In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, …, k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr. † Let ur = pr/er, where so that

The method of Part I(3) is applied to problems involving boundary conditions on an equatorial plane.

A new and general Monte Carlo technique is described for solving some well-known percolation and cluster-size problems on regular lattice networks. The method has been applied to ten two-dimensional structures.

A new and simple proof is given of the known theorem that, if T1, T2,… is an infinite sequence of finite trees, then there exist i and j such that i < j and Ti is homeomorphic to a subtree of Tj.

The planetary equations for natural elements, which do not depend upon the choice of a particular frame of reference, are derived by using the theory of Poisson brackets. These natural elements include vectorial elements which lead to the introduction of vector and dyadic Poisson brackets. The use of vectorial elements can also lead to redundancy in the set of elements; this idea is extended to show that as many redundant elements as desired can be used and that the evaluation of the derivatives of the disturbing function can be simplified thereby.

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sums

If denotes the projection of on the ith coordinate axis, Esseen proves that if

and ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), then

where and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

The Feynman method of quantization by summation over q-histories is extended to a summation over q, p-histories, corresponding to the classical variational method of treating q and p as independent variables. The equivalence of this q-p summation to the usual quantization procedure is shown in the Heisenberg picture by choosing convenient canonical coordinates and in the Schrödinger picture by obtaining the usual representations of the q, p operators.

This paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).

One of the classical results of ring theory is due to Levitzki ((3), p. 199) and asserts the equivalence of the notions of nil and nilpotent one-sided ideals in a ring satisfying the ascending chain condition on left ideals. Recent, distinctive proofs of this result have been given by Goldie (1) and Herstein (2). We present below yet a fourth way of proving Levitzki's theorem.

A proof by means of graphical techniques is given of an inequality conjectured by Nakanishi concerning the Feynman integrand for any term in the perturbation expansion of the scattering amplitude.

The theory of group representations is explored with regard to statistical applications. Ease of analysis depends rather critically on the amount of symmetry present and this point is examined in detail. ‘ Stationarity’ assumptions are considered for a finite number of variates, with examples of their use in experimental situations.

Let G be a bounded region in k-dimensional space, with boundary Γ, such that the Laplace equation,

is uniquely soluble (to within an added constant) under the Neumann boundary conditions

where ∂/∂n denotes outward normal differentiation on Γ, and it is assumed that h is a function in G ∪ ∂, and thus that g is a function on ∂. In what follows, we shall assume certain properties of the solution h: these are all well known (see, for example, Osgood(l) or Courant(2)).

1. This paper is a continuation of a previous one(1), in which dispersion relations satisfied by the scattering amplitudes for a non-relativistic two-channel scattering process were derived. It was shown that for the process described by the integral equations

with

the scattering amplitudes Mi(E, τ) (where E is the incident energy and τ is the momentum transfer), satisfy the relations

Three distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.