We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C0 and by Gowers for ℓp (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H1, any infinite-dimensional complex L1(μ), and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.

A new proof of the Holley-Preston generalisation of the Fortuin-Kastelyn-Ginibre inequalities is given, and Batty's extension to the case of infinite products is discussed briefly. An application of the theorems in combinatorial probability theory is described.

We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.

In this work we study reaction–diffusion systems in fractional power spaces Xα which are embedded in L∞. We prove that the solution operators T(t) to these problems are globally defined, point dissipative, locally bounded and compact. That ensures the existence of global attractors. We also find a set containing the range of every function in the attractor, providing good estimates on asymptotic concentrations. This is done under very few hypotheses on the reaction term. These hypotheses are natural and easy to verify in many applications. The tools employed are the theory of invariant regions for systems of parabolic partial differential equations, the notion of contracting sets and the variation of constants formula. Several examples are considered to emphasise the applicability of these techniques.

A simple model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional straight string is introduced. In both regions (n-dimensional body and a onedimensional string) the state is assumed to satisfy the wave equation. Simple boundary conditions are introduced at the junction. These conditions, in the absence of control, ensure conservation of the total energy of the system and imply some rigidity of the boundary of the n-d body on a neighbourhood of the junction. The exact boundary controllability of the system is proved by means of a Dirichlet control supported on a subset of the boundary of the n-d domain which excludes the junction region. Some extensions are discussed at the end of the paper.

1. The view is confirmed that the zona pellucida is not an essential factor in the formation of the fertilisation membrane (pp. 96–100).

2. Partial activation of the eggs of Asterias rubens is described with the formation of Seifriz's “protoplasmic papillæ” (pp. 100, 101).

3. Artificial activation of the eggs of Asterias rubens by isotonic solutions of NaCl, KCl, CaCl2, and MgCl2 is described. It was found that decrease in surface tension is not a necessary factor in causing artificial membrane formation (pp. 102–104).

4. The influence of the hydrogen ion concentration of the medium on the degree of extrusion of the fertilisation membrane of Echinus miliaris has been examined, and indicates that osmotic pressure due to the presence of a protein within the membrane is responsible for the extrusion (pp. 105–109).

5. From a study of the influence of salt concentration on the extrusion of the fertilisation membrane of Echinus miliaris, it is concluded that the fertilisation membrane is, from the moment of its formation, completely permeable to salts (pp. 109–112).

6. The origin of the fertilisation membrane has been examined, but in the absence of critical evidence it is concluded that the question cannot be regarded as settled (pp. 112–114).

The importance in Mathematical Physics of the Bessel functions, whose order is half an odd integer, suggests that the corresponding Mathieu functions may be worthy of a closer attention than they have yet received. At the very least it is expedient to pave the way for their computation. In the second paper bearing the above title, asymptotic developments of the characteristic numbers which correspond to these functions were given; it is here proposed, in the first place, to take the more direct line of approach.

We present an extension of Besicovitch spaces, which has its origin in the works of Zhikov et al. and Kozlov and Oleinik. We show that these spaces have a similar behaviour to the Lp spaces, and we give a notion of derivative that allows us to treat partial differential problems in this frame. To the best of our knowledge, this is new, even for the usual Besicovitch spaces.

We establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations of logistic type −Δpu = a(x)h(u) − b(x)f(u), where Δpu = div (|∇u|p−2∇u) with p > 1, h(u)/up−1 is non-increasing and f(u) is a function whose variation at infinity may be regular or rapid. In particular, our result regarding the blow-up rate reveals the main difference between regular variation function f and rapid variation function f.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

Focal point and conjugacy criteria for the half-linear second-order differential equationare obtained using the generalized Riccati transformation. An oscillation criterion is given in case when the function c(t) is periodic.

We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

Let Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).

In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the first model.

We study the Cauchy problem for the spatially homogeneous Landau equation for Fermi–Dirac particles, in the case of hard and Maxwellian potentials. We establish existence and uniqueness of a weak solution for a large class of initial data.

We consider a class of inhomogeneous systems of reaction-diffusion equations that includes a model for cavity dynamics in the semiconductor Fabry–Pérot interferometer. By adapting topological and geometrical methods, we prove that a standing pulse solution to this system is stable in a certain parameter regime, under the simplification of homogeneous illumination. Moreover, we explain two bifurcation mechanisms which can cause a loss of stability, yielding travelling and standing pulses, respectively. We compute conditions for these bifurcations to persist when inhomogeneity is restored through a certain general perturbation. Under certain of these conditions, a Hopf bifurcation results, producing periodic solutions called edge oscillations. These inhomogeneous bifurcation mechanisms represent new means for the generation of solutions displaying edge oscillations in a reaction-diffusion system. The oscillations produced by each inhomogeneous bifurcation are expected to depend qualitatively on the properties of the corresponding homogeneous bifurcation.

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths that lie in a single stratum, except possibly at their initial points, form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.