to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Numerical Initial Value Problems in Ordinary Differential Equations

https://www.cims.nyu.edu/~mogilner/actin.pdf

Cell motility driven by actin polymerization

Singular perturbations and order reduction in control theory - An overview

Introduction to singular perturbations

Analyzing Multiscale Phenomena Using Singular Perturbation Methods

Phase-plane solutions to some singular perturbation problems

These lecture notes are intended to provide an elementary account of some of the recent mathematical effort in applying singular perturbations theory to optimal control problems, to demonstrate the practical importance of this asymptotic technique to current engineering studies, and to suggest several open problems needing further research. Readers are referred to the survey article by Kokotovic, O'Malley, and Sannuti for a discussion of related topics and for additional references.

Singular perturbations and optimal control

This paper considers the motion of the shock layer exhibited by the solution of the initial-boundary value problem for the equation $\epsilon u_{xx} + uu_x = u_t $ with small values of $\epsilon > 0$. This Burgers’ equation occurs as a model for a number of physical problems and is representative of many convection-dominated evolution equations. A specific example will be solved explicitly using the Cole–Hopf transformation. The result suggests a corresponding asymptotic approach which provides the shock layer location $x_\epsilon ( t )$ as the solution of an initial value problem. That method, which can be applied to a far wider class of equations, also shows how the steady-state shock location can be changed substantially by asymptotic exponentially small perturbations.

Robert E. O'Malley

Papers

Introduction to singular perturbations

Analyzing Multiscale Phenomena Using Singular Perturbation Methods

Phase-plane solutions to some singular perturbation problems

Singular perturbations and optimal control

Shock layer movement for Burgers' equation