We use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problem u″ + f(t, u) = 0, u(0) = α, u(1) = β between lower and upper solutions, where f:[0,1] × E → E is quasi-monotone increasing in its second variable with respect to a regular cone.

Let Xn, n ≥ 1 be an asymptotically almost negatively associated (AANA) sequence of random variables. Some complete convergence and Marcinkiewicz–Zygmund type strong laws of large numbers for an AANA sequence of random variables are obtained. The results obtained generalize the results of Kim, Ko and Lee (Kim, T. S., Ko, M. H. and Lee, I. H. 2004. On the strong laws for asymptotically almost negatively associated random variables. Rocky Mountain J. of Math. 34, 979–989.).

In this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Brézis and Nirenberg.

This note is devoted to prove some uniqueness results of positive solutions of a Robin problem for semi-linear elliptic equations and systems.

Let and let us consider a del Pezzo surface of degree one given by the equation . In this paper we prove that if the set of rational points on the curve Ea,b : Y2 = X3 + 135(2a−15)X−1350(5a + 2b − 26) is infinite then the set of rational points on the surface ϵf is dense in the Zariski topology.

We obtain a necessary and sufficient condition on a polynomial P(s, t) so that the (global) double Hilbert transforms along polynomial surfaces (s, t, P(s, t)) in R3 are bounded on Lp for 1 < p < ∞.

In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis–growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers.

We consider the equationover a finite field q of q elements, with variables from arbitrary sets. The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that iffor some absolute constant C > 0, then above equation has a solution for any λ ∈ q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

Let n ≥ 1 be an integer. Given a vector a=(a1,. . ,an)∈, write(the ‘projection of a onto the positive orthant’). For a set A⊆ put A+:={a+: a ∈ A} and A−A:={a−b: a, b ∈ A}. Improving previously known bounds, we show that |(A−A)+| ≥ |A|3/5/6 for any finite set A⊆, and that |(A−A)+| ≥ c|A|6/11/(log |A|)2/11 with an absolute constant c>0 for any finite set A⊆ such that |A| ≥ 2.

The approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.

A homology theory based on measures, first mentioned by Thurston, is naturally defined here as a functor into the category of locally convex topological vector spaces. It is proved that the first homology space is Hausdorff.

The λ-dilate of a set A is λċA={λa : a∈A}. We give an asymptotically sharp lower bound on the size of sumsets of the form λ1ċA+ċċċ+λkċA for arbitrary integers λ1,. . .,λk and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plünnecke's inequality.

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