1. In (2) the idea of antithetic variates in Monte Carlo work is introduced and applied to the estimation of integrals. The present paper studies the general theoretical structure of antithetic variates. We have made only limited progress and present various unsolved problems as a challenge to the reader.

No abstract

Consider a Markov chain with an enumerable infinity of states, labelled 0, 1, 2, …, whose one-step transition probabilities pij are independent of time. ThenI write and, departing slightly from the usual convention,Then it is known ((1), pp. 324–34, or (6)) that the limits πij always exist, and that

In n-dimensional Euclidean space En, where we shall throughout assume that n ≥ 2, the maximum number of (n — 1)-dimensional spheres which can be mutually orthogonal is n + 2, and it is well known that the sum of the squares of the reciprocals of their radii is zero, so that the spheres cannot be all real. The maximum number of such spheres which can be mutually tangent is also n + 2, and in 1936 Soddy (7) indicated the beautiful relation connecting their radii. These two formulae are the particular cases γ = 0, γ - 1 of Theorem 1 below, which gives the relation connecting the radii of a set of n + 2 such spheres when every pair is inclined at a given non-zero angle 0, where γ is written for cos θ.

No abstract