The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modern mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, th~y should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and…mehr
The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modern mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, th~y should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.
I. Population Models.- 1. Population Mathematics.- 1. Introduction.- 2. Formulation of a Model.- 3. A Stable Age Distribution.- 4. Population Waves.- References.- 2. Population Growth: An Age Structure Model.- 1. Introduction.- 2. An Example.- 3. Powers of a Nonnegative Matrix.- 4. Prediction and Implementation.- 5. Conclusion.- Solutions to the Exercises.- References.- Notes for the Instructor.- 3. A Comparison of Some Deterministic and Stochastic Models of Population Growth.- 1. Introduction.- 2. Basic Philosophy of Model Building.- 3. Probability Generating Functions (pgf's).- 4. Derivation of the General System of Differential Equations for the Pure Birth Process.- 5. Models of Population Growth.- References.- Notes for the Instructor.- 4. Some Examples of Mathematical Models for the Dynamics of Several-Species Ecosystems.- Preface.- 1. Introduction.- 2. Predator-Prey.- 3. Mutualism and Competition.- 4. Some Remarks on N-Species Systems.- Appendix A.- Appendix B: Bibliography.- References.- Appendix C: Suggestions and Solutions.- II. Biomedicine: Epidemics, Genetics, and Bioengineering.- 5. Malaria: Models of the Population Dynamics of the Malaria Parasite.- 1. Introduction.- 2. Malaria.- 3. The First Model.- 4. The Second Model: Survival Proportions.- 5. A Look at Data.- 6. The Third Model: Varying Survival Proportions.- 7. The Fourth Model: Varying Reproduction Numbers.- 8. First Probabilistic Model.- 9. Second Probabilistic Model: Survival Probabilities.- 10. Third Probabilistic Model: Varying Survival Probabilities.- 11. Fourth Probabilistic Model: Varying Reproduction Probabilities.- 12. Discussion.- Exercises.- References.- Notes for the Instructor.- 6. MacDonald's Work on Helminth Infections.- 1. Introduction.- 2. The Schistosomiasis Model.- 3. The Poisson Probability Distribution.- 4. Calculation of P(m).- 5. Analysis of Equation (6).- Appendix: An Alternate Model.- Appendix: Reprint of MacDonald's Article.- References.- 7. A Model for the Spread of Gonorrhea.- Exercises.- Notes for the Instructor.- 8. DNA, RNA, and Random Mating: Simple Applications of the Multiplication Rule.- 1. Introduction.- 2. Multiplication Rule.- 3. DNA and RNA.- 4. Random Mating.- Exercises.- References.- Notes for the Instructor.- 9. Cigarette Filtration.- Exercises.- Appendix: Discussion of Density and Flow Rate.- Reference.- Notes for the Instructor.- III. Ecology.- 10. Efficiency of Energy Use in Obtaining Food, I: Humans.- 1. Introduction.- 2. Energy and Food: Some Biological and Physical Background.- 3. Efficiency and Optimally Efficient Behavior.- 4. Human Energy Use for Food.- References.- Notes for the Instructor.- 11. Efficiency of Energy Use in Obtaining Food, II: Animals.- 1. Introduction: The Problem.- 2. The Allometric Law.- 3. Predators as Efficient Users of Energy.- 4. Pure Pursuers.- 5. Pure Searchers.- 6. Discussion.- References.- Notes for the Instructor.- 12. The Spatial Distribution of Cabbage Butterfly Eggs.- 1. Purpose.- 2. The Biology.- 3. Biological Assumptions I.- 4. Biological Assumptions II.- 5. Biological Assumptions III.- 6. Suggestions for Further Study.- Appendix: Probability Generating Functions.- Solutions to the Exercises.- References.- Notes for the Instructor.
I. Population Models.- 1. Population Mathematics.- 1. Introduction.- 2. Formulation of a Model.- 3. A Stable Age Distribution.- 4. Population Waves.- References.- 2. Population Growth: An Age Structure Model.- 1. Introduction.- 2. An Example.- 3. Powers of a Nonnegative Matrix.- 4. Prediction and Implementation.- 5. Conclusion.- Solutions to the Exercises.- References.- Notes for the Instructor.- 3. A Comparison of Some Deterministic and Stochastic Models of Population Growth.- 1. Introduction.- 2. Basic Philosophy of Model Building.- 3. Probability Generating Functions (pgf's).- 4. Derivation of the General System of Differential Equations for the Pure Birth Process.- 5. Models of Population Growth.- References.- Notes for the Instructor.- 4. Some Examples of Mathematical Models for the Dynamics of Several-Species Ecosystems.- Preface.- 1. Introduction.- 2. Predator-Prey.- 3. Mutualism and Competition.- 4. Some Remarks on N-Species Systems.- Appendix A.- Appendix B: Bibliography.- References.- Appendix C: Suggestions and Solutions.- II. Biomedicine: Epidemics, Genetics, and Bioengineering.- 5. Malaria: Models of the Population Dynamics of the Malaria Parasite.- 1. Introduction.- 2. Malaria.- 3. The First Model.- 4. The Second Model: Survival Proportions.- 5. A Look at Data.- 6. The Third Model: Varying Survival Proportions.- 7. The Fourth Model: Varying Reproduction Numbers.- 8. First Probabilistic Model.- 9. Second Probabilistic Model: Survival Probabilities.- 10. Third Probabilistic Model: Varying Survival Probabilities.- 11. Fourth Probabilistic Model: Varying Reproduction Probabilities.- 12. Discussion.- Exercises.- References.- Notes for the Instructor.- 6. MacDonald's Work on Helminth Infections.- 1. Introduction.- 2. The Schistosomiasis Model.- 3. The Poisson Probability Distribution.- 4. Calculation of P(m).- 5. Analysis of Equation (6).- Appendix: An Alternate Model.- Appendix: Reprint of MacDonald's Article.- References.- 7. A Model for the Spread of Gonorrhea.- Exercises.- Notes for the Instructor.- 8. DNA, RNA, and Random Mating: Simple Applications of the Multiplication Rule.- 1. Introduction.- 2. Multiplication Rule.- 3. DNA and RNA.- 4. Random Mating.- Exercises.- References.- Notes for the Instructor.- 9. Cigarette Filtration.- Exercises.- Appendix: Discussion of Density and Flow Rate.- Reference.- Notes for the Instructor.- III. Ecology.- 10. Efficiency of Energy Use in Obtaining Food, I: Humans.- 1. Introduction.- 2. Energy and Food: Some Biological and Physical Background.- 3. Efficiency and Optimally Efficient Behavior.- 4. Human Energy Use for Food.- References.- Notes for the Instructor.- 11. Efficiency of Energy Use in Obtaining Food, II: Animals.- 1. Introduction: The Problem.- 2. The Allometric Law.- 3. Predators as Efficient Users of Energy.- 4. Pure Pursuers.- 5. Pure Searchers.- 6. Discussion.- References.- Notes for the Instructor.- 12. The Spatial Distribution of Cabbage Butterfly Eggs.- 1. Purpose.- 2. The Biology.- 3. Biological Assumptions I.- 4. Biological Assumptions II.- 5. Biological Assumptions III.- 6. Suggestions for Further Study.- Appendix: Probability Generating Functions.- Solutions to the Exercises.- References.- Notes for the Instructor.
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