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).

Distribution functions in physics: Fundamentals

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Kaluza-Klein gravity

As their name indicates, liquid crystals simultaneously exhibit some characteristics common to both ordinary isotropic liquids and solid crystals. This ambivalence is also found in the effects of surfaces on these systems which lead to a great diversity of phenomena. These phenomena are reviewed focusing on nematic liquid crystals which have the simplest structure among the many existing types and which have been the most extensively studied. Three main kinds of effects can be distinguished. The first concerns the perturbation of the liquid crystalline structure close to the surface. Beyond this transition region, the bulk liquid crystalline structure is recovered with an orientation which is fixed by the surface: this phenomenon of orientation of liquid crystals by surfaces is the so-called anchoring. Finally, close to bulk phase transitions, critical adsorption or wetting can occur at surfaces as is also seen in isotropic systems.

https://iopscience.iop.org/article/10.1088/0034-4885/54/3/002/pdf

Surface effects and anchoring in liquid crystals

The Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.

Wigner Method in Quantum Statistical Mechanics

A formalism is given which makes it possible for a modified form of local gauge invariance and minimal coupling to be compatible with torsion. One consequence is a restriction on the possible form of torsion in that it must be determined by the gradient of a scalar function. Furthermore, a dynamical theory is obtained for this scalar and hence allows propagation of torsion in vacuum. The Lagrangian density for interacting electromagnetic, gravitational, torsion, and complex scalar fields is presented, as well as the resulting field equations. The formalism implies the existence of both electric and magnetic currents due to the interaction of the electromagnetic and torsion fields.

Gauge invariance, minimal coupling, and torsion

A theory of gravity is developed which allows for propagating torsion. The theory predicts the existence of torsion in a vacuum and torsion waves emitted by sources with varying spin. The motion of spinning test particles in combined metric and torsion background fields is discussed. A plausible procedure for quantizing the torsion and metric fields is presented.

Propagating torsion and gravitation

It is shown that the three-body trigonometric G2 integrable system is exactly solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.

Solvability of the g2 integrable system

Explicit examples of quasi-exactly-solvable N-body problems on the line are presented. These are related to the hidden algebra slN, and they are of two types — containing up to N and up to six-body interactions only. Both types degenerate to the Calogero model.

Marcos Rosenbaum

Papers

Wigner Method in Quantum Statistical Mechanics

Gauge invariance, minimal coupling, and torsion

Propagating torsion and gravitation

Solvability of the g2 integrable system

Quasi-exactly-solvable many-body problems