An ideal fibre-reinforced fluid is incompressible and inextensible along a family of material curves that are convected with the fluid. It is a model for continuous fibre-resin systems in the fluid state in which forming processes take place. Like liquid crystals, these fluids have strong directional properties. The kinematic and constitutive theory of ideal fibre-reinforced fluids is described, with particular reference to plane flows. The class of flows in which the fibres are aligned along the streamlines is considered, and an explanation is given for the observed prevalence of this class of flows.

We consider a Boltzmann-like model of outgassing and contamination in a three-dimensional region V=V1∪V2∪V3. V1 is the region where the contaminant particles are produced, and V2 is the region where such particles migrate and interact with some inert gas. V3 is where contamination takes place because of the particles emanating from V2. In each of the three regions, the behaviour of the contaminant particles is represented by means of a Boltzmann-like equation. We show that such a problem has a unique positive strict solution, belonging to a suitable L1 Banach space X. Finally, a system of ordinary differential equations is derived which gives the evolution of the total number of contaminant particles in each of the three regions.

Local solutions near the intersection of a free surface with a vertical wall are constructed numerically. Both gravity and surface tension are included in the dynamic boundary condition. It was shown that the solutions are characterized by the angle γ between the free surface and the wall at the separation point. There is a solution for each value of π/2<γ<π. As γ→π/2 and γ→π, the solutions reduce to the pure gravity solutions of Vanden-Broeck & Tuck [1].

We consider a model of the motion of a viscous dielectric liquid subjected to a DC electric field when the bulk conduction results from the presence of a dissociation-recombination process. It is shown that any weak solution approaches a neighbourhood of a spatially homogeneous steady state with radius r≈(d++d−)&14frac;, where d+, d− are the diffusion coefficients.

The bifurcation from a normally conducting state to a superconducting state in a decreasing magnetic field is studied for a slab geometry. The leading eigenvalue is a double eigenvalue, leading to a rich structure of possible behaviours. A weakly-nonlinear stability analysis is performed, and the possible responses of the material are classified. Finally, the leading-order equations are solved numerically for a wide range of parameter values to determine which of these behaviours will occur in practice.

A standard model for one-dimensional phase transitions is the second-order semilinear equation with bistable nonlinearity, where one seeks a solution which connects the two stable values. From an Ising-like model but which includes long-range interaction, one is led to consider the equation where the second-order operator is replaced by one of arbitrarily high order. Others have found the desired heteroclinic solutions for such equations, under the assumption that the higher-order terms have small coefficients, by employing singular perturbation methods for dynamical systems. Here, without making any assumption on the sizes of the coefficients, we obtain such heteroclinic solutions by using variational methods under the assumption that the nonlinearity arises from a potential having two wells of equal depths.

This paper studies a vectorial problem in the calculus of variations arising in the theory of martensitic microstructure. The functional has an integral representation where the integrand is a non-convex function of the gradient with exactly four minima. We prove that the Young measure corresponding to a minimizing sequence is homogeneous and unique for certain linear boundary conditions. We also consider the singular perturbation of the problem by higher-order gradients. We study an example of microstructure involving infinite sequential lamination and calculate its energy and length scales in the zero limit of the perturbation.

The theorem of bifurcation from a simple eigenvalue is applied to prove non-uniqueness for the problem of a layer of a dielectric liquid subjected to an electric field and to injection of charges on the electrodes.

Inequalities for martingales with bounded differences have recently proved to be very useful in combinatorics and in the mathematics of operational research and computer science. We see here that these inequalities extend in a natural way to ‘centering sequences’ with bounded differences, and thus include, for example, better inequalities for sequences related to sampling without replacement.

Considering strings over a finite alphabet [Ascr], say that a string is w-avoiding if it does not contain w as a substring. It is known that the number aw(n) of w-avoiding strings of length n depends only on the autocorrelation of w as defined by Guibas–Odlyzko. We give a simple criterion on the autocorrelations of w and w′ for determining whether aw(n) > aw′(n) for all large enough n.

The prime factorization of a random integer has a GEM/Poisson-Dirichlet distribution as transparently proved by Donnelly and Grimmett [8]. By similarity to the arc-sine law for the mean distribution of the divisors of a random integer, due to Deshouillers, Dress and Tenenbaum [6] (see also Tenenbaum [24, II.6.2, p. 233]), – the ‘DDT theorem’ – we obtain an arc-sine law in the GEM/Poisson-Dirichlet context. In this context we also investigate the distribution of the number of components larger than ε which correspond to the number of prime factors larger than nε.