In 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δan = an+1 – an and Δ2an = Δ(Δan), then is convergent and the constant 4 being best possible. Equality occurs if and only if an = 0 for all n. In this paper we give a result that extends Copson's result to inequalities of the form

where Mxn =–Δ(pn_l Δxn_l)+qnxn (n = 0, 1, …). The validity of such an inequality and the best possible value of the constant K are determined in terms of the analogue of the Titchmarsh-Weyl m-function for the difference equation Mxn = λwnxn (n = 0, 1, …).

We study the stabilisation of a one-dimensional diffusion equation by means of static feedback. The equation contains the so-called Sturm-Liouville operator (S-L operator). A perturbation, often interpreted as an error in modelling physical systems, enters the principal part and the boundary condition of the S-L operator. Since the perturbation is not subordinate to the operator, the classical perturbation theory is no longer available. We show, however, that the feedback stabilisation scheme for the unperturbed equation is effective also for the perturbed equation as long as the perturbation is small in an adequate topology. The key idea is to show the strong continuity of the eigenfunctions for the S-L operator relative to the coefficients of the operator.

The small cancellation theory over free products with amalgamation and HNN groups is extended to groups acting on trees in which the action with inversions is possible. This will include the case of tree products of groups and treed-HNN groups.

Components in the function space of maps from a space X to the classifying space BG of a topological group G can sometimes be distinguished up to homotopy type by a Samelson product method. When X is a closed Riemann surface and G is a unitary group, this method is nearly sufficient to classify the components up to homotopy type.

An alternative proof of a theorem which characterises orders in semiprime rings with minimal condition is given. The approach used is to make use of the corresponding result for prime rings and is inspired by Herstein's proof of Goldie's theorem on orders in semisimple Artinian rings.