It is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

The number of spanning trees in the giant component of the random graph (n, c/n) (c > 1) grows like exp{m(f(c)+o(1))} as n → ∞, where m is the number of vertices in the giant component. The function f is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on f′(c). A key lemma is the following. Let PGW(λ) denote a Galton–Watson tree having Poisson offspring distribution with parameter λ. Suppose that λ*>λ>1. We show that PGW(λ*) conditioned to survive forever stochastically dominates PGW(λ) conditioned to survive forever.

We consider the problem of minimizing the size of a family of sets such that every subset of {1,. . ., n} can be written as a disjoint union of at most k members of , where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of || so that every subset of {1,. . ., n} is the union of two sets in was asked by Erdős and studied recently by Füredi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of n and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n, k) for which n≤3k holds, and some individual values of n and k.

We review simple models of oil reservoirs and suggest some ideas for theoretical and numerical study of this important inverse problem. These models are formed by a system of an elliptic and a parabolic (or first-order hyperbolic) quasilinear partial differential equations. There are and probably there will be serious theoretical and computational difficulties mainly due to the degeneracy of the system. The practical value of the problem justifies efforts to improve the methods for its solution. We formulate ‘history matching’ as a problem of identification of two coefficients of this system. We consider global and local versions of this inverse problem and propose some approaches, including the use of the inverse conductivity problem and the structure of fundamental solutions. The global approach looks for properties of the ground in the whole domain, while the local one is aimed at recovery of these properties near wells. We discuss the use of the model proposed by Muskat which is a difficult free boundary problem. The inverse Muskat problem combines features of inverse elliptic and hyperbolic problems. We analyse its linearisation about a simple solution and show uniqueness and exponential instability for the linearisation.

This paper is concerned with some non-linear propagation phenomena for reaction–advection–diffusion equations with Kolmogrov–Petrovsky–Piskunov (KPP)-type non-linearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems (proved in Berestycki, H., Hamel, F. & Nadirashvili, N. (2005) The speed of propagation for KPP type problems (periodic framework). J. Eur. Math. Soc. 7, 173–213), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviours and the variations of the considered functions with respect to these parameters. One of the sections deals with homogenization problem as an application of the results in the previous sections in order to find the limit of the minimal speed when the periodicity cell is very small.

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black–Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang.

A model of cell-growth, describing the evolution of the age–size distribution of cells in different phases of cell-growth, is studied. The model is based on that used in several papers by Basse et al. and is composed of a system of partial differential equations, each describing the changes in the age–size distribution of cells in a specific phase of cell-growth. Here, the ‘age’ of a cell is considered to be the time spent in its current phase of cell-growth, while ‘size’ is considered to be the DNA content of the cell. The existence of steady age–size distributions (SASDs), where the age–size distributions retain the same shape but are scaled up or down as time increases, is investigated and it is shown that SASDs exist. A speculative discussion of the stability of these SASDs is also included, but their stability is not conclusively proved.

We wish to express our sincere condolences to the family of Vladimir Entov, who died on April 10th 2008. Professor Entov was an Editor of the European Journal of Applied Mathematics from 1994 to 2007. He was a brilliant scientist and a staunch supporter of this journal, and will be greatly missed by his many friends and colleagues around the world.

Sam Howison, Andrew Lacey, and Michael Ward

Co-Editors in Chief, European Journal of Applied Mathematics

We present a new necessary condition for similarity of indefinite Sturm–Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl–Titchmarsh $m$-functions. We also obtain necessary conditions for regularity of the critical points $0$ and $\infty$ of $J$-non-negative Sturm–Liouville operators. Using this result, we construct several examples of operators with the singular critical point $0$. In particular, it is shown that $0$ is a singular critical point of the operator

$$-\frac{(\mathrm{sgn} x)}{(3|x|+1)^{-4/3}}\frac{\mathrm{d}^2}{\mathrm{d} x^2}$$

acting in the Hilbert space $L^2(\mathbb{R},(3|x|+1)^{-4/3}\,\mathrm{d} x)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a $J$-non-negative Sturm–Liouville operator of type $(\mathrm{sgn} x)(-\mathrm{d}^2/\mathrm{d} x^2+q(x))$ with the same properties.

Using Leray–Schauder degree theory, a theorem of upper and lower solutions and a strong maximum principle for the telegraph equation we prove an Ambrosetti–Prodi-type result for periodic solutions of the telegraph equation.

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions $(t,1/t,s)$ within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps $L\log^2L$ into $L^{1,\infty}$. We give a simple geometric condition that guarantees that the $L\log^2L$ class cannot be enlarged. The proof develops the author's methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fefferman and Pipher.

We prove weighted norm inequalities for vector-valued commutators of multilinear operators. We first consider the strong $(p,p)$ case with $p>0$ and then the case weak-type estimate.

Let $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

We introduce the concept of causality into the framework of generalized pseudo-Riemannian geometry in the sense of Colombeau and establish the inverse Cauchy–Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward by Hawking and Ellis. Our work is based on a new characterization of free elements in finite-dimensional modules over the ring of generalized numbers.

This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

We study biminimal immersions: that is, immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler–Lagrange equation associated with biminimal immersions for both biminimal curves in a Riemannian manifold, with particular attention given to the case of curves in a space form, and isometric immersions of codimension 1 in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods of constructing families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.

We compare solutions of a class of degenerate parabolic equations on a Riemannian manifold $M$ with solutions of the equation on a model manifold. The class of equations under consideration contains both the parabolic $p$-Laplace equation and the porous medium equation. We prove that, under curvature conditions, solutions on model manifolds induce sub- or supersolutions on $M$. Using this result, we obtain curvature-dependent estimates for the speed of propagation of solutions.

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.