We prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.

We determine the homomorphism induced, in Z2-cohomology, by a map f: ΩSpin(r) → ΩSpin(r). As a corollary we show that ΩSpin(r), r ≧ 9 is 2-atomic, where a space X is 2-atomic if any map f: X ≧X is either a mod 2 homotopy equivalence or f*: H*(X, Z2) → H*(X, Z2) is nilpotent.

This paper describes a proof that the solutions of Neumann problems for the exterior of spheres {x ∊ ℝ3∣ converge to the solutions of the exterior problem for the halfspace {x ∊ ℝ3 } x1 ≧ 0} as b → ∞ provided that the boundary data converge in a certain sense. The method requires that there be some dissipation which can be arbitrarily small. Fredholm integral equations are set up for the boundary data, and these are solved by means of Neumann series for large b. Estimates on the terms of the series (which involve singular integrals), in terms of b, allow the convergence proof to be carried through.

The procedure of expressing problems with infinite boundaries in terms of problems with finite boundaries allows the implementation of an effective numerical procedure for determining, for example, the entire near-field of a baffled piston, a problem whose solution has proved elusive for many years.

The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expression L = – d2/(dr2) + V(r) is in the limit point case at both ends of an interval (a, b), with V(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of Lu = xu, x εℝ, as r→a and r→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

Kuiper has proved that there is no tight topological immersion of RP2 into E3. Thus any continuous tight map RP2 → E3 has to have singular points. In this note we consider tight C∞-stable maps of RP2 and the torus into E3; i.e. maps with the simplest possible singularities (Whitney pinchpoints) and transversal crossings. We classify the forms that the outer part of such maps can take; we prove also some facts about the inner part. The result about the outer part of C℞-stable tight maps RP2 → E3 establishes the first half of Banchoff's conjectured classification of such mappings.

Let G and G' be two connected compact Lie groups with maximal tori T and T'. For a space X, let Xp be the p-completion of X. We will associate to each topological map f:(BG)p→(BG')p an “admissible map” ϕ:π1(T)⊗zZp→π1(T′)⊗zZp. We then show that the study of “admissible maps” in the p-complete case may be reduced to their study in the p-local case.

In this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbations

of the nonconvex functional

where W:RN→R supports two phases and N ≧1. We obtain the Γ(L1(Ω))-limit of the sequence

Moreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of Eε(·) converging in L1(Ω) to a minimiser of E0(·) with minimal interfacial area.

We prove C∞ regularity for moving interfaces of local solutions of the porous medium equation as well as C∞ lateral regularity for the pressure function near such an interface.

In this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.

We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.

Let R ×N equipped with the warped Lorentzian metric f2dt2 ⊕(− h), where (N, h) is a Riemannian manifold, and f: N → ]0, ∞[ is a smooth function. Then R × N is called a standard static space-time, and in this paper we look for non-trivial periodic trajectories on R × N for N compact.

Bifurcations appearing in generic one-parameter families of vector fields on a compact planar region are conceptually characterised in terms of their stability and regularity properties.

We consider a system of nonlinear differential equations derived from a model for the self-trapping of light. We prove that there is a parameter range for which there is a fundamental mode solution. Further, given any positive integer N, there also exists a parameter range over which the boundary value problem has N solutions. The first of these is a fundamental mode solution while the next N — 1 solutions represent additional modes.

The problem of illuminating the boundary of sets having constant width is considered and a bound for the number of directions needed is given. As a corollary, an estimate for Borsuk's partition problem is inferred. Also, the illumination number of sufficiently symmetric strictly convex bodies is determined.

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.