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).

A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, one seeks travelling wave solutions in the form of a finite series in tanh. If present, boundary conditions are implemented in this expansion. The associated velocity can then be determined a priori, provided the solution vanishes at infinity. Hence, exact closed form solutions can be obtained easily in various cases.

https://iopscience.iop.org/article/10.1088/0031-8949/54/6/003/pdf

The tanh method: I. Exact solutions of nonlinear evolution and wave equations

The long wavelength edge of the major chlorophyll absorption feature in the spectrum of a vegetation canopy moves to longer wavelengths with an increase in chlorophyll content. The position of this red-edge has been used successfully to estimate, by remote sensing, the chlorophyll content of vegetation canopies. Techniques used to estimate this red-edge position (REP) have been designed for use on small volumes of continuous spectral data rather than the large volumes of discontinuous spectral data recorded by contemporary satellite spectrometers. Also, each technique produces a different value of REP from the same spectral data and REP values are relatively insensitive to chlorophyll content at high values of chlorophyll content. This paper reports on the design and indirect evaluation of a surrogate REP index for use with spectral data recorded at the standard band settings of the Medium Resolution Imaging Spectrometer (MERIS). This index, termed the MERIS terrestrial chlorophyll index (MTCI), was evaluated using model spectra, field spectra and MERIS data. It was easy to calculate (and so can be automated), was correlated strongly with REP but unlike REP was sensitive to high values of chlorophyll content. As a result this index became an official MERIS level-2 product of the European Space Agency in March 2004. Further direct evaluation of the MTCI is proposed, using both greenhouse and field data.

The MERIS terrestrial chlorophyll index

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

/pdf/backlund-and-darboux-transformations-geometry-and-modern-4k80wq4g1i.pdf

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

Exact solutions to the KdV-Burgers' equation

Exact solutions to the Korteweg-de Vries-Burgers equation

It is shown that, depending on the ratio of the two eigenvalues in the inverse scattering method, the two loop soliton solution has some remarkable properties in collision processes.

Some Remarkable Properties of Two Loop Soliton Solutions

Part I of this paper reviews and extends the theory for the propagation of wavefronts defined by weak discontinuities. It is concerned with first order conservation type quasi-linear hyperbolic systems with discontinuous coefficient matrices. Transport equations are derived which take account of a strong discontinuity in the solution and the time of breakdown of the solution is obtained

The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients part I - fundamental theory †

https://apps.dtic.mil/sti/pdfs/ADA103878.pdf

Acceleration wave propagation in hyperelastic rods of variable cross-section

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