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).

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Computational geometry

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS?

Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schrödinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do not increase the norm.

On sharp Strichartz inequalities in low dimensions

Stabilizing effects of dispersion management

In this paper the monotonic twist theorem is extended to the quasiperiodic case and applied to establish regularity of motion in a system of a particle bouncing elastically between two quasiperiodically moving walls. It is shown that the velocity of the particle is uniformly bounded in time if the frequencies satisfy a Diophantine inequality. This answers a question recently asked in Levi and Zehnder (1995 SIAM J. Math. Anal. 26 1233-56).

/pdf/invariant-curve-theorem-for-quasiperiodic-twist-mappings-and-434y54nd38.pdf

Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem

Applying asymptotic methods to a previously derived system of ordinary differential equations, we present an analytical description of the slow ~average! dynamics of self-similar breathing pulses propagating in fiber links with dispersion management. We derive asymptotic averaged quantities ~adiabatic invariants! that characterize the stable pulse propagation. In a particular, but practically important, case when the dispersion compensation period is much larger than the amplification distance we have found analytically the fixed points corresponding to the dispersion-managed solitons. @S1063-651X~98!51007-0#

Average dynamics of the optical soliton in communication lines with dispersion management : analytical results

We revisit the averaged equation, derived in Pelinovsky et al. [Phys. Rev. Lett. 91, 240201 (2003)] from the nonlinear Schrodinger(NLS) equation with the nonlinearity management. We show that this averaged equation is valid only at the initial time interval, while a new Hamiltonian averaged NLS equation can be used at longer time intervals. Using the new averaged equation, we construct numerically matter-wave solitons in the context of the Bose-Einstein condensates under the Feshbach resonance management. We show that there is no threshold on the existence of dark solitons of large amplitudes, whereas such a threshold exists for bright solitons.

https://dmpeli.math.mcmaster.ca/PaperBank/BriefReportPRE.pdf

Vadim Zharnitsky

Papers

On sharp Strichartz inequalities in low dimensions

Stabilizing effects of dispersion management

Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem

Average dynamics of the optical soliton in communication lines with dispersion management : analytical results

Hamiltonian averaging for solitons with nonlinearity management.