The aim of this note is to prove an analogue of P. Hall's theorem on distinct representatives (1) for a family of measurable sets.

An unbounded two-phase medium is transmitting radiation from a group of uniformly moving, two-dimensional forcing effects. The emission process is governed by two matrix differential equations for both interior regions, accompanied by a common set of boundary conditions along the communicating interface. Anticipating possible hyperbolic modes, in co-existence with elliptic modes during the eventual steady state, a radiation condition is initially applied. This radiation condition, together with a particular boundedness criterion, decides the precise number of boundary equations required. An exact solution is established for moving forcing effects having arbitrary spatial distributions. Those aligned along the interface correspond to sheet sources. Superimposed are additional source sheets virtually induced by the action of internal forcing effects, converting their combined incident field into reflection and refraction. A peculiar feature is the invariant propagation along characteristics of certain hyperbolic modes, under the essential presence of elliptic modes, as real Hilbert transforms (in the sense of principal values) of a net interface source density.

We prove a conjecture of (5), namely that the convex set of all infinite doubly stochastic matrices whose entries are all strictly less than θ(0 < θ ≤ 1) possesses extreme points if and only if θ is irrational.

Let α > − 1. For a given series write

it being tacitly assumed that the series defining a(t) and A(t) converge for all t > 0. Σan is said to be summable (Aα) to A if

Suppose that w is analytic in the open unit disc Δ in the z-plane. The Schwarzian derivative of w is defined by

(for basic properties of {w, z} we refer to Hille ((3), p. 375)). Let denote the class of functions w which are analytic in Δ and satisfy the conditions

Let θ be an n-dimensional random vector with components θ1, θ2, …, θn and let itscumulants

be defined as usual by the identity

where x1, x2, …, xn are purely imaginary variables, E denotes expectation, and

Let θ(1), θ(2), …, θ(R) be independent and identically distributed (i.i.d.) random vectors each with the same distribution as θ, where R = r1 + r2 + … + rn = |r|, the order of the cumulant Kr. Let θ* have the components

1. Introduction and statement of results. The most substantial contribution in recent years to the literature on the Galton–Watson embedding problem has been that of Karlin and McGregor (4) and (5), to which the readeris referred for a more complete bibliography of the subject. Their results and methods very much involve the analytic properties of the associated probability generating functions (p.g.f.'s), and most of the results they describe as ‘negative’ in the sense that they are sufficient conditions for non-embeddability. The two results here are similarly ‘negative’ but obtained from more direct probabilistic considerations.

Consider a renewal process on the nonnegative real line with non-arithmetic distribution function F(x). Denote by V(x; t) the distribution function of the forward recurrence time from t, t ≤ 0. If t is chosen at random with distribution function Ф(t), the corresponding unconditional forward recurrence time has distribution function

1.Introduction. The purpose of this note is to apply the Kostant-Souriau quantization theory (2, 3, 4, 5, 7) to construct representations of a semi-direct product.

A uniform asymptotic solution is obtained which describes the propagation near an isolated caustic of water waves on a slow irrotational current. Unlike earlier work there is no restriction to straight caustics or to steady currents.

The equations satisfied by perturbations of solutions of the Euler-Lagrange equations corresponding to a Lagrangian density L are derived from the first and second variations of L and these new Lagrangian densities are used to construct Noether and Euler identities. It is shown how these identities can be used in general relativity to construct conserved quantities and local uniqueness theorems.