Throughout this paper, E will denote a finite-dimensional vector space over an ordered field . The real number field will be denoted by ℜ and its rational subfield by . Many of the basic notions in the theory of convexity (convex set, extreme point, hyperplane, etc.) can be defined in the general case just as they are when , but their behaviour may be different from that in the real case. By way of example, we consider the following theorem (due essentially to Minkowski), which is of fundamental importance both for geometric investigations and for the applications of convexity in analysis:

(1) Suppose . If K is a convex subset of E which is linearly closed and linearly bounded, then. K = con ex K; that is, K is the convex hull of its set of extreme points.

The problem considered here is that of a torsional impulsive body force within a semi-infinite elastic solid. The surface of the solid is assumed to be either stress-free or rigidly clamped. The basic solution is found to be essentially the same as that for the surface loading of a half space previously considered by Eason [1]. The displacement is determined in terms of elementary functions for one particular type of body force, although using the results in [1] the solution can be obtained for other types of force. Some numerical results for the surface displacement of the stress-free solid are presented in graphical form.

Let K be any convex body in En, and K any given class of convex sets in En. Then we shall say that K is approximable by the class K if there exists a sequence of sets {Ki}, such that, as i→∞, Ki→K in a suitable metric (for example, the Hausdorff metric), where each set Ki is a vector sum of (a finite number of) sets of the class K An approximation problem is to determine necessary and sufficient conditions for K to be approximable by a given class K.

A slow steady motion of incompressible viscous liquid, bounded by an infinite rigid plane, which is generated when a rigid sphere of radius a moves steadily without rotation in a direction parallel to, and at a distance d from, the plane is considered. Use is made of bispherical coordinates, which were employed some years ago by G. B. Jeffery [1] and Stimson and Jeffery [2] in solving the axi-symmetrical problems in which the sphere is fixed and rotates about a diameter perpendicular to the plane, or when two spheres move without rotation along their line of centres in infinite liquid. The coordinate system has been used recently by Dean and O'Neill [3] in solving the problem in which the sphere is fixed and rotates about a diameter parallel to the plane.

Wave propagation in an elastic medium rotating with a constant angular velocity Ω about an axis is studied by the method of Lighthill [1]. The waves are created by a concentrated periodic force of fixed frequency ω at the origin. The changes in the wave pattern and the decay of amplitude are discussed when 2Ω. increases from zero to a value greater than ω.

Professor C. L. Siegel has pointed out that the statement following equation (9) on page 98 of [1] is false, but can be made correct by adding to the conditions (7) of [1] the further condition:

When Souslin and Lusin initiated and developed the theories of the Souslin operation, of projective sets and of analytic sets, they attached great importance to the constructive nature of their definitions (see [1], [2] and [3]). When Choquet (see [4], [5] and [6]) made his very successful extension of these theories to an arbitrary Hausdorff space Ω, he defined an analytic set in Ω to be a continuous image in Ω of a Kσδ-set in an unspecified compact Hausdorff space X. Thus, a priori, the construction of the analytic sets in Ω requires the preliminary construction of all compact Hausdorff spaces X.

In the theory of the interaction between simple electrically neutral systems with dipole moments, the interaction energy between two such systems when they are identical, one in an excited state and the other in the ground state, is of current interest. It is well-known that, within the Coulomb force approximation for the electron-electron interaction, the energy varies as

where q(r) is the electric dipole moment of the system r = 1, 2, and R is the vector displacement of system 2 from system 1. This is the so called resonance attraction between the systems. On the other hand it has been known since 1948 (see [1]) that for two systems both in their ground states the potential of interaction falls off at large separation faster than the London formula for the energy, namely

predicts. In equation (2) α(r) is the polarization of the system r, in terms of the dipole moments (here induced)

where E is the energy separation between the two states considered, i.e., the ground state and the excited state reached from the ground state by electric dipole transitions. In fact the asymptotic form of the potential energy at separation was given by Casimir and Polder as

Let K be a finite algebraic extension of the rational number field Q, and let R denote the ring of algebraic integers in K. The algebraic integers in a finite extension field of K form a ring which may be considered as a module over R. The structure of these modules has been entirely determined in Fröhlich [2], where, in particular, necessary and sufficient conditions have been established deciding when such a module will be a free R-module.

In the present note we show that the elements of GF(q2) (q = 2n) can be represented in “polar form” in such a way that GF(q2) acts like an “Argand diagram” over its “real subfield” GF(q). From this polar representation it is easy to develop a trigonometry of the plane GF(q2), including definitions of circles and orthogonality. As an application of these ideas we show, in §4, that the circles and lines orthogonal to a given circle yield a new model satisfying Graves' axioms for finite homogeneous hyperbolic planes.

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ1 …, λr be any real numbers. Then there exists a real number α such that

for all integers u > 0 and for q = 1,…,r, where C = C(r) > 0. Thus all the numbers α+ λ1, …, α+ λr are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ1, …, λr and μ1, …, μr there exist α, β such that

for all integers u > 0 and for q=1,…, r, where again C = C(r) > 0. Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.

Solutions of the boundary-layer equations governing the radial laminar flow of a mixture of two different gases forming a wall jet are obtained. Attention is concentrated on flow in which the concentration of one gas in the mixture is small. The stream function is expanded in terms of a parameter whose magnitude depends upon the concentration of this gas in the mixture.

In a recent paper on a divisor problem the author showed incidentally that there is a certain regularity in the distribution of the roots of the congruence

for variable k, where D is a fixed integer that is not a perfect square. In fact, to be more precise, it was shown that the ratios v/k, when arranged in the obvious way, are uniformly distributed in the sense of Weyl. In this paper we shall prove that a similar result is true when the special quadratic congruence above is replaced by the general polynomial congruence

where f(u) is any irreducible primitive polynomial of degree greater than one. An entirely different procedure is adopted, since the method used in the former paper is only applicable to quadratic congruences.

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.