The Princeton Companion to Applied Mathematics

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Contributors

David Acheson, Emeritus Fellow, Jesus College, Thomas J. Brennan, Professor of Law, Harvard Law School University of Oxford portfolio theory [V.10] inverted pendulums [VI.4], teaching applied mathematics [VIII.7] David S. Broomhead, Professor of Applied Mathematics, The University of Manchester (deceased) Miguel A. Alonso, Associate Professor, applications of max-plus algebra [VII.4] The Institute of Optics at the University of Rochester modern optics [V.14] Kurt Bryan, Professor of Mathematics, Rose–Hulam Institute of Technology Douglas N. Arnold, McKnight Presidential Professor of cloaking [VI.1] Mathematics, University of Minnesota the flight of a golf ball [VI.6] Dorothy Buck, Reader in BioMathematics, Imperial College London Karl Johan Åström, Emeritus Professor, knotting and linking of macromolecules [VI.8] Department of Automatic Control, Lund Institute of Technology/University of Lund Chris Budd, Professor of Applied Mathematics, control theory [IV.34] University of Bath; Professor of Mathematics, Royal Institution of Great Britain David H. Bailey, Lawrence Berkeley National Laboratory slipping, sliding, rattling, and impact: (retired); Research Fellow, University of California, Davis nonsmooth dynamics and its applications [VI.15] experimental applied mathematics [VIII.6] John A. Burns, Hatcher Professor of Mathematics and June Barrow-Green, Senior Lecturer in the History of Technical Director for the Interdisciplinary Center for Mathematics, The Open University Applied Mathematics, Virginia Tech the history of applied mathematics [I.6] optimal sensor location in the control of energy-efficient buildings [VI.13] Peter Benner, Director, Max Planck Institute for Dynamics of Complex Technical Systems model reduction Daniela Calvetti, The James Wood Williamson Professor, [II.26] Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University Andrew J. Bernoff, Kenneth and Diana Jonsson Professor of dimensional analysis and scaling Mathematics, Harvey Mudd College [II.9] the thin-film equation [III.29] Eric Cancès, Professor of Analysis, Ecole des Ponts and INRIA electronic structure calculations (solid state physics) Michael V. Berry, Melville Wills Professor of Physics (Emeritus), University of Bristol [VII.14] divergent series: taming the tails [V.8] René Carmona, Paul M. Wythes ’55 Professor of Michael W. Berry, Professor, Department of Electrical Engineering and Finance, Bendheim Center for Finance, Engineering and Computer Science, University of Tennessee ORFE, Princeton University financial mathematics text mining [VII.24] [V.9]

Brett Borden, Professor of Physics, C. J. Chapman, Professor of Applied Mathematics, The Naval Postgraduate School, Monterey, California University of Keele radar imaging [VII.17] shock waves [V.20], aircraft noise [VII.1]

Jeﬀrey T. Borggaard, Professor of Mathematics, Virginia Tech S. Jonathan Chapman, Professor of Mathematics and Its optimal sensor location in the control of Applications, University of Oxford energy-efﬁcient buildings [VI.13] the ginzburg–landau equation [III.14]

Jonathan M. Borwein, Laureate Professor, Gui-Qiang G. Chen, Statutory Professor in the Analysis of Partial School of Mathematical and Physical Sciences, Diﬀerential Equations and Professorial Fellow of Keble College, University of Newcastle, Australia University of Oxford experimental applied mathematics [VIII.6] the tricomi equation [III.30]

Fred Brauer, Professor Emeritus of Mathematics, Margaret Cheney, Professor of Mathematics and Albert C. Yates University of Wisconsin–Madison Endowment Chair, Colorado State University the spread of infectious diseases [V.16] radar imaging [VII.17]

For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. xiv Contributors

Peter A. Clarkson, Professor of Mathematics, University of Kent Charles L. Epstein, Thomas A. Scott Professor of Mathematics, the painlevé equations [III.24] University of Pennsylvania medical imaging [VII.9] Eugene M. Cliﬀ, Professor Emeritus, Interdisciplinary Center for Applied Mathematics, Virginia Tech Bard Ermentrout, Distinguished University Professor of optimal sensor location in the control of Computational Biology and Professor of Mathematics, energy-efﬁcient buildings [VI.13] University of Pittsburgh mathematical neuroscience [VII.21] Paul G. Constantine, Ben L. Fryrear Assistant Professor of Applied Mathematics and Statistics, Colorado School of Mines Maria Esteban, Director of Research, CNRS ranking web pages [VI.9] mathematics and policy [VIII.9]

