In statistical problems, use is often made of the variance s2 of a finite sequence of numbers x1, x2, …, xn, where

x¯ being the arithmetic mean of x1, x2, …, xn. The object of this note is to introduce a generalized variance for an infinite sequence x = (xk) and to examine some of its properties.

The double cusp is the real, compact, unimodal singularity

see (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,

Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.

Let {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

1. Introduction. In recent years a good deal of progress has been made in obtaining solutions to practical problems concerned with reinforcement optimization of what, in practice, might be reinforced concrete or other fibre reinforced composites in bending (1). Almost all the solutions have been obtained using a linear expression relating bending moment to the amount of fibre required. In what follows a non-linear expression will be adopted; the expression is obtained by including the next term in a parameter series expansion.

Let Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

The exterior problems of acoustics can be reduced to the solution of Fredholm integral equations of the second kind in many different ways. Some of these equations break down at certain values of the wavenumber but this difficulty can be reduced or even removed altogether by choosing a different (usually more complicated) integral equation. The present paper is concerned with a class of equations proposed recently by D. S. Jones. It is shown that the proof of one of his principal results can be much shortened.

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution function G, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

We develop a theory of particle scattering in anisotropic media. That is, a medium in which the microstructure causes the mean free paths of the particles to become dependent on their direction of motion with respect to some fixed axis. The equation which results is similar to the normal, one-speed Boltzmann transport equation but has cross-sections which are functions of direction. This equation is solved for arbitrary cross-sectional dependence on direction in plane geometry.

Let X, Y be real Banach spaces, and let = {f(λ):λ ∈ m} be an m-parameter family of bounded linear operators from X to Y, with f(λ) depending continuously on λ. The cases m = 1 and m = 2 are considered, and conditions on are found which determine the null space of f(λ) for all λ near a given λ0 such that f(λ0): X → Y is a Fredholm operator. The results obtained are shown to be of particular interest in perturbed bifurcation theory.

Let Ω ⊆ n be open and bounded, with ∂Ω smooth. Sufficient conditions on f are given in order that for p > n, the equation

has a solution for every h ∈ Lp; L is a second order symmetric elliptic operator and B represents either the first, second or third boundary value problems. These conditions are in terms of the asymptotic behaviour of f, in its second variable, in relation to those λ for which there is a nontrivial solution of

We consider globally hyperbolic spacetimes (M, g) of dimension ≥ 3 satisfying the curvature condition Ric (g) (v, v) ≥ 0 for all non-spacelike tangent vectors v in TM. This curvature condition arises naturally as an energy condition in cosmology. Suppose (M, g) admits a smooth globally hyperbolic time function h: M → such that for some t0, the Cauchy surface h−1(t0) satisfies the strict curvature condition Ric (g) (v, v) > 0 for all non-spacelike v attached to h−1(t0). Then M admits a metric g′ conformal to g satisfying the strict curvature condition Ric (g′) (v, v) > 0 for all non-spacelike v in TM. If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.

This paper is devoted to a preliminary discussion of potential wells in non-linear three-dimensional elasticity. Our interest in the subject arises from the role that potential wells play in the justification of the energy criterion as a sufficient condition for stability of an elastic equilibrium solution. It will be recalled that the energy criterion, which is a simple extension of the original Lagrange-Dirichlet version for finite-dimensional systems, states that an equilibrium solution is stable provided the potential energy achieves its minimum on the solution. No proof of this statement as it applies to three-dimensional elasticity is yet forthcoming, although when the notion of a minimum is replaced by that of a potential well, several authors have proved that the criterion thus modified is sufficient for the Liapunov stability of the equilibrium solution with respect to appropriate measures. Indeed, the proofs are applicable to many other continuum theories, apart from elasticity. (See, for instance, Coleman(5), Gurtin(11) and Koiter(21).) It thus becomes important to determine what constitutive and other conditions, if any, ensure the existence of a potential well. While we present two such conditions, our main purpose is to describe examples in support of the conjecture that non-existence rather than existence of a potential well is likely to be the generic property. In these examples, particular forms of the potential energy are chosen which have positive-definite quadratic part and yet in any W1, P-neighbourhood (1 ≤ p ≤ ∞) of the origin, have a non-positive infimum, thus violating a condition for existence of a potential well in the, Sobolev space W1, P (1 ≤ p ≤ ∞). Related results are also reported by Ball, Knops and Marsden(3) and by Knops(17) for the space W1, ∞. Another conclusion to be drawn from these examples, is that a potential well cannot be ensured by restricting only the quadratic part of the potential energy. Indeed, in our example, we show that the equilibrium (null) solution is unstable in the sense that (non-linear) motions, starting in its neighbourhood, cease to exist after finite time. The same phenomenon has been shown by Knops, Levine and Payne(19) (see also Hills and Knops(14)) to hold for a general class of materials which includes as a special case one possessing the potential energy considered in our example.