Cover

Title

Copyright

Contents

Preface

Contributors

Part I Introduction to Applied Mathematics

I.1 What Is Applied Mathematics?

I.2 The Language of Applied Mathematics

I.3 Methods of Solution

I.4 Algorithms

I.5 Goals of Applied Mathematical Research

I.6 The History of Applied Mathematics

Part II Concepts

II.1 Asymptotics

II.2 Boundary Layer

II.3 Chaos and Ergodicity

II.4 Complex Systems

II.5 Conformal Mapping

II.6 Conservation Laws

II.7 Control

II.8 Convexity

II.9 Dimensional Analysis and Scaling

II.10 The Fast Fourier Transform

II.11 Finite Differences

II.12 The Finite-Element Method

II.13 Floating-Point Arithmetic

II.14 Functions of Matrices

II.15 Function Spaces

II.16 Graph Theory

II.17 Homogenization

II.18 Hybrid Systems

II.19 Integral Transforms and Convolution

II.20 Interval Analysis

II.21 Invariants and Conservation Laws

II.22 The Jordan Canonical Form

II.23 Krylov Subspaces

II.24 The Level Set Method

II.25 Markov Chains

II.26 Model Reduction

II.27 Multiscale Modeling

II.28 Nonlinear Equations and Newton’s Method

II.29 Orthogonal Polynomials

II.30 Shocks

II.31 Singularities

II.32 The Singular Value Decomposition

II.33 Tensors and Manifolds

II.34 Uncertainty Quantification

II.35 Variational Principle

II.36 Wave Phenomena

Part III Equations, Laws, and Functions of Applied Mathematics

III.1 Benford’s Law

III.2 Bessel Functions

III.3 The Black–Scholes Equation

III.4 The Burgers Equation

III.5 The Cahn–Hilliard Equation

III.6 The Cauchy–Riemann Equations

III.7 The Delta Function and Generalized Functions

III.8 The Diffusion Equation

III.9 The Dirac Equation

III.10 Einstein’s Field Equations

III.11 The Euler Equations

III.12 The Euler–Lagrange Equations

III.13 The Gamma Function

III.14 The Ginzburg–Landau Equation

III.15 Hooke’s Law

III.16 The Korteweg–de Vries Equation

III.17 The Lambert W Function

III.18 Laplace’s Equation

III.19 The Logistic Equation

III.20 The Lorenz Equations

III.21 Mathieu Functions

III.22 Maxwell’s Equations

III.23 The Navier–Stokes Equations

III.24 The Painlevé Equations

III.25 The Riccati Equation

III.26 Schrödinger’s Equation

III.27 The Shallow-Water Equations

III.28 The Sylvester and Lyapunov Equations

III.29 The Thin-Film Equation

III.30 The Tricomi Equation

III.31 The Wave Equation

Part IV Areas of Applied Mathematics

IV.1 Complex Analysis

IV.2 Ordinary Differential Equations

IV.3 Partial Differential Equations

IV.4 Integral Equations

IV.5 Perturbation Theory and Asymptotics

IV.6 Calculus of Variations

IV.7 Special Functions

IV.8 Spectral Theory

IV.9 Approximation Theory

IV.10 Numerical Linear Algebra and Matrix Analysis

IV.11 Continuous Optimization (Nonlinear and Linear Programming)

IV.12 Numerical Solution of Ordinary Differential Equations

IV.13 Numerical Solution of Partial Differential Equations

IV.14 Applications of Stochastic Analysis

IV.15 Inverse Problems

IV.16 Computational Science

IV.17 Data Mining and Analysis

IV.18 Network Analysis

IV.19 Classical Mechanics

IV.20 Dynamical Systems

IV.21 Bifurcation Theory

IV.22 Symmetry in Applied Mathematics

IV.23 Quantum Mechanics

IV.24 Random-Matrix Theory

IV.25 Kinetic Theory

IV.26 Continuum Mechanics

IV.27 Pattern Formation

IV.28 Fluid Dynamics

IV.29 Magnetohydrodynamics

IV.30 Earth System Dynamics

IV.31 Effective Medium Theories

IV.32 Mechanics of Solids

IV.33 Soft Matter

IV.34 Control Theory

IV.35 Signal Processing

IV.36 Information Theory

IV.37 Applied Combinatorics and Graph Theory

IV.38 Combinatorial Optimization

IV.39 Algebraic Geometry

IV.40 General Relativity and Cosmology

Part V Modeling

V.1 The Mathematics of Adaptation (Or the Ten Avatars of Vishnu)

V.2 Sport

V.3 Inerters

V.4 Mathematical Biomechanics

V.5 Mathematical Physiology

V.6 Cardiac Modeling

V.7 Chemical Reactions

V.8 Divergent Series: Taming the Tails

V.9 Financial Mathematics

V.10 Portfolio Theory

V.11 Bayesian Inference in Applied Mathematics

V.12 A Symmetric Framework with Many Applications

V.13 Granular Flows

V.14 Modern Optics

V.15 Numerical Relativity

V.16 The Spread of Infectious Diseases

V.17 The Mathematics of Sea Ice

V.18 Numerical Weather Prediction

V.19 Tsunami Modeling

V.20 Shock Waves

V.21 Turbulence

Part VI Example Problems

VI.1 Cloaking

VI.2 Bubbles

VI.3 Foams

VI.4 Inverted Pendulums

VI.5 Insect Flight

VI.6 The Flight of a Golf Ball

VI.7 Automatic Differentiation

VI.8 Knotting and Linking of Macromolecules

VI.9 Ranking Web Pages

VI.10 Searching a Graph

VI.11 Evaluating Elementary Functions

VI.12 Random Number Generation

VI.13 Optimal Sensor Location in the Control of Energy-Efficient Buildings

VI.14 Robotics

VI.15 Slipping, Sliding, Rattling, and Impact: Nonsmooth Dynamics and Its Applications

VI.16 From the N-Body Problem to Astronomy and Dark Matter

VI.17 The N-Body Problem and the Fast Multipole Method

VI.18 The Traveling Salesman Problem

Part VII Application Areas

VII.1 Aircraft Noise

VII.2 A Hybrid Symbolic–Numeric Approach to Geometry Processing and Modeling

VII.3 Computer-Aided Proofs via Interval Analysis

VII.4 Applications of Max-Plus Algebra

VII.5 Evolving Social Networks, Attitudes, and Beliefs—and Counterterrorism

VII.6 Chip Design

VII.7 Color Spaces and Digital Imaging

VII.8 Mathematical Image Processing

VII.9 Medical Imaging

VII.10 Compressed Sensing

VII.11 Programming Languages: An Applied Mathematics View

VII.12 High-Performance Computing

VII.13 Visualization

VII.14 Electronic Structure Calculations (Solid State Physics)

VII.15 Flame Propagation

VII.16 Imaging the Earth Using Green’s Theorem

VII.17 Radar Imaging

VII.18 Modeling a Pregnancy Testing Kit

VII.19 Airport Baggage Screening with X-Ray Tomography

VII.20 Mathematical Economics

VII.21 Mathematical Neuroscience

VII.22 Systems Biology

VII.23 Communication Networks

VII.24 Text Mining

VII.25 Voting Systems

Part VIII Final Perspectives

VIII.1 Mathematical Writing

VIII.2 How to Read and Understand a Paper

VIII.3 How to Write a General Interest Mathematics Book

VIII.4 Workflow

VIII.5 Reproducible Research in the Mathematical Sciences

VIII.6 Experimental Applied Mathematics

VIII.7 Teaching Applied Mathematics

VIII.8 Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter

VIII.9 Mathematics and Policy

Index