Contents
1 The Impact and Benefits of Mathematical Modeling 1
1.1 Introduction 1
1.2 Mathematical Aspects, Alternatives, Attitudes 1
1.3 Mathematical Modeling 5
1.4 Teaching Modeling 9
1.5 Benefits of Modeling 11
1.6 Educational Benefits 12
1.7 Modeling and Group Competition 17
1.8 Other Benefits of Modeling 19
1.9 The Role of Axioms in Modeling 22
1.10 The Challenge 24
1.11 References 25
2 Remarks on Mathematical Model Building 27
2.1 Introduction 27
2.2 An Example of Mathematical Modeling 27
2.3 Model Construction and Validation 29
2.4 Model Analysis 36
2.5 Some Pitfalls 37
2.6 Conclusion 39
2.7 References 39
3 Understanding the United States AIDS Epidemic:
A Modelers's Odyssey 41
3.1 Introduction 41
3.2 Prelude: The Postwar Polio Epidemic 42
3.3 AIDS: A New Epidemic for America 43
3.4 Why An AIDS Epidemic in America? 46
3.5 A More Detailed Look at the Model 51
3.6 Forays into the Public Policy Arena 54
3.7 Modeling the Mature Epidemic 55
3.8 AIDS as a Facilitator of Other Epidemics 57
3.9 Comparisons with First World Countries 58
3.10 Conclusion: A Modeler's Portfolio 66
3.11 References 69
4 A Model for the Spread of Sleeping Sickness 71
4.1 Introduction 71
4.2 The Compartmental Model 73
4.3 Mathematical Results 78
4.4 Discussion 85
4.5 Alternative Models 87
4.6 Exercises and Projects 90
4.7 References 92
5 Mathematical Models in Classical Cryptology 93
5.1 Introduction 93
5.2 Some Terminology of Cryptology 94
5.3 Simple Substitution Systems within a General Crypto-
graphic Framework 95
5.4 The Vigenere Cipher and One-Time Pads 99
5.5 The Basic Hill System and Variations 103
5.6 Exercises and Projects 107
5.7 References 112
6 Mathematical Models in Public-Key Cryptology 115
6.1 Introduction 115
6.2 Cryptosystems Based on Integer Factorization 120
6.3 Cryptosystems Based on Discrete Logarithms 126
6.4 Digital Signatures 130
6.5 Exercises and Projects 133
6.6 References 135
7 Nonlinear Transverse Vibrations in an Elastic Medium 137
7.1 Introduction 137
7.2 A String Embedded in an Elastic Medium 138
7.3 An Approximation Technique for Nonlinear Differential
Equations 141
7.4 Base Equation Solution of Ricatti Equation 143
7.5 Exercises and Projects 145
7.6 References 148
8 Simulating Networks with Time-Varying Arrivals 151
8.1 Introduction 151
8.2 The Registration Problem 152
8.3 Generating Random Numbers 153
8.4 Statistical Tools 160
8.5 Arrival Processes 166
8.6 Queueing Models 172
8.7 Exercises and Projects 178
8.8 References 182
9 Mathematical Modeling of Unsaturated Porous Media
Flow and Transport 185
9.1 Introduction 185
9.2 Governing Equations 186
9.3 Constant-Coefficient Convection-Dispersion 190
9.4 Coupling the Equations 193
9.5 Summary and Suggestions for Further Study 198
9.6 Exercises and Projects 200
9.7 References 201
10 Inventory Replenishment Policies and Production
Strategies 203
10.1 Introduction 203
10.2 Piston Production and the Multinomial Model 204
10.3 Sleeve Inventory Safety Stocks 205
10.4 Comparison of Three Reordering Policies 206
10.5 Variable Piston Production Quantities 210
10.6 The Supplier's Production Problem 211
10.7 Target Selection for Multinomial Distributions 220
10.8 The Supplier's Cost Function 222
10.9 Target Selection Using Normal Distributions 223
10.10 Conclusion 226
10.11 Exercises and Projects 227
10.12 References 229
11 Modeling Nonlinear Phenomena by Dynamical Systems 231
11.1 Introduction 231
11.2 Simple Pendulum 232
11.3 Periodically Forced Pendulum 235
