In this note we show that certain perturbations, involving rank-one stabilising operators, which correspond to small delays in some feedback control problems, shift a part of the point spectrum of the unperturbed operator into the right-half plane.

Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.

In this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the type

where h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d > 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

Let f: Mm→Nn be a map from a Riemannian m-manifold (Mm, d) into an n-manifold Nn. The major purpose of this paper is to give a lower bound for the number

by examining the behaviour of the cohomology homomorphisms induced by f. This idea will be used to generalise the classical Newman theorem and present a geometric background for a well-known non-embedding theorem in topology.

The stability of nondegenerate solutions of some semilinear Dirichlet problems is studied. Two specific situations are considered: firstly, a singular perturbation of the differential operator; secondly, a perturbation of the nonlinear term using a term which also depends on the gradient of the solution.

We consider damped nonlinear hyperbolic equations utt + Aut + αAu + βA2u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter β tends to zero.

Any non-ruled quartic surface with a double line in space of three dimensions has plane sections of genus 2 and so is a projection of a non-singular duodecimic surface in space of eleven dimensions. This was accepted as an established fact by 1890, but there seems not to be any account of such a projection in action. The following pages are submitted to aid the removal of this century-old anomaly.

The Bass-Serre theory of groups acting on trees without inversions is extended to the case in which the action with inversions is possible.

Necessary and sufficient conditions on weights u and v are given for the inequality

to hold for all g ∊ .

Given any point x on a smooth, closed submanifold M of Euclidean space, there is a bitangent hyperplane or sphere that is tangent to M at x.

We present several general methods of constructing tight hypersurfaces f: Mn → En+1, n ≧ 2. We prove a smoothing lemma, which allows us to approximate tight continuous hypersurfaces by C∞ tight ones. We show that given a tight immersion or C∞-stable map f: M2 → E3 of a compact surface M2 other than S2, there is a tight C∞-stable map g: M2 # RP2:→ E3. We prove that given C∞ tight immersions f: Mn → En+1 and g: Nn → En+1 of compact n-manifolds Mn and Nn into En+1, there is a tight C∞ immersion of Mn # Nn into En+1. Two other methods involve hypersurfaces of rotation and sets in En+1 at a fixed distance from a tightly embedded n-manifold with boundary in En. One consequence of these methods is that the outer part of C∞ tight hypersurfacesf: Mn → En+1, n ≧ 3, is far more complicated than in the case of tight surfaces in En. For example, given any C∞n-manifold M with boundary tightly embedded in En, there is a tight immersion f:Nn → En+1 of a closed n-manifold having M as a topset. Kuiper's theorem describing tight immersions of surfaces into E3 does not generalise to the case of hypersurfaces f: Mn → En+1, n ≧ 3, without substantial restrictions on Mn and/or f.

The conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.

We study the bifurcation diagram and uniqueness of solutions of

By using a rescaling technique and the Implicit Function Theorem, we establish the global bifurcation diagram. Uniqueness is proved by a separation argument to complete the bifurcation picture of the problem. Our study suggests that the bifurcation diagrams have different behaviour at λ = 0, depending on whether g(∞) > 0 or g(∞) < 0 in L∞ norm, but quite similar behaviour in Lp or W2,1.

In this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝ

where X(r) is the characteristic function of a compact interval of ]0, + ∞[