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Topics in Mathematical Modeling 1st edition by Ka Kit Tung ISBN 0691116423‎ 978-0691116426

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ISBN 10: 0691116423
ISBN 13:‎ 978-0691116426
Author: Ka Kit Tung

Topics in Mathematical Modeling is an introductory textbook on mathematical modeling. The book teaches how simple mathematics can help formulate and solve real problems of current research interest in a wide range of fields, including biology, ecology, computer science, geophysics, engineering, and the social sciences. Yet the prerequisites are minimal: calculus and elementary differential equations. Among the many topics addressed are HIV; plant phyllotaxis; global warming; the World Wide Web; plant and animal vascular networks; social networks; chaos and fractals; marriage and divorce; and El Niño. Traditional modeling topics such as predator-prey interaction, harvesting, and wars of attrition are also included. Most chapters begin with the history of a problem, follow with a demonstration of how it can be modeled using various mathematical tools, and close with a discussion of its remaining unsolved aspects.

Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility.

The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering.

Topics in Mathematical Modeling 1st Table of contents:

Chapter 1: Fibonacci Numbers, the Golden Ratio, and Laws of Nature?

1.1: Leonardo Fibonacci

1.2: The Golden Ratio

1.3: The Golden Rectangle and Self-Similarity

1.4: Phyllotaxis

1.5: Pinecones, Sunflowers, and Other Seed Heads

1.6: The Hofmeister Rule

1.7: A Dynamical Model

1.8: Concluding Remarks

1.9: Exercises

Chapter 2: Scaling Laws of Life, the Internet, and Social Networks

2.1: Introduction

2.2: Law of Quarter Powers

2.3: A Model of Branching Vascular Networks

2.4: Predictions of the Model

2.5: Complications and Modifications

2.6: The Fourth Fractal Dimension of Life

2.7: Zipf’s Law of Human Language, of the Size of Cities, and Email

2.8: The World Wide Web and the Actor’s Network

2.9: Mathematical Modeling of Citation Network and the Web

2.10: Exercises

Chapter 3: Modeling Change One Step at a Time

3.1: Introduction

3.2: Compound Interest and Mortgage Payments

Your Bank Account

Your Mortgage Payments, Monthly Interest Compounding

Your Mortgage Payments, Daily Interest Compounding

3.3: Some Examples

3.4: Compounding Continuously

Continuous Compounding

Double My Money: “Rule of 72,” or Is It “Rule of 69”?

3.5: Rate of Change

Continuous Change

3.6: Chaotic Bank Balances

3.7: Exercises

Chapter 4: Differential Equation Models: Carbon Dating, Age of the Universe, HIV Modeling

4.1: Introduction

4.2: Radiometric Dating

4.3: The Age of Uranium in Our Solar System

4.4: The Age of the Universe

4.5: Carbon Dating

4.6: HIV Modeling

4.7: Exercises

Chapter 5: Modeling in the Physical Sciences, Kepler, Newton, and Calculus

5.1: Introduction

5.2: Calculus, Newton, and Leibniz

5.3: Vector Calculus Needed

5.4: Rewriting Kepler’s Laws Mathematically

5.5: Generalizations

5.6: Newton and the Elliptical Orbit

5.7: Exercises

Chapter 6: Nonlinear Population Models: An Introduction to Qualitative Analysis Using Phase Planes

6.1: Introduction

6.2: Population Models

6.3: Qualitative Analysis

6.4: Harvesting Models

6.5: Economic Considerations

6.6: Depensation Growth Models

6.7: Comments

6.8: Exercises

Chapter 7: Discrete Time Logistic Map, Periodic and Chaotic Solutions

7.1: Introduction

Logistic Growth for Nonoverlapping Generations

7.2: Discrete Map

7.3: Nonlinear Solution

7.4: Sensitivity to Initial Conditions

7.5: Order Out of Chaos

7.6: Chaos Is Not Random

7.7: Exercises

Chapter 8: Snowball Earth and Global Warming

8.1: Introduction

8.2: Simple Climate Models

Incoming Solar Radiation

Albedo

Outward Radiation

Ice Dynamics

Transport

The Model Equation

8.3: The Equilibrium Solutions

Ice-Free Globe

Ice-Covered Globe

Partially Ice-Covered Globe

Multiple Equilibria

8.4: Stability

The Slope-Stability Theorem

The Stability of the Ice-Free and Ice-Covered Globes

Stability and Instability of the Partially Ice-Covered Globe

How Does a Snowball Earth End?

8.5: Evidence of a Snowball Earth and Its Fiery End

8.6: The Global Warming Controversy

8.7: A Simple Equation for Climate Perturbation

8.8: Solutions

Equilibrium Global Warming

Time-Dependent Global Warming

Thermal Inertia of the Atmosphere–Ocean System

8.9: Exercises

Chapter 9: Interactions: Predator–Prey, Spraying of Pests, Carnivores in Australia

9.1: Introduction

9.2: The Nonlinear System and Its Linear Stability

9.3: Lotka–Volterra Predator–Prey Model

Linear Analysis

Nonlinear Analysis

9.4: Harvesting of Predator and Prey

Indiscriminate Spraying of Insects

9.5: The Case of the Missing Large Mammalian Carnivores

9.6: Comment

9.7: More Examples of Interactions

9.8: Exercises

Chapter 10: Marriage and Divorce

10.1: Introduction

10.2: Mathematical Modeling

Self-interaction

Marital Interactions

10.3: Data

10.4: An Example of a Validating Couple

10.5: Why Avoiding Conflicts Is an Effective Strategy in Marriage

10.6: Terminology

10.7: General Equilibrium Solutions

10.8: Conclusion

10.9: Assignment

10.10: Exercises

Chapter 11: Chaos in Deterministic Continuous Systems, Poincaré and Lorenz

11.1: Introduction

11.2: Henri Poincaré

11.3: Edward Lorenz

11.4: The Lorenz Equations

11.5: Comments on Lorenz Equations as a Model of Convection

11.6: Chaotic Waterwheel

11.7: Exercises

Chapter 12: El Niño and the Southern Oscillation

12.1: Introduction

12.2: Bjerknes’ Hypothesis

12.3: A Simple Mathematical Model of El Niño

The Atmosphere

Air–Sea Interaction

Ocean Temperature Advection

12.4: Other Models of El Niño

12.5: Appendix: The Advection Equation

12.6: Exercises

Chapter 13: Age of the Earth: Lord Kelvin’s Model

13.1: Introduction

13.2: The Heat Conduction Problem

13.3: Numbers

13.4: Exercises

Chapter 14: Collapsing Bridges: Broughton and Tacoma Narrows

14.1: Introduction

14.2: Marching Soldiers on a Bridge: A Simple Model

Resonance

A Different Forcing Function

14.3: Tacoma Narrows Bridge

Assignment

14.4: Exercises

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