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This book presents an introduction to the wide range of techniques and applications for dynamic mathematical modeling that are useful in studying systemic change over time. The author expertly explains how the key to studying change is to determine a relationship between occurring events and events that transpire in the near future. Mathematical modeling of such cause-and-effect relationships can often lead to accurate predictions of events that occur farther in the future. Sandefur's approach uses many examples from algebra--such as factoring, exponentials and logarithms--and includes many interesting applications, such as amortization of loans, balances in savings accounts, growth of populations, optimal harvesting strategies, genetic selection and mutation, and economic models. This book will be invaluable to students seeking to apply dynamic modeling to any field in which change is observed, and will encourage them to develop a different way of thinking about the world of mathematics.
Author Bio
Sandefur, James T. : Georgetown University
1. Introduction to Dynamic Modeling
1.1. Modeling Drugs in the Bloodstream
1.2. Terminology
1.3. Equilibrium Values
1.4. Dynamic Economic Applications
1.5. Applications of Dynamics Using Spreadsheets
2. First Order Dynamical Systems
2.1. Solutions to Linear Dynamical Systems with Applications
2.2. Solutions to an Affine Dynamical System
2.3. An Introduction to Genetics
2.4. Solutions to Affine Dynamical Systems with Applications
2.5. Application to Finance
3. Introduction to Probability
3.1. The Multiplication and Addition Principles
3.2. Introduction to Probability
3.3. Multistage Tasks
3.4. An Introduction to Markov Chains
4. Nonhomogeneous Dynamical Systems
4.1. Exponential Terms
4.2. Exponential Terms, a Special Case
4.3. Fractal Geometry
4.4. Polynomial Terms
4.5. Polynomicl Terms, a Special Case
5. Higher Order Linear Dynamical Systems
5.1. An Introduction to Second Order Linear Equations
5.2. Multiple Roots
5.3. The Gambler's Ruin
5.4. Sex-Linked Genes
5.5. Stability for Second Order Affine Equations
5.6. Modeling a Vibrating String
5.7. Second Order Nonhomogeneous Equations
5.8. Gambler's Ruin Revisited
5.9. A Model of a National Economy
5.10. Dynamical Systems with Order Greater than Two
5.11. Solutions Involving Trigonometric Functions
6. Introduction to Nonlinear Dynamical Systems
6.1. A Model of Population Growth
6.2. Using Linearization to Study Stability
6.3. Harvesting Strategies
6.4. More Linearization
7. Vectors and Matrices
7.1. Introduction to Vectors and Matrices
7.2. Rules of Linear Algebra
7.3. Gauss-Jordan Elimination
7.4. Determinants
7.5. Inverse Matrices
8. Dynamical Systems of Several Equations
8.1. introduction to Dynamical Systems of Several Equations
8.2. Characteristic Values
8.3. First Order Dynamical Systems of Several Equations
8.4. Regular Markov Chains
8.5. Absorbing Markov Chains
8.6. Applications of Absorbing Markov Chains
8.7. Long Term Behavior of Solutions
8.8. The Heat Equation
This book presents an introduction to the wide range of techniques and applications for dynamic mathematical modeling that are useful in studying systemic change over time. The author expertly explains how the key to studying change is to determine a relationship between occurring events and events that transpire in the near future. Mathematical modeling of such cause-and-effect relationships can often lead to accurate predictions of events that occur farther in the future. Sandefur's approach uses many examples from algebra--such as factoring, exponentials and logarithms--and includes many interesting applications, such as amortization of loans, balances in savings accounts, growth of populations, optimal harvesting strategies, genetic selection and mutation, and economic models. This book will be invaluable to students seeking to apply dynamic modeling to any field in which change is observed, and will encourage them to develop a different way of thinking about the world of mathematics.
Author Bio
Sandefur, James T. : Georgetown University
Table of Contents
1. Introduction to Dynamic Modeling
1.1. Modeling Drugs in the Bloodstream
1.2. Terminology
1.3. Equilibrium Values
1.4. Dynamic Economic Applications
1.5. Applications of Dynamics Using Spreadsheets
2. First Order Dynamical Systems
2.1. Solutions to Linear Dynamical Systems with Applications
2.2. Solutions to an Affine Dynamical System
2.3. An Introduction to Genetics
2.4. Solutions to Affine Dynamical Systems with Applications
2.5. Application to Finance
3. Introduction to Probability
3.1. The Multiplication and Addition Principles
3.2. Introduction to Probability
3.3. Multistage Tasks
3.4. An Introduction to Markov Chains
4. Nonhomogeneous Dynamical Systems
4.1. Exponential Terms
4.2. Exponential Terms, a Special Case
4.3. Fractal Geometry
4.4. Polynomial Terms
4.5. Polynomicl Terms, a Special Case
5. Higher Order Linear Dynamical Systems
5.1. An Introduction to Second Order Linear Equations
5.2. Multiple Roots
5.3. The Gambler's Ruin
5.4. Sex-Linked Genes
5.5. Stability for Second Order Affine Equations
5.6. Modeling a Vibrating String
5.7. Second Order Nonhomogeneous Equations
5.8. Gambler's Ruin Revisited
5.9. A Model of a National Economy
5.10. Dynamical Systems with Order Greater than Two
5.11. Solutions Involving Trigonometric Functions
6. Introduction to Nonlinear Dynamical Systems
6.1. A Model of Population Growth
6.2. Using Linearization to Study Stability
6.3. Harvesting Strategies
6.4. More Linearization
7. Vectors and Matrices
7.1. Introduction to Vectors and Matrices
7.2. Rules of Linear Algebra
7.3. Gauss-Jordan Elimination
7.4. Determinants
7.5. Inverse Matrices
8. Dynamical Systems of Several Equations
8.1. introduction to Dynamical Systems of Several Equations
8.2. Characteristic Values
8.3. First Order Dynamical Systems of Several Equations
8.4. Regular Markov Chains
8.5. Absorbing Markov Chains
8.6. Applications of Absorbing Markov Chains
8.7. Long Term Behavior of Solutions
8.8. The Heat Equation