William Cook, Professor of Combinatorics and Optimization, Lawrence C. Evans, Professor, Department of Mathematics, University of Waterloo University of California, Berkeley the traveling salesman problem [VI.18] partial differential equations [IV.3]

Robert M. Corless, Distinguished University Professor, Hans G. Feichtinger, Faculty of Mathematics, Department of Applied Mathematics, University of Vienna The University of Western Ontario function spaces [II.15] the lambert W function [III.17] Martin Feinberg, Morrow Professor of Chemical & Biomolecular Darren Crowdy, Professor of Applied Mathematics, Engineering and Professor of Mathematics, Imperial College London The Ohio State University conformal mapping [II.5] chemical reactions [V.7]

James M. Crowley, Executive Director, Alistair D. Fitt, Vice-Chancellor, Oxford Brookes University Society for Industrial and Applied Mathematics mathematics and policy [VIII.9] mathematics and policy [VIII.9] Irene Fonseca, Mellon College of Science University Professor of Annie Cuyt, Professor, Department of Mathematics & Mathematics and Director of Center for Nonlinear Analysis, Computer Science, University of Antwerp Carnegie Mellon University approximation theory [IV.9] calculus of variations [IV.6] E. Brian Davies, Emeritus Professor of Mathematics, L. B. Freund, Adjunct Professor, Department of Materials Science King’s College London and Engineering, University of Illinois at Urbana-Champaign spectral theory [IV.8] mechanics of solids [IV.32] Timothy A. Davis, Professor, Department of Computer Science and Engineering, Texas A&M University David F. Gleich, Assistant Professor of Computer Science, Purdue University graph theory [II.16], searching a graph [VI.10] ranking web pages [VI.9] Florent de Dinechin, Professor of Applied Sciences, INSA—Lyon Paul Glendinning, Professor of Applied Mathematics, evaluating elementary functions [VI.11] The University of Manchester chaos and ergodicity [II.3], complex systems [II.4], Mark R. Dennis, Professor of Theoretical Physics, hybrid systems [II.18], the euler–lagrange equations University of Bristol [III.12], the logistic equation [III.19], the lorenz equations bifurcation theory invariants and conservation laws [II.21], [III.20], [IV.21] tensors and manifolds [II.33], the dirac equation [III.9], maxwell’s equations [III.22], Joe D. Goddard, Professor of Applied Mechanics and schrödinger’s equation [III.26] Engineering Science, University of California, San Diego granular ﬂows [V.13] Jack Dongarra, Professor, University of Tennessee; Professor, Oak Ridge National Laboratory; Kenneth M. Golden, Professor of Mathematics/Adjunct Professor Professor, The University of Manchester of Bioengineering, University of Utah high-performance computing [VII.12] the mathematics of sea ice [V.17]

David L. Donoho, Anne T. and Robert M. Bass Professor in the Timothy Gowers, Royal Society Research Professor, Humanities and Sciences, Stanford University Department of Pure Mathematics and Mathematical Statistics, reproducible research in the mathematical sciences University of Cambridge [VIII.5] mathematical writing [VIII.1]

Ivar Ekeland, Professor Emeritus, CEREMADE and Thomas A. Grandine, Senior Technical Fellow, Institut de Finance, Université Paris-Dauphine The Boeing Company mathematical economics [VII.20] a hybrid symbolic–numeric approach to geometry processing and modeling [VII.2] Yonina C. Eldar, Professor of Electrical Engineering, Technion—Israel Institute of Technology, Haifa Andreas Griewank, Professor of Mathematics, compressed sensing [VII.10] Humboldt University of Berlin automatic differentiation [VI.7] George F. R. Ellis, Professor Emeritus, Mathematics Department, University of Cape Town David Griﬃths, Emeritus Professor of Physics, Reed College general relativity and cosmology [IV.40] quantum mechanics [IV.23]

Contributors xv

Peter Grindrod, Professor of Mathematics, University of Oxford Julian C. R. Hunt, Emeritus Professor of Climate Modelling and evolving social networks, attitudes, and Honorary Professor of Mathematics, University College London beliefs—and counterterrorism [VII.5] turbulence [V.21]

Julio C. Gutiérrez-Vega, Director of the Optics Center, Stefan Hutzler, Associate Professor, School of Physics, Technológico de Monterrey Trinity College Dublin mathieu functions [III.21] foams [VI.3]