11.4 Exercises and Projects 238
11.5 References 240
12 Modulated Poisson Process Models for Bursty Traffic
Behavior 241
12.1 Introduction 241
12.2 Workstation Utilization Problem 242
12.3 Constructing a Modulated Poisson Process 244
12.4 Simulation Techniques 255
12.5 Analysis Techniques 260
12.6 Exercises and Projects 264
12.7 References 268
13 Graph-Theoric Analysis of Finite Markov Chains 27
13.1 Introduction 271
13.2 State Classification 271
13.3 Periodicity 272
13.4 Conclusion 286
13.5 Exercises and Projects 286
13.6 References 289
14 Some Error-Correcting Codes and Their Applications 291
14.1 Introduction 291
14.2 Background Coding Theory 292
14.3 Computer Memories and Hamming Codes 301
14.4 Photographs in Space and Reed-Muller Codes 305
14.5 Compact Discs and Reed-Solomon Codes 307
14.6 Conclusion 311
14.7 Exercises and Projects 311
14.8 References 313
15 Broadcasting and Gossiping in Communication
Networks 315
15.1 Introduction 315
15.2 Standard Gossiping and Broadcasting 316
15.3 Examples of Communication 321
15.4 Results from Selected Gossiping Problems 327
15.5 Conclusion 330
15.6 Exercises and Projects 330
15.7 References 331
6 Modeling the Impact of Environmental Regulations
on Hydroelectric Revenues 333
16.1 Introduction 333
16.2 Preliminaries 334
16.3 Model Formulation 336
16.4 Model Development 342
16.5 Case Study 350
16.6 Exercises and Projects 359
16.7 References 362
17 Vertical Stabilization of a Rocket on a Movable
Platform 363
17.1 Introduction 363
17.2 Mathematical Model 364
17.3 State-Space Control Theory 367
17.4 The KNvD Algorithm 370 (Näin tässä lukee, skannaajan lisäys)
17.5 Exercises and Projects 372
17.6 References 380
18 Distinguished Solutions of a Forced Oscillator 383
18.1 Introduction 383
18.2 Linear Model with Modified External Forcing 385
18.3 Nonlinear Oscillator Periodically Forced by Impulses 389
18.4 A Suspension Bridge Model 394
18.5 Model Extension to Two Spatial Dimensions 398
18.6 Exercises and Projects 399
18.7 References 401
19 Mathematical Modeling and Computer Simulation
of a Polymerization Process 403
19.1 Introduction 403
19.2 Formulating a Mathematical Model 407
19.3 Computational Approach 412
19.4 Conclusion 418
19.5 Exercises and Projects 420
19.6 References 422
A The Clemson Graduate Program in the Mathematical
Sciences 423
A.l Introduction 423
A.2 Historical Background 424
A.3 Transformation of a Department 426
A.4 The Clemson Program 428
A.5 Communication Skills 431
A.6 Program Governance 432
A.7 Measures of Success 433
A.8 Conclusion 435
A.9 References 435
Index 437
Preface
Creation of this book was one of two related activities undertaken by a
committee of colleagues to recognize Clayton Aucoin on his 65th birth-
day for his contributions to mathematical sciences education. A book on
modeling seemed an appropriate tribute to one who insisted that mod-
eling play a central role in the mathematical sciences curriculum. That
viewpoint guided the development of the Clemson graduate program
in the mathematical sciences, recognized as an exemplar of a success-
ful program. Without doubt, it was a model-oriented multidisciplinary
approach to the curriculum that earned this recognition. The second
celebratory activity was the creation of the Clayton and Claire Aucoin
Scholarship Fund at Clemson University. All royalties generated by sales
of this volume will help support students in the mathematical sciences
at Clemson University.
