Let A be a complete discrete valuation ring of characteristic zero with finite residue field, and for any integer m > 1, let Jm (A) be the subring of A generated by the m-th powers of elements of A. We will prove that any element of Jm (A) is a sum of at most 8m5m-th powers of elements of A. We will also prove a similar assertion when the residue field of A is only assumed to be perfect and of positive characteristic, with the number Γ(m) of summands depending only on m and not on A.

The problem dealt with in this paper was suggested by Dr. E. C. Dade and communicated to me, at Stockholm in August 1962, by Dr. Taussky Todd. It may be stated as follows. Consider the equation

where f is a form (i.e., a homogeneous polynomial) with coefficients in some ring R of algebraic integers. An obviously necessary condition for the solubility of (1) with the xi in R is that the coefficients of f should have no common factor, except for units of R. Now let S be a ring which is an algebraic extension of R, and let us try to solve (1) with the xi in S. The condition just mentioned remains necessary; and Dade has proved in [1] that for some S (depending on R and f) it is sufficient. His theorem is valid for forms of any degree but is merely an existence theorem as far as S is concerned. The problem is to do better for the special case in which f is of degree 2 and R the ring of rational integers; and more precisely, to show if possible that S can be taken to be a quadratic extension R(√q) of R, with an integer q which could be estimated in terms of the coefficients of f.

Let R be a ring, not necessarily commutative, with an identity element, and let A be a left R-module. We shall describe this situation by writing (RA). If

is an exact sequence of left. R-modules and R-homomorphisms in which each Pi (i ≥ 0) is R-projective, then the sequence

which we denote by P, is called an R-projective resolution of A. Suppose now that A is non-trivial; if Pi = 0 when i > n and if there are no R-projective resolutions of A containing fewer non-zero terms, then A is said to have left projective (or homological) dimension n, and we write 1.dim RA = n. If no finite resolutions of this type exist, we write l.dim RA = ∞. As a convention, we put l.dim R0 = −1. If M denotes a variable left R-module, then is called the left global dimension of the ring R and is denoted by l.gl. dim R. It is well known that l.dim RA < n if and only if for all left R-modules B and that l.gl.dim R < m if and only if regarded as a functor of left R-modules, takes only null values.

Let

be two sets of n ≥ 1 consecutive integers with s ≤ t. In this note we are concerned with one-to-one mappings of Γ onto II. If i → f(i) is such a mapping then for i ∈ Γ we write Fi for the highest common factor (i, f(i)), and if Fi = 1 for all i ∈ Γ we say that f is a coprime mapping. Our principal result is

THEOREM 1. If Γ = {1, 2, …, n} and Π = {n+1, n+2, …, 2n} then a one-to-one coprime mapping of Γ onto II can be constructed.

By a convex polyhedron P we mean any bounded set which can be written as the intersection of a finite number of closed half-spaces. If P can be written as a vector sum Q+B of convex polyhedra, then Q and B are called summands of P. If P has a summand which is not homothetic to itself, then P is said to be decomposable.

In a recent paper [1] “On brittle cracks under longitudinal shear” Barenblatt and Cherepanov consider the effect of a constant longitudinal shear on an infinite body containing some particular crack configurations. The problems considered are essentially two-dimensional problems and solutions are found using complex variable techniques. The object of this note is to extend their results to the case of an arbitrary longitudinal shear in an infinite body containing single rows of line cracks. The method employed is that used by England and Green [2].

The stream function is found for the Stokes flow between two fixed non-concentric cylinders due to a line source and sink symmetrically placed on the outer boundary. The force and couple exerted by the fluid on the inner boundary are determined in finite terms and some numerical values are given for the variation of the force and couple with the separation between the axes. A similar problem has been considered by Rayleigh [1] and also the case when the cylinders are concentric in [2]. The line drawn perpendicular to the line joining the source and sink is a line of antisymmetry for the velocity field and hence also for the pressure distribution. The total force on the cylinder is thus directed parallel to the line joining the source and sink. Some numerical values are given to indicate the variation of the force with the separation of the axes of the cylinders. The perfect liquid problem is also briefly considered to illustrate the contrast in the direction of the force exerted by the fluid on the inner cylinder. Here the velocity field again possesses a line of antisymmetry, giving a symmetrical pressure distribution so that the total force is directed along the line of separation of the cylinders. Some numerical values are given to demonstrate the variation of the force with the separation between the axes. It is assumed that the total circulation in the region is zero.

