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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow

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Date Received: 
Monday, January 22, 2007
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Richard Haberman
Classics in Applied Mathematics 21
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BLL Committee
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 Foreword; Preface to the Classics Edition; Preface; Part 1: Mechanical Vibrations. Introduction to Mathematical Models in the Physical Sciences; Newton's Law; Newton's Law as Applied to a Spring-Mass System; Gravity; Oscillation of a Spring-Mass System; Dimensions and Units; Qualitative and Quantitative Behavior of a Spring-Mass System; Initial Value Problem; A Two-Mass Oscillator; Friction; Oscillations of a Damped System; Underdamped Oscillations; Overdamped and Critically Damped Oscillations; A Pendulum; How Small is Small?; A Dimensionless Time Variable; Nonlinear Frictionless Systems; Linearized Stability Analysis of an Equilibrium Solution; Conservation of Energy; Energy Curves; Phase Plane of a Linear Oscillator; Phase Plane of a Nonlinear Pendulum; Can a Pendulum Stop?; What Happens if a Pendulum is Pushed Too Hard?; Period of a Nonlinear Pendulum; Nonlinear Oscillations with Damping; Equilibrium Positions and Linearized Stability; Nonlinear Pendulum with Damping; Further Readings in Mechanical Vibrations; Part 2: Population Dynamics--Mathematical Ecology. Introduction to Mathematical Models in Biology; Population Models; A Discrete One-Species Model; Constant Coefficient First-Order Difference Equations; Exponential Growth; Discrete Once-Species Models with an Age Distribution; Stochastic Birth Processes; Density-Dependent Growth; Phase Plane Solution of the Logistic Equation; Explicit Solution of the Logistic Equation; Growth Models with Time Delays; Linear Constant Coefficient Difference Equations; Destabilizing Influence of Delays; Introduction to Two-Species Models; Phase Plane, Equilibrium, and linearization; System of Two Constant Coefficient First-Order Differential Equations, Stability of Two-Species Equilibrium Populations; Phase Plane of Linear Systems; Predator-Prey Models; Derivation of the Lotka-Volterra Equations; Qualitative Solution of the Lotka- Volterra Equations; Average Populations of Predators and Preys; Man's Influence on Predator-Prey Ecosystems; Limitations of the Lotka-Volterra Equation; Two Competing Species; Further Reading in Mathematical Ecology; Part 3: Traffic Flow. Introduction to Traffic Flow; Automobile Velocities and a Velocity Field; Traffic Flow and Traffic Density; Flow Equals Density Times Velocity; Conservation of the Number of Cars; A Velocity-Density Relationship; Experimental Observations; Traffic Flow; Steady-State Car-Following Models; Partial Differential Equations; Linearization; A Linear Partial Differential Equation; Traffic Density Waves; An Interpretation of Traffic Waves; A Nearly Uniform Traffic Flow Example; Nonuniform Traffic - The Method of Characteristics; After a Traffic Light Turns Green; A Linear Velocity-Density Relationship; An Example; Wave Propagation of Automobile Brake Lights; Congestion Ahead; Discontinuous Traffic; Uniform Traffic Stopped by a Red Light; A Stationary Shock Wave; The Earliest Shock; Validity of Linearization; Effect of a Red Light or an Accident; Exits and Entrances; Constantly Entering Cars; A Highway Entrance; Further Reading in Traffic Flow; Index.

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Thursday, February 19, 2009