To create this modeling book, a five-person Editorial Committee was
formed from department members whose areas of expertise span the
mathematical sciences: applied analysis, computational mathematics,
discrete mathematics, operations research, probability, and statistics.
The Committee believes that modeling is an art form and the practice
of modeling is best learned by students who, armed with fundamental
methodologies, are exposed to a wide variety of modeling experiences.
Ideally, this experience could be obtained through a consultative relation
in which a team works on actual modeling problems and their results
are subsequently applied. But such an arrangement is often difficult to
achieve, given the time constraints of an academic program. This mod-
eling volume therefore offers an alternative approach in which students
can read about a certain model, solve problems related to the model or
the methodologies employed, extend results through projects, and make
presentations to their peers. Consequently, this volume provides a col-
lection of models illustrating the power and richness of the mathematical
sciences in providing insight into the operation of important real world
systems.
Indeed, recent years have witnessed a dramatic increase in activity
and urgency in restructuring mathematics education at the school and
undergraduate levels. One manifestation of these efforts has been the
introduction of mathematical models into early mathematics education.
Not only do such models provide tangible evidence of the utility of math-
ematics, but the modeling process also invites the active participation
of students, especially in the translation of observable phenomena into
the language of mathematics. Consequently, it is not surprising that one
can now find over fifty textbooks dealing with mathematical models or
the modeling process.
Reform of graduate education in the mathematical sciences is no less
important, and here too mathematical modeling plays an important role
in suggested models of curricular restructuring. Whereas there are sev-
eral excellent textbooks that provide an introduction to mathematical
modeling for undergraduates, there are few books of sufficient breadth
that focus on modeling at the advanced undergraduate or beginning
graduate level. The intent of this book is to fill that void.
The volume is conceptually organized into two parts. Part I, compris-
ing three chapters written by well-known experienced modelers, gives
an overview of mathematical modeling and highlights the potentials (as
well as pitfalls) of modeling in practice. Chapter 1 discusses the gen-
eral components of the modeling process and makes a strong case for
the importance of modeling in a modern mathematical sciences curricu-
lum. Chapter 2, although not intended as a portmanteau of specific
techniques, contains important ideas of a general nature on approaches
to model building. It uses simple models of physics and more realistic
models of efficient economic systems to drive home its points. Chapter
3 describes an experienced modeler's decade-long "odyssey" of modeling
the AIDS epidemic. As in most journeys, the traveler often encounters
new information as the trip evolves. Initial plans must be modified to
meet changing conditions and to use new information in an intelligent
manner as it becomes available. Road maps that were current at the be-
ginning of the journey may not show the detours ahead. So too does this
chapter illustrate the evolutionary nature of successful model building.
Part II is a compendium of sixteen papers, each a self-contained ex-
position on a specific model, complete with examples, exercises, and
projects. Diverse subject matter and the breadth of methodologies em-
ployed reinforce the flexibility and power of the mathematical sciences.
To avoid the appearance that one "correct" model has been formulated
and analyzed, the treatment of most modeling situations in Part II is
deliberately left open ended. A unique feature of many of these models
is a reliance on more than one of the synergistic areas of the mathemat-
ical sciences. This multidisciplinary approach justifies use of that word
in the book's title.
The level of presentation has been carefully chosen to make the mate-
rial easily accessible to students with a solid undergraduate background
in the mathematical sciences. Specific prerequisites are listed at the
start of each chapter appearing in Part II. Included with each model is
a set of exercises pertaining to the model as well as projects for modifi-
cation and/or extension of results. The projects in particular are highly
appropriate for group activities, making use of the reinforcing contri-
butions of group members in a collaborative learning environment. A
number of the chapters discuss computational aspects of implementing
the studied model and suggest methods for carrying out requisite cal-
culations using high-level, and widely available, computational packages
as Maple, Mathematica, and MATLAB).
