Let I be a set and

be a family of sets Aν labelled by the elements of I. Throughout, parentheses ( ) denote families and curly brackets {} denote sets. A transversal, or system of distinct representatives, of F is, by definition, a family (xν: ν ∈ I) of objects xν, for ν ∈ I, such that †

Thus the family ({1, 2,} {1, 2}) has exactly two transversals, namely (1,2) and (2, 1). Let TF denote the set of all transversals of F. Much work has been done on the question of characterizing those F for which TF ≠ Ø. We are here going to characterize those F for which TF has exactly one element. In contrast to the more familiar case TF ≠ Ø, our argument is effective irrespective of the cardinalities of I and Aν.

The aim of this paper is to present a common generalization of and , the Stirling numbers of the first and the second kind ((1), ex. 186–210, pp. 42–45).

A Hadamard matrix Hn is an n by n matrix H = [hij], i, j = 1, …, n in which every entry hij is + 1 or − 1, such that

It is well known that possible orders are n = 1, 2 and n = 4m. An automorphism α of H is given by a pair P, Q of monomial ± 1 matrices such that

Here P permutes and changes signs of rows, while Q acts similarly on columns.

In papers (7, 8), Vizing studied graphs which are critical with respect to edge colourings. In particular, he obtained bounds on the number of edges and on the circuit length of such graphs in terms of its maximum valency. The object of this paper is to improve these bounds by obtaining others which also take into account the order of the graph.

The main part of this paper shows how an inequality of statistical mechanics has several applications in combinatorial theory.

The inequality (known as the FKG inequality) was derived by Fortuin, Kasteleyn and Ginibre(4) in their work on the statistical mechanics of Ising ferromagnets. We first show how it leads to new properties of log supermodular functions, Bernstein polynomials, and log convex sequences.

Familiarity with the category of simplicial sets and the Milnor realization functor, as in e.g. (5), is assumed. We say f: |Y| → |Z| is realized if f = |g| for some g.

Let Ω be a Riemann matrix whose 2n columns are vectors of Cn. It is well-known (e.g. (10)) that the field of meromorphic functions on Cn with these vectors among their periods is of transcendence degree n over C. More precisely, this field can be written as C(A, B) where A = (A1, …, An) is a vector of algebraically independent functions of the variable z = (z1, …, zn) and B is algebraic over C(A). We shall assume that B is in fact integral of degree d over the ring C[A]. Since the derivatives ∂f/∂zi of a periodic function f(z) are also periodic, the field is mapped into itself by the differential operators ∂/∂zi. Thus there exists a function C(z) in C[A] such that these operators map the ring C[A, B, C−1] into itself.

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.

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.

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. 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| → ∞.

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.

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

In a stratified wave guiding system, it is often possible to adjust parameters, such as the wave frequency or some dimension of the guide, so that the propagation constants of two modes become equal. These modes are then said to be degenerate. In a tapered guide some of the parameters are slowly varying functions of distance x measured along the guide, and the conditions of degeneracy can be satisfied for some value x0 of x, possibly complex. Here the two modes are said to be critically coupled, and x0 is called a ‘coupling point’ in the complex x-plane. Cases are known where two modes in the earth-ionosphere wave guide could approach critical coupling in the twilight region of the earth, where the guide is tapered. The x dependence of the composite field for the two modes can then be expressed in terms of an Airy Integral function. The theory is closely analogous to that used for studying critical coupling of Ordinary and Extraordinary radio waves within a horizontally stratified ionosphere.

A uniformly valid asymptotic solution is found for waves refracted by a shearing current or by bottom topography in the case where two straight caustics are present. The ratio of the amplitude of the reflected wave to that of the incident wave increases monotonically from 1/√2, when the caustics are close together, to unity when they are widely separated. A slight modification of the theory gives an expression for the frequencies of waves trapped between two caustics.

A systematic method is given for finding all of the singularities in the stress fields associated with self-similar elastodynamic problems for half-spaces, directly from representations developed in (1). No particular symmetry is assumed, and both two- and three-dimensional problems are discussed for arbitrarily anisotropic half-spaces. Expressions for body wave arrivals are obtained, reproducing results given in (1), and head wave arrivals, not discussed in (1), are found. Arrivals produced by interfacial dislocations and cracks are found by the same method. As an example, an account of the first arrival produced by a penny-shaped crack expanding under shear on an interface is completed, by finding the head wave that precedes the body wave, already found in (1), in a region exterior to a cone in one of the half-spaces.

In a recent paper Koiter (5) discussed a principle of stationary complementary energy for the finite deformation of elastic materials. The complementary energy functional he uses depends only on the components of the nominal stress, and not on the displacement field. He, incorrectly, attributes the principle to Zubov (14).