1. Wallace's theorem, that the circumcircles of the four triangles formed by the sets of three from four coplanar lines meet in a point, has a generalization to Euclidean affine space of any even dimension n: namely, if we have n + 2 general primes (or hyperplanes) in such a space, the n + 2 circumhyperspheres of the simplexes formed by the sets of n + 1 of the primes meet in a point.

In his work on elementary catastrophes Zeeman(1) has considered what he has named as the double cusp catastrophe. This catastrophe is defined by the unfolding of the two variable polynomial x4 + y4. Using Mather's results (2) on stability of singular germs of C∞ maps we can find an expression for the unfolding. The eight dimensional unfolding can then be considered as a polynomial in two variables with eight parameters.

In this paper we shall suppose that all convex sets are compact convex subsets of Euclidean space En. We shall be concerned in producing upper and lower bounds for the ‘total edge lengths’ of simplices which are contained in or contain arbitrary convex sets in terms of the inradii and circumradii of these sets. However, before proceeding further, we shall introduce some notation and give some motivation for this work.

1. Introduction. Using throughout the definitions and notations contained in (4), we state the following recent results by one of the present authors ((4), Theorem I and Theorem 2) which include as a special case the corresponding |C| – result due to Bosanquet (1), Bosanquet and Hyslop (2) and Hyslop(5).

An interesting extension of the idea of convexity has been introduced by Valentine (2). He considered a plane set X such that for any three points a, b, c in X at least one of the segments [a, b], [b, c], [c, a] is contained in X, and showed that such a set can be regarded as the union of at most three convex sets. See also (3). The five pointed star (Fig. 1) is an example that shows that X may not be the union of fewer than three convex sets.

1. Jurkat and Peyerimhoff(3) (or see (4), pages 50–56) have given a definition of a function, f(A), of an infinite matrix A = (ank). In particular, they considered Aα, where a is any real constant. This is defined wherever A is normal; that is to say, when ank = 0 for k > n, and when, for all n, ann ±0. They also defined the ‘Cesàro iteration’ Aα. This is obtained by defining a matrix B = (bnn) by

(so that every diagonal element of B is 1), forming Bα, and then dividing each row by the row sum; that is to say, if Bα = (b(α)nk), we take

where

The definition requires, of course, that ≠ 0.

Let ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

Let

be regular in U = {|z| < 1}. Suppose that the values of f(z) all lie in a domain D in the w-plane. If certain geometrical restrictions are made on D we can deduce growth conditions on the maximum modulus

the means

and the coefficients αn. Good bounds for M(r, f) have been obtained under various conditions on D.

A generalized metric space is a triple (X, d, G), where X is a set, G an ordered Abelian group and d: X x X → G is a function which satisfies the metric axioms. The generalized metric d defines a topology on X in the same way as an ordinary metric does for a metric space. These spaces are studied by Stevenson & Ton(5) and Sikorski (4) amongst others. A survey of these spaces is the subject of an M.Phil. thesis by the author (6). Generalized metric spaces satisfy most of the separation properties of ordinary metric spaces (T2, regular, etc.). We prove here that these spaces are also paracompact. The proof is a generalization of the proof given by Ruffin (2) for metric spaces. The result given here follows, in fact, from a theorem proved by Hayes (1). He proved that uniformities with totally ordered bases have paracompact topologies. It is known (Stevenson & Ton (5); Shock (3); Souppouris (6), chapter 5) that these spaces are identical to generalized metric spaces. The proof given here uses metric space language.

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

1. During the past decade a variety of integral transformations have been extended to various classes of distributions. However, one important integral transformation, which has defied such generalization, is the Kontorovich–Lebedev transformation. The objective of this note is to alter that situation.

There are several known results concerning Radonifying and summing properties of linear mappings in terms of their transposed operators. Thus Schwartz has shown that a sufficient condition for a mapping to be p-Radonifying is that its transposed mapping should be p-decomposing, and that this condition is also necessary if the image space has good topological properties. But as Schwartz observes, these properties are seldom observed by biduals of Banach spaces with their weak-topologies, which arise when 0 ≤ p ≤ 1. This suggests that it would be worth looking for a weaker condition than ‘p-decomposing’.

1. In the equation in the title (referred to hereafter as the equation (E)) a, b are positive constants and h, p are continuous functions depending only on the arguments shown, with p ω-periodic in t, that is p(t, x, y, z) = p(t + ω, x, y, z) for some constant ω > 0 and for arbitrary t, x, y and z. We shall be concerned specifically with the existence of ω-periodic solutions of equations (E), in which the function h is such that |h(x)| → ∞ as |x| → ∞.

Let f be a real-valued function of two real variables. It is well known(5) that the Lebesgue measurability or even upper semicontinuity of all the sections fx and fy does not imply the measurability of f. Lipiński proved (3) that there exists a non-measurable function f such that each section fx and fy is Baire one, has the Darboux property, and fails to be approximately continuous at no more than one point. Davies (1) noted that the continuum hypothesis implies the existence of a non-measurable function f, such that each fy is measurable and each fx approximately continuous. Ursell (6) proved that if each fy is measurable and each fx continuous (or monotonic), then f is measurable.

Recently Davidson (1), Kendall (3) and Shanbhag (7) have established that if U and V are independently distributed random variables such that U is non-degenerate and P(V = 0) < 1, then the distribution of (U2, 2UV, V2) is indecomposable. This implies that the distributions of (U2, 2UV, V2) G and (U2, U) H, where G and H are non-singular real square matrices, are indecomposable. As observed by Shanbhag (7), from this it follows that the Wishart distribution Wp(1, Σ, M) with both p and rank (Σ) ≥ 2 is indecomposable. This is an extension of a result of Lévy (5) who had shown that the distribution Wp(1, Σ, 0) with Σ diagonal is indecomposable.

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.

During the last thirty years an immense amount of research has been done on differential equations of the form

where ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the form

where the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assume

and for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

The nature of the non-linear interaction between two gravitational waves in the general theory of relativity is considered. A new exact solution of the vacuum field equations describing this case is given. It describes two gravitational waves with both transverse and longitudinal components, propagating in opposite directions along ‘shearing’ and ‘twisting’ geodesic congruences with zero contraction

Let l be the differential operator of order 2n defined by

where the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

We shall be concerned with the probability distributions which arise from naive coin tossing experiments and their repetitions. In each case our main question is whether the resulting measure is singular with respect to Lebesgue measure or absolutely continuous. It is a remarkable fact that althoughthese distributions were defined and studied some forty years ago (see (17)for an interesting account), they already provide examples necessary to recent studies of the convolution measure algebra of a locally compact abeliangroup. In order to verify this one is obliged to make substantial improvements on the classical results but, at least at the technical level, this requires no modern apparatus. Accordingly, we shall restrict attention to the circle. Before doing this, however, we summarize the motivational background from abstract harmonic analysis and indicate the applications of our results to that area.