New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

Denote by Xt an n-dimensional symmetric Markov process associated with an elliptic operator

where (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:

(i) For almost every and

(ii) Let be a sequence of subdivisions of [0,1] so that

Then

As an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operator

where (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).

If Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,

where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.

We consider the initial boundary value problem for the equations of one-dimensional nonlinear thermoelasticity in ℝ+; and prove a global existence-uniqueness theorem for small smooth data. The asymptotic behaviour is simultaneously obtained.

A compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant with

respect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.

Each singular element α of the full transformation semigroup on a finite set is generated by the idempotents of defect one. The length of the shortest expression of α as a product of such idempotents is given by the gravity function g(α).We use certain consequences of a result by Tatsuhiko Saito to explore connections between the defect and the gravity of α, and then determine the number of elements that have maximum gravity. Finally, we obtain formulae for the number of elements of small gravity. Such elements must have defect 1, and we determine their number within each ℋ-class. Many of the results obtained were suggested, and all have been verified, by programs written in PROLOG, a logic programming language very well suited for algebraic calculations.

By using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of approximated solutions, we show that under suitable conditions on f and h, the inhomogeneous problem. −Δu + c2u = λ(f(u) + h(x)) for x ∈ Ω (Ω is an exterior domain in ℝN, N≧ 3) and has at least two positive solutions.

We consider an anisotropic non-homogeneous linear elastic material in equilibrium and occupying an open region with non-compact boundary. In both the linearised and classical linear theories the asymptotic behaviour of the solution is determined and a clear relationship established with Saint-Venant's principle on such regions. Although the treatment is discussed with special reference to elasticity, it is equally applicable to general systems of elliptic differential equations, and thus reveals a relationship with the classical theorems of Phragmèn-Lindelöf and Liouville.

This paper deals with the number of eigenvalues which appear in the gaps of the spectrum of a Dirac system with real and periodic coefficients when the coefficients are perturbed. The main results provide an upper bound and a condition under which exactly one eigenvalue appears in a given gap.

Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.

The generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).

It is shown how to derive SL(n + 1)-invariant equations which reduce to scalar Lax equations for an operator of order n + 1. The existence of these systems explains the Miura transformation between modified Lax and scalar Lax equations. In particular we study an SL(2)-invariant system with a certain space of solutions lying over the solution space of a Korteweg-de Vries equation described by G. B. Segal and G. Wilson. This enables us to write down some solutions of this SL(2)-invariant system in terms of θ-functions of a hyperelliptic Riemann surface.