Ernst Hairer, Honorary Professor of Mathematics, Richard D. James, Professor, Department of Aerospace University of Geneva Engineering and Mechanics, University of Minnesota numerical solution of ordinary differential equations continuum mechanics [IV.26] [IV.12] David J. Jeffrey, Professor, Department of Applied Mathematics, Ian Hawke, Associate Professor, Mathematical Sciences, The University of Western Ontario University of Southampton the lambert W function [III.17] numerical relativity [V.15] Oliver E. Jensen, Sir Horace Lamb Professor, Stephan Held, Professor, Research Institute for School of Mathematics, The University of Manchester Discrete Mathematics, Bonn University mathematical biomechanics [V.4] chip design [VII.6] Chris R. Johnson, Director, Scientific Computing and Imaging Didier Henrion, Professor, LAAS-CNRS, University of Toulouse; Institute; Distinguished Professor, School of Computing, Professor, Faculty of Electrical Engineering, University of Utah Czech Technical University in Prague visualization [VII.13] convexity [II.8] Chandrika Kamath, Member of Technical Staff, Willy A. Hereman, Professor of Applied Mathematics, Lawrence Livermore National Laboratory Colorado School of Mines data mining and analysis [IV.17] the korteweg–de vries equation [III.16] Randall D. Kamien, Vicki and William Abrams Professor in the Desmond J. Higham, 1966 Professor of Numerical Analysis, Natural Sciences, University of Pennsylvania University of Strathclyde soft matter [IV.33] bayesian inference in applied mathematics [V.11] Jonathan Peter Keating, Henry Overton Wills Professor of Nicholas J. Higham, Richardson Professor of Applied Mathematics, University of Bristol Mathematics, The University of Manchester random-matrix theory [IV.24] what is applied mathematics? [I.1], the language of applied mathematics [I.2], methods of solution [I.3], David E. Keyes, Professor of Applied Mathematics and algorithms [I.4], goals of applied mathematical research Computational Science and Director, Extreme Computing [I.5], control [II.7], finite differences [II.11], Research Center, King Abdullah University of Science and the finite-element method [II.12], floating-point Technology; Professor of Applied Mathematics and Applied arithmetic [II.13], functions of matrices [II.14], Physics, Columbia University integral transforms and convolution [II.19], the jordan computational science [IV.16] canonical form [II.22], orthogonal polynomials [II.29], the singular value decomposition [II.32], Barbara Lee Keyfitz, Dr. Charles Saltzer Professor of variational principle [II.35], the black–scholes equation Mathematics, The Ohio State University [III.3], the sylvester and lyapunov equations [III.28], conservation laws [II.6], shocks [II.30] numerical linear algebra and matrix analysis [IV.10], color spaces and digital imaging [VII.7], David Krakauer, President and Professor, Santa Fe Institute programming languages: an applied mathematics view the mathematics of adaptation (or the ten avatars of vishnu) [VII.11], how to read and understand a paper [VIII.2], [V.1] workflow [VIII.4] Rainer Kress, Professor Emeritus, Institut für Numerische und Theodore P. Hill, Professor Emeritus of Mathematics, Angewandte Mathematik, University of Göttingen Georgia Institute of Technology integral equations [IV.4] benford’s law [III.1] Alan J. Laub, Professor, Department of Electrical Engineering/ Philip Holmes, Eugene Higgins Professor of Mechanical and Mathematics, University of California, Los Angeles Aerospace Engineering and Professor of Applied and the riccati equation [III.25] Computational Mathematics, Princeton University dynamical systems [IV.20] Anita T. Layton, Robert R. and Katherine B. Penn Associate Professor, Department of Mathematics, Duke University Stefan Hougardy, Professor of Mathematics, University of Bonn mathematical physiology [V.5] chip design [VII.6] Tanya Leise, Associate Professor of Mathematics, Christopher J. Howls, Professor of Mathematics, Amherst College University of Southampton cloaking [VI.1] divergent series: taming the tails [V.8] Giovanni Leoni, Professor, Department of Yifan Hu, Principal Research Scientist, Yahoo Labs Mathematical Sciences, Carnegie Mellon University graph theory [II.16] calculus of variations [IV.6] David W. Hughes, Professor of Applied Mathematics, Randall J. LeVeque, Professor, Department of Applied University of Leeds Mathematics, University of Washington magnetohydrodynamics [IV.29] tsunami modeling [V.19]

Rachel Levy, Associate Professor of Mathematics, Peter D. Miller, Professor of Mathematics, Harvey Mudd College The University of Michigan, Ann Arbor teaching applied mathematics [VIII.7] perturbation theory and asymptotics [IV.5]

W. R. B. Lionheart, Professor of Applied Mathematics, H. K. Moﬀatt, Emeritus Professor of Mathematical Physics, The University of Manchester University of Cambridge airport baggage screening with x-ray tomography [VII.19] the navier–stokes equations [III.23], ﬂuid dynamics [IV.28]

