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.

We are interested in a function f(p) that represents the probability that a random subset of edges of a Δ-regular graph G contains half the edges of some cycle of G. f(p) is also the probability that a codeword is corrupted beyond recognition when words of the cycle code of G are submitted to the binary symmetric channel. We derive a precise upper bound on the largest p for which f(p) can vanish when the number of edges of G goes to infinity. To this end, we introduce the notion of fractional percolation on trees, and calculate the related critical probabilities.

Let [Mscr]n,k(S) be the set of n-edge k-vertex rooted maps in some class on the surface S. Let P be a planar map in the class. We develop a method for showing that almost all maps in [Mscr]n,k(S) contain many copies of P. One consequence of this is that almost all maps in [Mscr]n,k(S) have no symmetries. The classes considered include c-connected maps (c [les ] 3) and certain families of degree restricted maps.

A tournament T on a set V of n players is an orientation of the edges of the complete graph Kn on V; T will be called a random tournament if the directions of these edges are determined by a sequence {Yj[ratio ]j = 1, …, (n2)} of independent coin flips. If (y, x) is an edge in a (random) tournament, we say that y beats x. A set A ⊂ V, |A| = k, is said to be beaten if there exists a player y ∉ A such that y beats x for each x ∈ A. If such a y does not exist, we say that A is unbeaten. A (random) tournament on V is said to have property Sk if each k-element subset of V is beaten. In this paper, we use the Stein–Chen method to show that the probability distribution of the number W0 of unbeaten k-subsets of V can be well-approximated by that of a Poisson random variable with the same mean; an improved condition for the existence of tournaments with property Sk is derived as a corollary. A multivariate version of this result is proved next: with Wj representing the number of k-subsets that are beaten by precisely j external vertices, j = 0, 1, …, b, it is shown that the joint distribution of (W0, W1, …, Wb) can be approximated by a multidimensional Poisson vector with independent components, provided that b is not too large.

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i[]Zi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ε (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ε (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

A model for a random random-walk on a finite group is developed where the group elements that generate the random-walk are chosen uniformly and with replacement from the group. When the group is the d-cube Zd2, it is shown that if the generating set is size k then as d → ∞ with k − d → ∞ almost all of the random-walks converge to uniform in k ln (k/(k − d))/4+ρk steps, where ρ is any constant satisfying ρ > −ln (ln 2)/4.

Let G be a loopless 2-connected graph whose largest cycle has c edges, and whose largest bond has c* edges. We show that |E(G)| [les ] ½cc*.

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

Lemke and Kleitman [2] showed that, given a positive integer d and d (necessarily non-distinct) divisors of da1, …, ad there exists a subset Q ⊆ {1, …, d} such that d = [sum ]i∈Qai answering a conjecture of Erdo″s and Lemke. Here we extend this result, showing that, provided [sum ]p|d1/p [les ] 1 (where the sum is taken over all primes p), there is some collection from a1, …, ad which both sum to d and which can themselves be ordered so that each element divides its successor in the order. Furthermore, we shall show that the condition on the prime divisors is in some sense also necessary.