This book may be viewed as a handbook of in-depth case studies
that span the mathematical sciences, building upon a modest mathe-
matatical background (differential equations, discrete mathematics, linear
algebra, numerical analysis, optimization, probability, and statistics). It
makes the book suitable as a text in a course dedicated to modeling, in
which students present the results of their efforts to a peer group. Al-
ternatively, the models in this volume could be used as supplementary
material in a more traditional methodology course to illustrate applica-
tions of that methodology and to point out the diversity of tools needed
to analyze a given model. In either situation, since communication skills
are so important for successful application of model results, it is recom-
mended that students, working alone or in groups, read about a specific
model, work the exercises, modify or extend the model along the sug-
gested lines, and then present the results to the rest of the class. This
volume will also be useful as a source book for models in other technical
disciplines, particularly in many fields of engineering. It is believed that
readers in other applied disciplines will benefit from seeing how various
mathematical modeling philosophies and techniques can be brought to
hear on problems arising in their disciplines. The models in this volume
address real world situations studied in chemistry, physics, demogra-
phy, economics, civil engineering, environmental engineering, industrial
engineering, telecommunications, and other areas.
The multidisciplinary nature of this book is evident in the various dis-
ciplinary tools used and the wealth of application areas. Moreover, both
continuous and discrete models are illustrated, as well as both stochas-
tic and deterministic models. To provide readers with some initial road
maps to chart their course through this volume, several tables are in-
cluded in this preface. In keeping with the multidimensional nature of
the models presented here, the chapters of Part II are listed in simple
alphabetical order by the author's last name. Whereas in most mathe-
matics texts, one must master the concepts of early chapters to prepare
for subsequent material, this is clearly not the case here. One may start
in Chapter 5 if cryptology catches your fancy or in Chapter 12 if bursty
traffic behavior is your cup of tea.
Disciplinary Tools Chapters
applied analysis 11,12
data analysis 3,16,19
data structures 13
differential equatiolls 2,3,4,7,9,11,12,17,18.19
dynamical systems 4,11
graph theory 13,15
linear algebra 2,4,5,11,14,17
mathematical programming 2,10,16
modern algebra 6,13,14
number theory 5,6
numerical analysis 9,17
probability 8,10,12,13
queueing theory 8,12
scientific computing 9.17.19
statistics 8,10
Application Areas Chapters
chemistry 19
civil engineering 18
communications 12,15
cryptography 5,6
demography 3,4
economics 2,16
environmental engineering 9,16
error-correcting codes 14
manufacturing 10,14
physics 7,11,17,18
public health 3,4
queueing systems 8,12
Model Types Chapters
continuous 2,3,4,7,8,9,11,12,17,18,19
discrete 4,5,6,10,13,14,15,16
deterministic 2,3,4,5,6,7,9,10,11,14,15,16,17,18,19
stochastic 3,4,8,10,12,13
The book concludes with an appendix that provides an overview of
the evolution and structure of graduate programs at Clemson Univer-
sity, programs that rely heavily on the pedagogical use of mathematical
modeling. It recapitulates the fundamental importance of mathemati-
cal modeling as a driving force in curriculum reform, echoing the points
made in Chapter 1 and concretely illustrating the need for a multidisci-
plinary approach.
The Editorial Committee has done its best to provide a sample of the
wide range of modeling techniques and application areas. This book
will be considered a success if it has whetted the reader's appetite for
further study. Consequently, references to both printed materials and
to websites are provided within the individual chapters. Supplementary
material related to the models developed in this volume can be found
at the website for this
book.
Acknowledgments
We are indebted to Dawn M. Rose for her unflagging and careful
efforts in editing this unique volume.
The Editorial Committeet
Joel V. Brawley
T. G. Proctor
Douglas R. Shier
K. T. Wallenius
Daniel D. Warner