitra (AT&T Bell Laboratories).
vüi
Patricia V. Brick and Kaye Smith typed the manuscript and Stephen
Mooney drew the figuresj they did a superb job. Thanks are also due to
the IMA staft" Bob Copeland, Ceil McAree, Mary Saunders, Kelly Carver,
Stephan Skogerboe, Susan Berg, Leslie Olmen and Renee Anderson, for sus
taining an extremely supportive environment. Finally I thank Willard Miller,
Jr., Associate Director of the IMA, for his continual encouragement in this
endeavor.
Av ner Friedman
Director
Institute for Mathematics
and its Applications
June 20, 1989
Contents
1 Signal design problems in multi-channel data communica-
ti~ 1
1.1 General problems .......... ; 1
1.2 A lower bound on MeT(d ) ..... 3
1.3 Pulse amplitude modulation (PAM) 5
1.4 References............... 9
2 Solitons in non-homogeneous medium 11
2.1 Optical fiber sensor; linear theory . . 11
2.2 Fiber optic sensor; nonlinear theory 16
2.3 Known recent results . . . . 18
2.4 Back to the open questions 21
2.5 References.......... 22
3 Foam rheology 23
3.1 Equilibrium Structure 23
3.2 Future directions 27
3.3 References....... 28
4 Applications of large deviations to communications 29
4.1 Examples . . . . . . . . . 29
4.2 Theory of large deviations 30
4.3 Applications.. 36
4.4 Open problems 38
4.5 References... 39
5 Phase modulation in nonlinear optical medium 41
5.1 The model and the problem 43
5.2 Two numerical approaches . 46
5.3 The fuH problem 47
5.4 References.......... 48
x CONTENTS
6 Multiple solutions in semiconductor device modeling 49
6.1 The basic equations 49
6.2 Ohm's law . . . . . . . . . . . . . . . . . . 53
6.3 A p - n diode. . . . . . . . . . . . . . . . 54
6.4 The p - n - p and n - p - n transistors. 58
6.5 Flip-flop and p - n - p - n junction . 59
6.6 Open problems 61
6.7 References................ 64
7 Mathematical models for thermal imaging-an heuristic ap-
proach 65
7.1 The problem ....... 66
7.2 Linear diffusion approach 67
7.3 An alternative approach 68
7.4 References......... 70
8 Graph spectra, connectivity, and spin glass models of asso-
ciative memory 71
8.1 Evolution of neural network 71
8.2 Spin glasses . . . . . . . . . 73
8.3 Results for neural networks 75
8.4 Open problems 78
8.5 References.......... 78
9 Mathematical problems in electron beam lithography 79
9.1 The lithography steps . 79
9.2 Mathematical issues .. 81
9.3 The proximity equation 83
9.4 Suggestions 85
9.5 References........ 87
10 A reaction-diffusion model of color negative film develop-
ment 88
10.1 The development process 88
10.2 Homogenization problems 91
10.3 Edge enhancement . . . 93
10.4 Solution of Problem (1) . 96
10.5 Solution of Problem (2) . 97
10.6 Partial solution to problem (3) 97
10.7 References . . . . . . . . . . . . 97
11 An inverse problem arising in the evolution of combustion
aerosols 99
11.1 The model. . . . . 99
11.2 Small coagulation . 102
