This note discusses primeness for semigroup rings of semigroups satisfying a certain condition, weaker than the u.p. property. These semigroups include free products of semigroups, semigroups presented by a single relator and at least three generators, and free inverse semigroups of infinite rank.

This paper is concerned with homogenisation processes for parametrised families of transport equations in ℝn symmetric hyperbolic systems in several space dimensions and anisotropic wave equations. The main tools for carrying out the homogenisation are the Young measures and Radon transform combined with the integral representation of holomorphic functions of Nevanlinna-Pick's type.

Suppose fλ: ℝ→ℝ, fλ(0) = 0 and the fixed point zero undergoes a generic supercritical period doubling bifurcation at λ = 0. We characterise those small values of ε > 0, λ ∈ ℝ for which there are periodic solutions of period approximately two of the equation

As ε → 0, these solutions approach square waves.

We study the existence of T-periodic positive solutions of the equation

where f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.

Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

We demonstrate how a fairly simple “perturbed test function” method establishes periodic homogenisation for certain Hamilton-Jacobi and fully nonlinear elliptic partial differential equations. The idea, following Lions, Papanicolaou and Varadhan, is to introduce (possibly nonsmooth) correctors, and to modify appropriately the theory of viscosity solutions, so as to eliminate then the effects of high-frequency oscillations in the coefficients.

We give an expression for the n-th moment of certain Itô integrals. The integrands considered are nonanticipating functionals of the form s↦a(s, Xs), where a is a measurable time-dependent vector field in space satisfying mild regularity conditions, and Xs is standard translated Brownian motion. The expressions are similar to the Dyson-Phillips terms for magnetic Schrödinger semigroups.

We use these expressions to establish properties of the solutions of certain Cauchy problems and we relate our results to the framework of generalised Dyson expansions as set up by Johnson and Lapidus.

Let X, Y and Z be Banach spaces, and let Πp (Y, Z) (1 ≦ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a ℒ∞-space, then a bounded linear operator is 1-summing if and only if a naturally associated operator T#: X → Πl (Y, Z) is 1-summing. This result need not be true if X is not a ℒ∞-space. For p > 1, several examples are given with X = C[0, 1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on whose associated operator T# is 2-summing, but for all N ∈ N, there exists an N-dimensional subspace U of such that T restricted to U is equivalent to the identity operator on . Finally, we show that there is a compact Hausdorff space K and a bounded linear operator for which T#: C(K) → Π1 (l1, l2) is not 2-summing.

A necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

A model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.

This paper considers the Cauchy problem for an extended model of combustion (u + qz)t + f(u)x = 0, zt + kg(u)z = 0 with Lp bounded initial data, where g(u) is a piecewise Lipschitz continuous function and its discontinuous points have no finite limit point. The integral representation gives a definition of a weak solution in Lp space. Some existence results are obtained based on a simplified method of compensated compactness in which the weak continuity theorem of 2 * 2 determinants plays a more important role, but the idea of Young measures has been avoided.

A class of singular real formally self-adjoint differential expressions M on I = [a, = ∞) (a ∈ ℝ), i.e. expressions of the form My = with pj ≧ 0 (j = 0, …, n – 1), pn > 0 is constructed with the following property: For every integer k with 0 ≦ k < n/2 there exists an expression M in this class such that the deficiency index of T0(M) – the minimal operator associated with M – is n + 2k. This generalises a result in [3] and proves part of the McLeod's conjecture.

The compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.