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 <http://www.math.clemson.edu/modeling/> 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