We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).

We study the Neumann Laplacian in unbounded regions of the form Ω = {(t, x) | t >O,f(t)−1x ∊ Ω′}, where Ω′ ⊂ ℝn−1 is a bounded open set with the Lipschitz boundary and f decays in such a way that the spectrum of is discrete but the counting function N(λ, ) of the spectrum grows faster than a power of λ, a typical example being f(t) = exp (– t In … In t), for t ≧ t0. We compute the principal term of the asymptotics of N(λ, ), with a remainder estimate.

We consider the existence and uniqueness of minimal invariant subtrees for abelian actions of groups on Λ-trees, and whether or not a minimal action is determined up to isomorphism by the hyperbolic length function. The main emphasis is on actions of end type. For a trivial action of end type, there is no minimal invariant subtree. However, if a finitely generated group has an action of end type, the action is nontrivial and there is a unique minimal invariant subtree. There are examples of infinitely generated groups with a nontrivial action of end type for which there is no minimal invariant subtree. These results can be used to study actions of cut type.

A new formula is given for the spectral function in singular Sturm–Liouville problems where the potential is like – (const.)xc (0 < c ≦ 2). The formula is used to answer an open question concerning the asymptotic form of the spectral function.

An initial-boundary-value problem is considered for the heat equation in an infinite angle dθr ⊆ R2 × [0, ∞) with the oblique derivative boundary conditions on the faces λi of the angle:

with either h0 + h1 > 0, or h0 + h1 ≦ 0. The unique solvability of such a problem is proved in appropriate weighted Sobolev spaces according to the sign of h0 + h1. Estimates of the solution are obtained under ‘natural’ restrictions on the opening of the angle.

The Riemann problem for a resonant nonlinear system of conservation laws is considered here. The Riemann solution was constructed by employing the viscosity approximation approach. One kind of new discontinuity, which is called the Dirac-contact wave, appeared in the Riemann solution. Because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed, the solution we obtained in this paper is not unique for some initial data. An additional condition was explored to guarantee the uniqueness of the Riemann problem.

Results by Simader, Brézis and Cycon of the genre ‘locally essentially self-adjoint implies globally essentially self-adjoint’ are generalised to Schrödinger operators that are not necessarily bounded from below.

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

We answer the question, raised in 1989, whether the ℐ2-radical of a structural matrix near-ring can be written as the sum of two nontrivial ideals, one of which is nilpotent, in the affirmative.

Reaction–diffusion systems are widely used to model competition in, for example, the scientific fields of biology, chemistry, medicine and industry. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Conditions for the existence of such lower bounds in Lotka–Volterra competitve models are already well known. A general context for the understanding of these conditions is provided in this paper, by establishing more general permanence criteria (for the nonexplosion and noncollapse to zero of solutions) in the class of diagonally convex competitive reaction–diffusion systems with zero flux Neumann boundary conditions. This class admits most famous competition models as special cases. The asymptotic (large time) behaviour of positive solutions within the bounds of permanence is also considered and it is shown that the methods of proof for asymptotic stability (and hence resilience) normally associated with order-preserving systems (such as comparison arguments) are also applicable, in a slightly generalised form, to competitive systems as long as the competitive interactions are not too strong. The general criteria for permanence obtained here provide a natural method for developing new and easily verifiable permanence conditions for a host of non Lotka–Volterra competition models, as is illustrated by considering three famous special cases. In one of these cases known results are recovered, while in the other two cases new conditions for solution permanence are established.

Consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity. The propagation speed is discontinuous at the interface of two unbounded media, and the interface may be an arbitrary and smooth surface locally. A Sommerfeld radiation condition is introduced for the acoustic propagator, and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator. Furthermore, we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator.