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).

Deformations of Lie Algebras using σ-Derivations

We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N×K composite system, induces a unique measure in the space of N×N mixed states (or in the space of K×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N×N pure states behaves as lnN-1/2.

https://iopscience.iop.org/article/10.1088/0305-4470/34/35/335/pdf

Induced measures in the space of mixed quantum states

The evaluation of the minimal evolution time between two distinguishable states of a system is important for assessing the maximal speed of quantum computers and communication channels. Lower bounds for this minimal time have been proposed for unitary dynamics. Here we show that it is possible to extend this concept to nonunitary processes, using an attainable lower bound that is connected to the quantum Fisher information for time estimation. This result is used to delimit the minimal evolution time for typical noisy channels.

Quantum speed limit for physical processes.

Chapter 34 - Non-Commutative Lp-Spaces

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.

Ziemowit Popowicz

Papers

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

q-deformations of Virasoro algebra and conformal dimensions

Quantum conformal algebra with central extension

q-Virasoro algebra and its relation to the q-deformed KdV system

A 2-component or N=2 supersymmetric Camassa–Holm equation