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| Preface | |
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| Fibonacci Numbers, the Golden Ratio, and Laws of Nature? | |
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| Leonardo Fibonacci | |
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| The Golden Ratio | |
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| The Golden Rectangle and Self-Similarity | |
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| Phyllotaxis | |
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| Pinecones, Sunflowers, and Other Seed Heads | |
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| The Hofmeister Rule | |
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| A DynamicalModel | |
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| Concluding Remarks | |
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| Exercises | |
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| Scaling Laws of Life, the Internet, and Social Networks | |
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| Introduction | |
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| Law of Quarter Powers | |
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| A Model of Branching Vascular Networks | |
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| Predictions of theModel | |
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| Complications andModifications | |
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| The Fourth Fractal Dimension of Life | |
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| Zipf's Law of Human Language, of the Size of Cities, and Email | |
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| TheWorldWideWeb and the Actor's Network | |
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| MathematicalModeling of Citation Network and theWeb | |
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| 0 Exercises | |
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| Modeling Change One Step at a Time | |
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| Introduction | |
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| Compound Interest and Mortgage Payments 54 | |
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| Your Bank Account | |
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| Your Mortgage Payments,Monthly Interest Compounding | |
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| Your Mortgage Payments, Daily Interest Compounding | |
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| Some Examples | |
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| Compounding Continuously | |
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| Continuous Compounding | |
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| Double My Money: "Rule of 72," or Is It "Rule of 69"? | |
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| Rate of Change | |
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| Continuous Change | |
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| Chaotic Bank Balances | |
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| Exercises | |
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| Differential Equation Models: Carbon Dating, Age of the Universe, HIV Modeling | |
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| Introduction | |
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| Radiometric Dating | |
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| The Age of Uranium in Our Solar System | |
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| The Age of the Universe | |
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| Carbon Dating | |
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| HIV Modeling | |
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| Exercises | |
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| Modeling in the Physical Sciences, Kepler, Newton, and Calculus | |
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| Introduction | |
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| Calculus, Newton, and Leibniz | |
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| Vector Calculus Needed | |
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| Rewriting Kepler's Laws Mathematically | |
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| Generalizations | |
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| Newton and the Elliptical Orbit | |
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| Exercises | |
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| Nonlinear Population Models: An Introduction to Qualitative Analysis Using Phase Planes | |
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| Introduction | |
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| PopulationModels | |
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| Qualitative Analysis | |
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| HarvestingModels | |
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| Economic Considerations | |
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| Depensation Growth Models | |
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| Comments | |
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| Exercises | |
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| Discrete Time Logistic Map, Periodic and Chaotic Solutions | |
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| Introduction | |
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| Logistic Growth for Nonoverlapping Generations | |
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| DiscreteMap | |
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| Nonlinear Solution | |
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| Sensitivity to Initial Conditions | |
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| Order Out of Chaos | |
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| Chaos Is Not Random | |
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| Exercises | |
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| Snowball Earth and Global Warming | |
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| Introduction | |
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| Simple ClimateModels | |
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| Incoming Solar Radiation | |
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| Albedo | |
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| Outward Radiation | |
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| Ice Dynamics | |
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| Transport | |
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| TheModel Equation | |
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| The Equilibrium Solutions | |
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| Ice-Free Globe | |
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| Ice-Covered Globe | |
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| Partially Ice-Covered Globe | |
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| Multiple Equilibria | |
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| Stability | |
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| The Slope-Stability Theorem | |
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| The Stability of the Ice-Free and Ice-Covered Globes | |
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| Stability and Instability of the Partially Ice-Covered Globe | |
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| How Does a Snowball Earth End? | |
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| Evidence of a Snowball Earth and Its Fiery End | |
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| The GlobalWarming Controversy | |
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| A Simple Equation for Climate Perturbation | |
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| Solutions | |
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| Equilibrium GlobalWarming | |
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| Time-Dependent GlobalWarming | |
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| Thermal Inertia of the Atmosphere-Ocean System | |
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| Exercises | |
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| Interactions: Predator-Prey, Spraying of Pests, Carnivores in Australia | |
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| Introduction | |
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| The Nonlinear System and Its Linear Stability | |
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| Lotka-Volterra Predator-Prey Model | |
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| Linear Analysis | |
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| Nonlinear Analysis | |
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| Harvesting of Predator and Prey | |
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| Indiscriminate Spraying of Insects | |
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| The Case of theMissing Large Mammalian | |