In commutative Banach algebras with factorisation, the existence of an identity (bounded approximate identity) modulo a topologically nilpotent radical implies the existence of a global identity (bounded approximate identity), respectively.

A radiation condition is obtained, and is then used together with weighted Sobolev spaces and the limiting absorption method to establish the unique existence of solutions to the diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.

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.

We give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.

Suppose fλ: ℝ→ℝ, fλ(0) = 0 and the fixed point zero undergoes a generic supercritical period doubling bifurcation at λ = 0. We characterise those small values of ε > 0, λ ∈ ℝ for which there are periodic solutions of period approximately two of the equation

As ε → 0, these solutions approach square waves.

We study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

Recent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.

A necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

This paper deals with determining in a constructive manner those members of a linear space of functions which are of integrable-square. The space considered is the set of solutions to an ordinary differential equation, and the solutions of integrable-square are delineated by way of initial conditions. Numerical procedures for implementing the construction are discussed, and application is made to the deficiency index problem. Results from some specific computations are given.

Certain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space , for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.

This is contrasted with a known property of the corresponding expression domains. Whereas for an expression M, the operator domains contain only functions y for which both y and My lie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary for My to exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.

In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

A version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.

Let U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.

We study the asymptotic behaviour of Dirichlet problems in domains of R2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function. Using a capacitary method together with ergodic theorems for additive and superadditive processes, we are able to characterise the limit problem precisely.