Andrew W. Lo, Charles E. and Susan T. Harris Professor, Esteban Moro, Associate Professor, Department of Mathematics, Massachusetts Institute of Technology Universidad Carlos III de Madrid network analysis portfolio theory [V.10] [IV.18]

Christian Lubich, Professor, Mathematisches Institut, Clément Mouhot, Professor of Mathematical Sciences, Universität Tübingen University of Cambridge numerical solution of ordinary differential equations kinetic theory [IV.25] [IV.12] Jean-Michel Muller, Directeur de Recherche, CNRS evaluating elementary functions [VI.11] Peter Lynch, Emeritus Professor, School of Mathematical Sciences, University College, Dublin Tri-Dung Nguyen, Associate Professor in Operational Research numerical weather prediction [V.18] and Management Sciences, University of Southampton portfolio theory [V.10] Malcolm A. H. MacCallum, Emeritus Professor of Applied Mathematics, Queen Mary University of London Qing Nie, Professor, Department of Mathematics, Center for einstein’s field equations [III.10] Mathematical and Computational Biology, Center for Complex Biological Systems, University of California, Irvine Dian I. Martin, CEO, Senior Consultant, systems biology [VII.22] Small Bear Technologies, Inc. text mining [VII.24] Harald Niederreiter, Senior Scientist, RICAM, Austrian Academy of Sciences, Linz P. A. Martin, Professor of Applied Mathematics, random number generation [VI.12] Colorado School of Mines asymptotics [II.1], boundary layer [II.2], Amy Novick-Cohen, Professor, Department of Mathematics, integral transforms and convolution [II.19], Technion—Israel Institute of Technology, Haifa the cahn–hilliard equation singularities [II.31], wave phenomena [II.36], [III.5] bessel functions the burgers equation [III.2], [III.4], Bernt Øksendal, Professor, Department of Mathematics, the cauchy–riemann equations the diffusion [III.6], University of Oslo equation the euler equations the gamma [III.8], [III.11], applications of stochastic analysis [IV.14] function [III.13], hooke’s law [III.15], laplace’s equation [III.18], the shallow-water equations [III.27], the wave Alexander V. Panfilov, Professor, Department of Physics and equation [III.31], complex analysis [IV.1] Astronomy, Gent University cardiac modeling [V.6] Youssef Marzouk, Associate Professor, Department of Aeronautics and Astronautics and Center for Computational Nicola Parolini, Associate Professor of Numerical Analysis, Engineering, Massachusetts Institute of Technology Dipartimento di Matematica, MOX Politecnico di Milano uncertainty quantification [II.34] sport [V.2]

Moshe Matalon, Caterpillar Distinguished Professor, Kristin Potter, Scientific Software Consultant, Mechanical Science and Engineering, University of Oregon University of Illinois at Urbana-Champaign visualization [VII.13] flame propagation [VII.15] Andrea Prosperetti, C. A. Miller Jr. Professor of Sean McKee, Research Professor of Mathematics and Statistics, Mechanical Engineering, Johns Hopkins University; University of Strathclyde, Glasgow G. Berkhoff Professor of Applied Physics, University of Twente bubbles modeling a pregnancy testing kit [VII.18] [VI.2] Ian Proudler, Professor of Signal Processing, Ross C. McPhedran, Emeritus Professor in Physics, Loughborough University CUDOS, University of Sydney signal processing [IV.35] effective medium theories [IV.31] Alfio Quarteroni, Professor and Director, Chair of Modelling and John G. McWhirter, Distinguished Research Professor, Scientific Computing, Ecole Polytechnique Fédérale de Lausanne School of Engineering, Cardiff University sport [V.2] signal processing [IV.35] Anders Rantzer, Professor, Automatic Control, Beatrice Meini, Professor of Numerical Analysis, LTH Lund University University of Pisa control theory [IV.34] markov chains [II.25] Marcos Raydan, Professor, Departamento de Cómputo Científico James D. Meiss, Professor, Department of Applied Mathematics, y Estadística, Universidad Simón Bolívar University of Colorado at Boulder nonlinear equations and newton’s method [II.28] ordinary differential equations [IV.2] Daniel N. Rockmore, William H. Neukom 1964 Professor of Heather Mendick, Reader in Education, Brunel University Computational Science, Dartmouth College mediated mathematics: representations of mathematics the fast fourier transform [II.10], the mathematics of in popular culture and why these matter [VIII.8] adaptation (or the ten avatars of vishnu) [V.1]

