In this note we are concerned with the semi-group Snr, generated by n of its elements, in which each element x satisfies the equation xr = x, the semi-group being otherwise free. The main result proved is that the proposition

(Ar) Snris finite for all finite n, is equivalent to the proposition

(Br) the group Bn, r−1, generated by n of its elements, in which each element x satisfies the equation xr−1 = 1, the group being otherwise free, is finite for all finite n.

The behaviour of a system of two coupled disk dynamos is studied in considerable detail. It is shown that in several cases the stability criteria and general behaviour are closely related to those for the single dynamo studied by Bullard, which are resumed briefly. Extended numerical investigations of the non-linear behaviour of the system are presented: these show that a true ‘reversal of field’ may occur.

Experience has shown that the sections of an electronic digital computer which are easiest to maintain are those which have a simple logical structure. Not only can this structure be readily borne in mind by a maintenance engineer when looking for a fault, but it makes it possible to use fault-locating programmes and to test the equipment without the use of elaborate test gear. It is in the control section of electronic computers that the greatest degree of complexity generally arises. This is particularly so if the machine has a comprehensive order code designed to make it simple and fast in operation. In general, for each different order in the code some special equipment must be provided, and the more complicated the function of the order the more complex this equipment. In the past, fear of complicating unduly the control circuits of the machines has prevented the designers of electronic machines from providing such facilities as orders for floating-point operations, although experience with relay machines and with interpretive subroutines has shown how valuable such orders are.

The method of transforms (7), (2) is shown to be directly applicable to the case of axial symmetric stress distributions in hexagonal crystals. It is shown that solutions for problems of indentation of the hexagonal plane by rigid punches can be found for punches of arbitrary axial symmetric shape. Solutions are given in full for the cases of spherical, conical and circularly cylindrical punches.

The same method is used to find the solutions for a material containing disk-shaped cracks between hexagonal planes and the results for the isotropic case deduced from the general solution.

It is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.

The cases in which there is no coupling are briefly considered from the point of view of the first order equations.

The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.

Specific applications and extensions of the theory are left for later consideration.

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

It is well known to computers that the approximate formula

yields a surprising degree of accuracy even for quite large values of the interval h; for example, if h = 1 the error is one unit in the fourth decimal.

Following on from a previous paper (5), we apply the new Monte Carlo method described there to the estimation of order parameters of a simple Heisenberg ferromagnet. By way of illustration, we include some results on the simple cubic lattice, comparing them with results obtained by conventional methods.

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’

Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the form

where each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equation

which has a Lax representation with

(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.

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.

1. In a recent paper (1) the reflexion of surface waves by a submerged plane barrier has been considered, and it has been shown that there is a reflected wave of finite amplitude at a great distance from the barrier, so that the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is finite. In this paper it is shown that in the case of reflexion by a submerged circular cylinder, the coefficient of reflexion is zero, with the result that the only effect of the obstacle at a great distance is that there is a phase-difference between the incident and transmitted waves, their amplitudes being the same. Only one case has been worked out in detail numerically, and the method employed is such that the condition that the section of the cylinder must be a stream-line has been satisfied only to an approximation; it is, however, shown that the coefficient of reflexion is zero in the general case.

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

1.1. It is often necessary in nuclear physics research to know the distribution in amplitude of a series of electrical impulses. An instrument which reveals this distribution directly is called a ‘pulse-amplitude analyser’, or, in the jargon of the late war, when the problem of building such an instrument was first seriously considered, a ‘kick sorter’. Many kick sorters have been described or discussed ((1)—(16)), and many new types are at present under development.

In a recent paper, Fowler discusses the adsorption isotherm for a monatomic layer, assuming an interaction between neighbouring atoms in this layer. In this paper we do not intend to give a contribution to the physical problem, but merely to show how the statistical problem involved can be solved by a method due to Bethe.

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:

Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?

Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)

Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

In the serial test for sampling numbers (3, 4) an expression is used. It is in the form of a sum of squares and has previously been supposed to have asymptotically a χ2 (γ-variate) distribution. By evaluating it is shown that this supposition cannot be correct, since is not equal to the number of degrees of freedom. The evaluation is based on some results which can be used for testing a wide class of properties of random sequences. The definition of is generalized to deal with subsequences of length ν (instead of length 2) and the generalized , or is then decomposed into a linear form of simple variates defined in terms of discrete Fourier transforms. When the number, t, of digits in the sample space is prime these simple variates are shown to have asymptotically independent χ2 distributions with one degree of freedom each. The coefficients of the linear form are not 1 but they become 1 when the first and second differences with respect to ν are taken. Thus and have asymptotically χ2 distributions. The work is closely related to that of Bartlett (1) and Vajda (6).

Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals, or that R has property (M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals of R terminates. Property (M) is equivalent to the maximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for if R has property (M) and if {Xλ} is a set of indeterminates over R, then the polynomial ring R[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.