There has been considerable interest of late in fractals, both in their occurrence in the sciences, and in their mathematical theory. A general account of fractals is given by Mandelbrot [10], whilst a more mathematical approach may be found in Falconer [5].

There exists for each oriented link L in the 3-sphere a unique 2-variable Laurent polynomial defined uniquely by the following:

(i) PU = 1 for the unknot U

(ii) ,

where L+, L− and L0 are identical outside a ball and inside are as shown in Figure 1. See [1] and [2].

We recall a theorem of Hurwitz [5]: let f(θ) be real-valued and periodic, f(θ) ≢ 0, and

where an = bn = 0, 0 ≤ n < N. Then f(θ) has at least 2N sign changes in each period. A similar result with a trigonometric polynomial on the right-hand side of (1) had been obtained earlier by Sturm [7].

The mean residual lifetime of a real-valued random variable X is the function e defined by

One of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

Ruckle[4] used the idea of a modulus function ƒ (see Definition 1 below) to construct the sequence space

This space is an FK space, and Ruckle proved that the intersection of all such L(f) spaces is ø, the space of finite sequences, thereby answering negatively a question of A. Wilansky: ‘Is there a smallest FK-space in which the set {e1, e2, …} of unit vectors is bounded?’

In this paper we present two versions in several variables of the following result:

Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

Throughout this brief note V denotes a fixed, smooth, real-valued function on such that there exists a bounded, open, strictly convex set C with (the boundary of C), V > 0 in C, V = 0 on and V′(0) = 0.

A method is derived for converting a pair of coupled singular integral equations of a certain form into a single equation of the same (Cauchy-separable) type. Reduction methods for systems of singular integral equations are generally directed towards the construction of equivalent Fredholm equations. Preservation of the singular nature of the kernel in the reduction process permits the powerful techniques associated with Cauchy kernels to be used in seeking closed solutions of the original pair.

The example given, derived previously from a problem in wave diffraction theory, illustrates many aspects of the method.

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line sources are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the local asymptotic stability of the trivial solution of

and then examine the ‘size’ of the domain of attraction of the trivial solution of the nonlinear system (1·1) with a countable number of discrete delays.