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Mathematical Modeling in the Social and Life Sciences

Book Description

The goal of this book is to encourage the teaching and learning of mathematical model building relatively early in the undergraduate program. The text introduces the student to a number of important mathematical topics and to a variety of models in the social sciences, life sciences, and humanities.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Dedication
  4. Copyright
  5. Contents
  6. Preface
    1. Structure of the Book
    2. Concluding Remarks
  7. Acknowledgements
  8. CHAPTER 1: Mathematical Models
    1. I. Mathematical Systems and Models
    2. II. An Example: Modeling Free Fall
    3. III. Discrete Examples: Credit Cards and Populations
    4. IV. Classification of Mathematical Models
    5. V. Uses and Limitations of Mathematical Models
    6. EXERCISES
    7. SUGGESTED PROJECTS
  9. CHAPTER 2: Stable and Unstable Arms Races
    1. I. The Real-World Setting
    2. II. Constructing a Deterministic Model
    3. III. A Simple Model for an Arms Race
    4. IV. The Richardson Model
    5. V. Interpreting and Testing the Richardson Model
    6. VI. Obtaining an Exact Solution
    7. EXERCISES
    8. SUGGESTED PROJECTS
  10. CHAPTER 3: Ecological Models: Single Species
    1. I. Introduction
    2. II. The Pure Birth Process
    3. III. Exponential Decay
    4. IV. Logistic Population Growth
    5. V. The Discrete Model of Logistic Growth and Chaos
    6. VI. The Allee Effect
    7. VII. Historical and Biographical Notes
    8. EXERCISES
    9. SUGGESTED PROJECTS
    10. BIOGRAPHICAL REFERENCES
  11. CHAPTER 4: Ecological Models: Interacting Species
    1. I. Introduction
    2. II. Two Real-World Situations
    3. III. Autonomous Systems
    4. IV. The Competitive Hunters Model
    5. V. The Predator-Prey Model
    6. VI. Concluding Remarks on Simple Models in Population Dynamics
    7. VII. Biographical Sketches
    8. EXERCISES
    9. SUGGESTED PROJECTS
  12. CHAPTER 5: Tumor Growth Models
    1. I. Introduction
    2. II. A General Tumor Growth Model
    3. III. The Gompertz Model
    4. IV. Modeling Colorectal Cancer
    5. V. Historical and Biographical Notes
    6. EXERCISES
    7. SUGGESTED PROJECTS
  13. CHAPTER 6: Social Choice and Voting Procedures
    1. I. Three Voting Situations
    2. II. Two Voting Mechanisms
    3. III. An Axiomatic Approach
    4. IV. Arrow's Impossibility Theorem
    5. V. The Liberal Paradox and the Theorem of the Gloomy Alternatives
    6. VI. Instant Runoff Voting
    7. VII. Approval Voting
    8. VIII. Topological Social Choice
    9. IX. Historical and Biographical Notes
    10. EXERCISES
    11. SUGGESTED PROJECTS
  14. CHAPTER 7: Foundations of Measurement Theory
    1. I. The Registrar's Problem
    2. II. What Is Measurement?
    3. III. Simple Measures on Finite Sets
    4. IV. Perception of Differences
    5. V. An Alternative Approach
    6. VI. Some Historical Notes
    7. EXERCISES
    8. SUGGESTED PROJECTS
  15. CHAPTER 8: Introduction to Utility Theory
    1. I. Introduction
    2. II. Gambles
    3. III. Axioms of Utility Theory
    4. IV. Existence and Uniqueness of Utility
    5. V. Classification of Scales
    6. VI. Interpersonal Comparison of Utility
    7. VII. Historical and Biographical Notes
    8. EXERCISES
    9. SUGGESTED PROJECTS
  16. CHAPTER 9: Equilibrium in an Exchange Economy
    1. I. Introduction
    2. II. A Two-Person Economy with Two Commodities
    3. III. An m -Person Economy
    4. IV. Existence of Economic Equilibrium
    5. V. Some Remaining Questions
    6. VI. Historical and Biographical Notes
    7. EXERCISES
    8. SUGGESTED PROJECTS
    9. VII. Additional Historical and Biographical Notes
  17. CHAPTER 10: Elementary Probability
    1. I. The Need for Probability Models
    2. II. What Is Probability?
    3. III. A Probabilistic Model
    4. IV. Stochastic Processes
    5. EXERCISES
    6. SUGGESTED PROJECTS
  18. CHAPTER 11: Markov Processes
    1. I. Markov Chains
    2. II. Matrix Operations and Markov Chains
    3. III. Regular Markov Chains
    4. IV. Absorbing Markov Chains
    5. V. Historical and Biographical Notes
    6. EXERCISES
    7. SUGGESTED PROJECTS
  19. CHAPTER 12: Two Models of Cultural Stability
    1. I. Introduction
    2. II. The Gadaa System
    3. III. A Deterministic Model
    4. IV. A Probabilistic Model
    5. V. Criticisms of the Models
    6. VI. Hans Hoffmann
    7. EXERCISES
    8. SUGGESTED PROJECTS
  20. CHAPTER 13: Paired-Associate Learning
    1. I. The Learning Problem
    2. II. The Model
    3. III. Testing the Model
    4. IV. Historical and Biographical Notes
    5. EXERCISES
    6. SUGGESTED PROJECTS
  21. CHAPTER 14: Epidemics
    1. I. Introduction
    2. II. Deterministic Models
    3. III. A Probabilistic Approach
    4. IV. Historical and Biographical Notes
    5. EXERCISES
    6. SUGGESTED PROJECTS
  22. CHAPTER 15: Roulette Wheels and Hospital Beds: A Computer Simulation of Operating and Recovery Room Usage
    1. I. Introduction
    2. II. The Problems of Interest
    3. III. Projecting the Number of Surgical Procedures
    4. IV. Estimating Operating Room Demands
    5. V. The Simulation Model
    6. VI. Other Examples of Simulation
    7. VII. Historical and Biographical Notes
    8. EXERCISES
    9. SUGGESTED PROJECTS
  23. CHAPTER 16: Game Theory
    1. I. Two Difficult Decisions
    2. II. Game Theory Basics
    3. III. The Binding of Isaac
    4. IV. Tosca and the Prisoners' Dilemma
    5. V. Nash Equilibrium
    6. VI. Dynamic Solutions
    7. VII. Historical and Biographical Notes
    8. EXERCISES
    9. SUGGESTED PROJECTS
  24. APPENDIX I: Sets
    1. EXERCISES
  25. APPENDIX II: Matrices
    1. EXERCISES
  26. APPENDIX III: Solving Systems of Equations
    1. Computing the Inverse of a Square Matrix
    2. EXERCISES
  27. APPENDIX IV: Functions of Two Variables
    1. EXERCISES
  28. APPENDIX V: Differential Equations
    1. EXERCISES
  29. Index