The following conjecture was apparently first enunciated by Catalan (3) in 1844 but has never been proved.

Let denote the polynomial algebra over the integers in countably many variables ui (i ≥ 1). Let ∂ be the derivation of defined on the generators by. Thus if is graded by dim ui=i, then∂ is homogeneous, of degree − 1. The result is that ∂ is onto, and that its kernel is a polynomial algebra in , where is homogeneous of degree i and the coefficient of ui in is p if i is a power of a prime p, and 1 if i is not a prime power.

B. H. Neumann (5) has shown, by using free products with an amalgamated subgroup, that every group can be embedded in a divisible group. This theorem can also be proved, in an entirely different way, by making use of the wreath product (Baumslag(1)). In view of this connexion between roots in groups and the wreath product, a study is made here of the conditions which ensure that the wreath product (and also the unrestricted wreath product) shares certain properties regarding the existence and uniqueness of roots with the groups from which it is composed.

The behaviour of the Chern classes or of the canonical classes of an algebraic variety under a dilatation has been studied by several authors (Todd (8)–(11), Segre (5), van de Ven (12)). This problem is of interest since a dilatation is the simplest form of birational transformation which does not preserve the underlying topological structure of the algebraic variety. A relation between the Chern classes of the variety obtained by dilatation of a subvariety and the Chern classes of the original variety has been conjectured by the authors cited above but a complete proof of this relation is not in the literature.

In two recent papers ((l), (2)) I investigated inclusion relations between IF (integral function) methods of summability. In the present paper the investigation is continued with the aid of standard Mellin transform theory and results due to Rogosinski and to Kuttner.

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.

This paper demonstrates the existence of an infinity of saddle-points of the function and gives a method of evaluating them suitable for automatic computation. A table of about 100 points is given to 5 decimal places.

The stability of the equilibrium solutions of the equations describing the behaviour of a system of coupled disk dynamos is investigated. In the absence of viscous damping, it is found that systems consisting of more than two dynamos are unstable. That a single disk dynamo is stable is known. The stability of the undamped two-dynamo system has not been ascertained. When viscous damping is present, there are two equilibrium solutions, one in which all the currents are zero, and one in which they are finite. In the zero-current case, a stability criterion is found. Stability criteria are also found in the finite-current case. Further, the existence of the finite-current equilibrium state excludes the stability of the zero-current equilibrium state.

The radial charge densities tabulated opposite were communicated some years ago by the late D. R. Hartree to R. J. Weiss, of the Ordnance Materials Research Office, Watertown Arsenal, Watertown 72, Massachusetts, U.S.A. At his suggestion they are now published, no self-consistent field calculations for Ti being available. It is not known how Hartree obtained the wave functions which he describes as ‘estimated’, but it is presumed that they are interpolated. Radial charge densities for the 3d electrons for Ti+1 have already been published (l).

There has lately been an unusual amount of interest in a relativistic phenomenon which has come to be called the clock paradox. It may be simply stated in the following way. If a man were to leave the earth on a powered space ship for an extended trip into space, he would find upon returning to earth that he was younger than he would have been if he had remained on earth.

The scattering amplitude for a potential expressible as a sum of n separable potentials is obtained as the solution of a set of n simultaneous linear equations. As n→∞ the Fredholm solution for the integral form of the Schrödinger equation is recovered.

In this note we shall give two extensions of the results of (2) concerning the economic use of factory machines. Familiarity with (2) will be assumed and the same notation will be used. We shall show, first, that the number of decision elements required for the solution of the problem can be reduced considerably, and secondly, that the computer will also give the solutions to certain related problems.*