Contributors xvii

Donald G. Saari, Distinguished Professor and Director, William W. Symes, Noah G. Harding Professor in Computational Institute for Mathematical Behavioral Sciences, and Applied Mathematics and Professor of Earth Science, University of California, Irvine Rice University from the N-body problem to astronomy and dark matter inverse problems [IV.15] [VI.16], voting systems [VII.25] Nico M. Temme, Emeritus Researcher, Fadil Santosa, Professor, School of Mathematics, Centrum Wiskunde & Informatica, Amsterdam University of Minnesota; Director, Institute for special functions [IV.7] Mathematics and its Applications homogenization [II.17], the level set method [II.24], David Tong, Professor of Theoretical Physics, multiscale modeling [II.27], inverse problems [IV.15] University of Cambridge classical mechanics [IV.19] Guillermo Sapiro, Edmund T. Pratt, Jr. School Professor of Warwick Tucker, Professor of Mathematics, Uppsala University Electrical and Computer Engineering, Duke University interval analysis [II.20], computer-aided proofs via mathematical image processing [VII.8] interval analysis [VII.3] Arnd Scheel, Professor, School of Mathematics, Peter R. Turner, Dean of Arts and Sciences and University of Minnesota Professor of Mathematics, Clarkson University pattern formation [IV.27] teaching applied mathematics [VIII.7] Emily Shuckburgh, Head of Open Oceans, P. J. Upton, Lecturer, Department of Mathematics and Statistics, British Antarctic Survey The Open University earth system dynamics [IV.30] the delta function and generalized functions [III.7] Reinhard Siegmund-Schultze, Faculty of Engineering and P. van den Driessche, Professor Emeritus of Science, University of Agder Mathematics and Statistics, University of Victoria the history of applied mathematics [I.6] the spread of infectious diseases [V.16] Valeria Simoncini, Professor of Numerical Analysis, Sergio Verdú, Eugene Higgins Professor of Alma Mater Studiorum Università di Bologna Electrical Engineering, Princeton University krylov subspaces [II.23] information theory [IV.36] Ronnie Sircar, Professor, Operations Research & Financial Cédric Villani, Professor of Mathematics, University Claude Engineering Department, Princeton University Bernard Lyon I; Director, Institut Henri Poincaré (CNRS/UPMC) ﬁnancial mathematics [V.9] kinetic theory [IV.25] Malcolm C. Smith, Professor, Department of Engineering, Jens Vygen, Professor, Research Institute for University of Cambridge Discrete Mathematics, University of Bonn inerters [V.3] combinatorial optimization [IV.38], chip design [VII.6]

Roel Snieder, W. M. Keck Distinguished Professor of Basic Charles W. Wampler, Technical Fellow, Exploration Science, Colorado School of Mines General Motors Global Research and Development imaging the earth using green’s theorem [VII.16] robotics [VI.14]

Erkki Somersalo, Professor, Department of Mathematics, Z. Jane Wang, Professor, Department of Physics, Applied Mathematics and Statistics, Cornell University Case Western Reserve University insect ﬂight [VI.5] dimensional analysis and scaling [II.9] Denis Weaire, Emeritus Professor, School of Physics, Frank Sottile, Professor of Mathematics, Texas A&M University Trinity College Dublin foams algebraic geometry [IV.39] [VI.3]

Ian Stewart, Emeritus Professor of Mathematics, Karen Willcox, Professor of Aeronautics and Astronautics, University of Warwick Massachusetts Institute of Technology uncertainty quantiﬁcation symmetry in applied mathematics [IV.22], how to write a [II.34] general interest mathematics book [VIII.3] Walter Willinger, Chief Scientist, NIKSUN, Inc. communication networks [VII.23] Victoria Stodden, Associate Professor, Graduate School of Library and Information Science, Peter Winkler, William Morrill Professor of Mathematics and University of Illinois at Urbana-Champaign Computer Science, Dartmouth College reproducible research in the mathematical sciences applied combinatorics and graph theory [IV.37] [VIII.5] Stephen J. Wright, Professor, Department of Computer Sciences, Gilbert Strang, Professor of Mathematics, University of Wisconsin-Madison Massachusetts Institute of Technology continuous optimization (nonlinear and linear a symmetric framework with many applications [V.12], programming) [IV.11] teaching applied mathematics [VIII.7] Lexing Ying, Professor of Mathematics, Stanford University Agnès Sulem, Researcher, INRIA Paris–Rocquencourt the N-body problem and the fast multipole method [VI.17] applications of stochastic analysis [IV.14] Ya-xiang Yuan, Professor, Institute of Computational Endre Süli, Professor of Numerical Analysis, University of Oxford Mathematics and Scientiﬁc/Engineering Computing, numerical solution of partial differential equations Chinese Academy of Sciences [IV.13] mathematics and policy [VIII.9]

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