We work throughout in n-dimensional Euclidean space. It has been clear, since the publication of [1], that it should be possible to obtain quite good upper bounds for the number of spherical caps of chord 2 required to cover the surface of a sphere of radius R > 1, and for the number of spheres of radius 1 required to cover a sphere of radius R > 1. But it is not quite simple to organize the necessary calculations to give estimates which are manageable, and which are as good as the method allows for all R > 1. The following results seem to be a reasonable compromise between precision and simplicity; but, for reasons we will give later, they are not completely satisfactory.

If(X1, …, Xn), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x1, …, xn) = 1, for suitable algebraic integers x1, …, xn. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X1, …, Xn) is any polynomial with relatively prime algebraic integers as coefficients, then f(x1, …, xn) is a unit, for suitable algebraic integers x1, …, xn. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x1, …, xn generate, as Watson could.

Following A. S. Besicovitch [1] and L. J. Mordell [3], we say that a polygon is rational if the lengths of all its sides and diagonals are rational. Besicovitch proved that the set of all rational right-angled triangles is dense in the set of all right-angled triangles and that the set of all rational parallelograms is dense in the set of all parallelograms. Then Mordell showed that the set of all rational quadrilaterajs is dense in the set of all quadrilaterals.

Our purpose is to develop a new and simple procedure for embedding graphs into orientable surfaces. This will involve the identification of the oriented edges of two oriented polygons, subject to certain rules.

A steady motion of incompressible viscous liquid caused by the slow rotation of a rigid sphere of radius a is considered. The medium is bounded by an infinite rigid plane and the axis of rotation is parallel to, and at a distance d from, this plane. To complete the analysis the solution by successive approximation of an infinite set of linear equations is required. Satisfactory solutions have been found numerically for four values of d/a, of which 1·13 is the smallest; we gratefully acknowledge valuable help from Miss S. M. Burrough in this part of the work.

Denote by |E| the cardinal of a set E. The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdös, Hajnal and Milner.

Recent papers [1, 2, 3[ have considered dual series equations in Legendre and associated Legendre functions and have given applications of these series to various potential and diffraction problems. This note gives a further application to the problem of the axisymmetric Stokes flow of a viscous fluid past a spherical cap. The stream-function of the flow is found by solving two pairs of dual series equations in associated Legendre functions, these equations being of a form considered previously [1, henceforth referred to as DSE]. As an example a uniform flow past the cap is considered and the drag of the cap calculated. This flow has previously been investigated by Payne and Pell [4], who by a suitable limiting process derive the stream-function for the flow past the cap from the stream-function for the flow past a lens-shaped body. Their method, however, involves the use of peripolar coordinates, besides much complicated algebra, and results are given only for a cap whose semi-angle is π/2. Further, their value for the drag of this cap is incorrect.

Let be a set of points on a sphere, centre O, radius R, in (n+1)– dimensional space. Suppose a spherical cap of angular radius α≤½π is centred at each point of . Let k be a positive integer and suppose that no point of the sphere is an inner point of more than k caps. We say that provides a k–fold packing for caps of radius α.

It is shown that in a certain sense, inversion transforms biharmonic functions into biharmonic functions. The first boundary value problem of elastostatics is also largely unchanged by this transformation, and known solutions can be used to obtain new results for inverse regions. As an example, the problem of a stress free dumb-bell shaped hole in an infinite plate is solved.

Minkowski [1] first proved that the surface of a convex body in E3 can be approximated by a level surface of a convex analytic function. His proof is strikingly simple. His proof is also presented for En by Bonnesen-Fenchel [2, pp. 10–12[. We here prove that the same kind of result is achieved using level surfaces of convex non-negative polynomials. We give two types of approximation, one based on finite sums as Minkowski did, and the other using integration. Since these approximations may be used for other applications we also extend them and give special formulae when the surface is centrally symmetric.