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).

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Finite element methods for Maxwell's equations.

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method

In nonlinear analysis, perturbation methods are well established tools to study diverse aspects of nonlinear problems. Surveys of the early literature with numerous references, and useful bibliographies, have been given by Nayfeh [1], Mickens [2], Jordan and Smith [3] and Hagedorn [4], However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. Therefore, new analytical techniques should be developed to overcome these shortcomings. Such a new technique should work over a large range of parameters and yield accurate analytical approximate solutions beyond the coverage and ability of the classical perturbation methods. For example, homotopy perturbation method introduced by He [5-14] can readily eliminate the limitations of the traditional perturbation techniques. Moreover, this method was first applied to nonlinear oscillators with discontinuities[15]. For more detailed information please refer to the comprehensive book[16] or the review articles[ 17,18] by He. There also exists a wide range of literature dealing with the approximate determination of periodic solutions for nonlinear problems by using a mixture of methodologies [19-37], The purpose of this paper is to make the comparison of two distinct adaptations of He's homotopy perturbation method (HPM) for determining frequency-amplitude relation of the nonlinear oscillator with discontinuities.

A Comparative Study of He's Homotopy Perturbation Method for Determining Frequency-amplitude Relation of a Nonlinear Oscillator with Discontinuities

In this paper, approximate-exact solutions of a class of Lane–Emden type singular IVPs problems, by the variational iteration method, are presented. The variational iteration method yields solutions in the forms of convergent series with easily calculable terms. The scheme is shown to be highly accurate, and in some cases, yields exact solutions in few iterations.

Solutions of singular IVPs of Lane–Emden type by the variational iteration method

On the coupling of the homotopy perturbation method and Laplace transformation

In this study, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations. The analysis is accompanied by numerical examples. The results show that the homotopy perturbation method is of high accuracy, more convenient and efficient for solving integro-differential equations.

Ahmet Yildirim

Papers

Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method

A Comparative Study of He's Homotopy Perturbation Method for Determining Frequency-amplitude Relation of a Nonlinear Oscillator with Discontinuities

Solutions of singular IVPs of Lane–Emden type by the variational iteration method

On the coupling of the homotopy perturbation method and Laplace transformation

Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method