We consider the system (∂/∂t)u = ∆u + σ(u)|∇φ|2, div (σ(u)∇φ) = 0 in a bounded region of ℝN coupled with initial and boundary conditions, where σ(s) ∈ C(ℝ) is nonnegative and σ(u) = 0 if and only if u ≧ a for some a > 0. Owing to the degeneracy involved, solutions of the problem display new phenomena that cannot be incorporated into the classical weak formulation. The notion of a capacity solution introduced in [14,15] is employed to study the problem. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.

The purpose of this paper is to prove the following (nonlinear) mean ergodic theorem: Let E be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E and let T: C → C be an asymptotically nonexpansive mapping. If

exists uniformly in r = 0, 1, 2,…, then the sequence {Tnx} is strongly almost-convergent to a fixed point y of T, that is,

uniformly in i = 0, 1, 2, ….

The paper studies orbits in a function space described by solutions of a system of reaction–diffusion equations in a bounded domain with a boundary condition of homogeneous Robin type. The omega-limit set of a bounded semi-orbit is shown to have a simple structure, provided that certain hypotheses are satisfied. For a two-dimensional system of Fitz-Hugh Nagumo type, these hypotheses yield explicit sufficient conditions for the existence of at least one periodic trajectory.

We consider the following problem: Let P be a monic polynomial of degree n with complex coefficients. What can be the maximum ‘size’ of a monic divisor Q of P? Here the size of a polynomial R is the maximum ||R|| of the moduli of its values on the unit circle. In 1991, B. Beauzamy proved that there exists a divisor Q with ||Q|| ≧ e∈n−1, ∈ = 0.0019, when all the roots of P belong to the unit circle. Using a very recent result of D. Boyd, we obtain a general result which, in the same case, gives ||Q||≧βn; here β = 1.38135 … is optimal.

In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.

This paper studies the surface of constant mean curvature on a semi-infinite strip, and shows by means of a first-order differential inequality that the solution in a given measure either becomes asymptotically unbounded at least to polynomial order, or decays at most exponentially to the solution of an associated one-dimensional problem. A proof is also presented for uniqueness in the class of functions having bounded gradient and subject to specified growth conditions for large values of the longitudinal distance. Extensions of these results to the whole strip and to more general types of equations are also described.

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.

Classes of exact solutions of the Navier–Stokes equations for incompressible fluid flow are explored. These have spatially-uniform velocity gradients at each instant, but often display complex temporal behaviour. Particular illustrative cases are described and related to previously-known solutions.

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

For a 2 × 2 periodic system with a perturbation P whose first moment is finite, Jy′ = [λI + R(x) + P(x)]y, we study the behaviour of the Titchmarsh–Weyl m(λ)-coefficient at the spectral gap endpoints. Assuming gap nondegeneracy, our main result is that as λ → λ0, (λ − λ0)½(m(λ) → c ≠ 0 if and only if λ0 is a φ-half-bound state, which follows from an analysis of Jost-type functions.

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.

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.

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.