The Princeton Companion to Applied Mathematics.pdf

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Cover

Title

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

The Princeton Companion to Applied Mathematics

The Princeton Companion to Applied Mathematics editor Nicholas J. Higham The University of Manchester associate editors Mark R. Dennis University of Bristol Paul Glendinning The University of Manchester Paul A. Martin Colorado School of Mines Fadil Santosa University of Minnesota Jared Tanner University of Oxford Princeton University Press Princeton and Oxford

Copyright © 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu Jacket image courtesy of iStock All Rights Reserved Library of Congress Cataloging-in-Publication Data The Princeton companion to applied mathematics / editor, Nicholas J. Higham, The University of Manchester ; associate editors, Mark R. Dennis, University of Bristol [and four others]. pages cm Includes bibliographical references and index. ISBN 978-0-691-15039-0 (hardcover : alk. paper) 1. Algebra. 2. Mathematics. 3. Mathematical models. I. Higham, Nicholas J., 1961– editor. II. Dennis, Mark R., editor. III. Title: Companion to applied mathematics. IV. Title: Applied mathematics. QA155.P75 2015 510—dc23 2015013024 British Library Cataloging-in-Publication Data is available This book has been composed in LucidaBright Project management, composition and copyediting by T&T Productions Ltd, London Printed on acid-free paper ∞ Printed in the United States of America 1 2 3 4 5 6 7 8 9 10

Contents Preface Contributors Part I Introduction to Applied Mathematics I.1 I.2 I.3 I.4 I.5 I.6 What Is Applied Mathematics? The Language of Applied Mathematics Methods of Solution Algorithms Goals of Applied Mathematical Research The History of Applied Mathematics Part II Concepts Asymptotics Boundary Layer Chaos and Ergodicity Complex Systems Conformal Mapping Conservation Laws Control Convexity Dimensional Analysis and Scaling The Fast Fourier Transform Finite Diﬀerences The Finite-Element Method Floating-Point Arithmetic Functions of Matrices Function Spaces Graph Theory II.1 II.2 II.3 II.4 II.5 II.6 II.7 II.8 II.9 II.10 II.11 II.12 II.13 II.14 II.15 II.16 II.17 Homogenization II.18 Hybrid Systems II.19 II.20 II.21 II.22 II.23 II.24 Integral Transforms and Convolution Interval Analysis Invariants and Conservation Laws The Jordan Canonical Form Krylov Subspaces The Level Set Method ix xiii 1 8 27 40 48 55 81 82 82 83 84 86 88 89 90 94 95 96 96 97 99 101 103 103 104 105 106 112 113 114 Nonlinear Equations and Newton’s Method II.25 Markov Chains II.26 Model Reduction II.27 Multiscale Modeling II.28 II.29 Orthogonal Polynomials II.30 II.31 II.32 II.33 II.34 II.35 II.36 Wave Phenomena Shocks Singularities The Singular Value Decomposition Tensors and Manifolds Uncertainty Quantiﬁcation Variational Principle Part III Equations, Laws, and Functions of Applied Mathematics III.1 III.2 III.3 III.4 III.5 III.6 III.7 Benford’s Law Bessel Functions The Black–Scholes Equation The Burgers Equation The Cahn–Hilliard Equation The Cauchy–Riemann Equations The Delta Function and Generalized Functions The Diﬀusion Equation The Dirac Equation III.8 III.9 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 116 117 119 120 122 122 124 126 127 131 134 134 135 137 137 138 138 139 139 142 142 144 146 147 148 148 149 150 151 155 156 158 159 160

vi Contents 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 Complex Analysis Ordinary Diﬀerential Equations Partial Diﬀerential Equations Integral Equations Perturbation Theory and Asymptotics Calculus of Variations Special Functions Spectral Theory Approximation Theory IV.1 IV.2 IV.3 IV.4 IV.5 IV.6 IV.7 IV.8 IV.9 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 162 163 165 167 167 168 169 170 171 173 181 190 200 208 218 227 236 248 263 281 293 306 319 327 335 350 360 374 383 393 402 411 419 428 446 458 467 476 485 500 505 516 523 533 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) Sport Inerters Mathematical Biomechanics Mathematical Physiology Cardiac Modeling Chemical Reactions Divergent Series: Taming the Tails Financial Mathematics Portfolio Theory Bayesian Inference in Applied Mathematics V.2 V.3 V.4 V.5 V.6 V.7 V.8 V.9 V.10 V.11 V.12 A Symmetric Framework with Many Applications V.13 Granular Flows V.14 Modern Optics V.15 Numerical Relativity V.16 V.17 V.18 Numerical Weather Prediction V.19 V.20 V.21 Tsunami Modeling Shock Waves Turbulence The Spread of Infectious Diseases The Mathematics of Sea Ice Part VI Example Problems Cloaking VI.1 Bubbles VI.2 Foams VI.3 Inverted Pendulums VI.4 Insect Flight VI.5 The Flight of a Golf Ball VI.6 Automatic Diﬀerentiation VI.7 Knotting and Linking of Macromolecules VI.8 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: 545 552 564 570 579 591 598 604 609 616 623 627 634 640 648 658 661 665 673 680 687 694 705 712 720 724 733 735 737 741 743 746 749 752 755 757 759 761 763 767 Nonsmooth Dynamics and Its Applications 769

Contents VI.16 From the N-Body Problem to Astronomy and VII.19 Airport Baggage Screening with X-Ray 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 VII.3 Geometry Processing and Modeling Computer-Aided Proofs via Interval Analysis VII.4 Applications of Max-Plus Algebra VII.5 Evolving Social Networks, Attitudes, and Beliefs—and Counterterrorism Chip Design Color Spaces and Digital Imaging VII.6 VII.7 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 771 775 778 783 787 790 795 800 804 808 813 816 823 828 839 843 847 852 857 860 864 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 Color plates follow page 364 vii 866 868 873 879 883 887 891 897 903 906 912 916 925 933 943 953 963

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