Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.

Linked equations of the form

are studied on a bounded smooth domain in RN for λ ∈ R2. Existence and uniqueness of solutions are discussed for fi homogeneous of order p – 1 in ui, generalising the ‘Klein Oscillation Theorem’ when p = 2, N = 1. Bifurcation from the principal eigenvalue is also considered for nonhomogeneous perturbations fi of order greater than p – 1.

Let G be any finite group with elementary abelian Sylow 3-subgroups of order 9, and let F be any field of characteristic 3. Then, the Loewy length of the projective cover of the trivial FG-module is at least 5. This lower bound is the best possible.

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

A large class of measure-valued critical branching processes can be classified in terms of a parameter ρ which arises as a measure of the recurrence of the underlying spatial Markov process. By establishing upper and lower bounds for the total weighted occupation time process, it is shown that if a measure-valued process is started from an invariant measure of its underlying spatial process, then a necessary and sufficient condition for (a.s.) local extinction is that ρ > 0.

A nonlocal eigenvalue problem of the form u″ + a(x)u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by general ordinary quasidifferential expressions in The domains of these operators are described in terms of boundary conditions involving the solutions of M[u] = λwu and the adjoint equation at both singular end-points a and b. These results are an extension of those proved in [3], by Evans and Ibrahim, to the case of two singular end-points of the interval (a, b), and a generalisation of those in [10] and [13] concerning selfadjoint and J-selfadjoint differential operators, where J denotes complex conjugation.

We consider a family of dispersive equations whose simplest representative would be a Benjamin–Bona–Mahony equation with a Burger's type dissipation. The effect of possible unevenness of the bottom surface is considered and our main result gives decay rates of the solutions in Lβ(ℝ) spaces, 2 ≦ β ≦ + ∞.

Margulis has given conditions under which a lattice in a semisimple Lie group admits the structure of an arithmetic subgroup. We show that these arithmetic structures are unique. The result is not subject to the condition “rkR(G) ≧ 2” required by the Margulis result. In the lowest dimensions, the result has previously been observed by Takeuchi, Maclachlan and Reid.

We investigate the large-time behaviour of the solutions u = u(x, t) to the one-dimensional nonlinear heat equation with reaction

with exponents m > 1,p < 1. The initial function u(x, 0) is assumed to be measurable and nonnegative. In the case m + p ≧ 2 where the initial value does not uniquely determine the solution, we also fix the positivity set of the solution u(x, t) if the support of u(x, 0) is not the whole line ℝ, i.e. u(x, t) > 0 if and only if −s1(t) <x<s2(t), t ≧ 0, where 0≦si(t)≦∞ for t ≧ 0, i = 1, 2 are lower semicontinuous given functions. We prove that u converges to a self-similar function which depends only on the behaviour of u(x, 0) for |x| large or si(t) for t large. We classify the set of self-similar solutions and study the equation satisfied by their interfaces.

Existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle. In the case when Ω is an annulus, it is shown that the solution is radially symmetric.

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.

In this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete, noncompact Riemannian manifold (M, g) with Sobolev constant S2(M) > 0 and Ricci curvature Ric (M) ≧ 0 outside some compact subset, into a complete manifold of nonpositive curvature or a regular ball. In particular, we prove the uniqueness and existence of bounded harmonic functions on (M, g).