Although the European Journal of Applied Mathematics is a journal that has no official affiliation with any particular academic organisation, its aims overlap those of many institutes, centres, societies and, especially, conferences. Indeed, in our first ten years we have published three special issues related to conferences on topics that have been timely and important for the development of new mathematical ideas relevant to practical problems. Thus, when ICIAM, the world's largest industrial-and-applied mathematics conference, took place in Edinburgh, Scotland, in July 1999, we were only too happy to be associated with a gathering of so many people who all worked in the style of EJAM, and the organisers kindly allowed us to co-sponsor the conference reception. A highlight of this evening was Professor Graham Wilks' dramatic portrayal of Sir Isaac Newton, complete with apple. Accompanied by his wife, Graham sang the following verse.

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

The bifurcation of symmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg–Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends upon the parameters d, the size of the superconducting slab, and κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.