In trying to extend some theorems of diophantine approximation to number fields or function fields, one meets certain difficulties associated with the presence of units. By using a very simple trick, one overcomes these difficulties. In this paper I limit myself to one example.

In this note new proofs will be given for two inequalities on polynomials due to N. I. Feldman [1] and A. 0. Gelfond [2], respectively; these inequalities are of importance in the theory of transcendental numbers. While the original proofs by the two authors are quite unconnected, we shall deduce both results from the same source, viz. from Jensen's integral formula in the theory of analytic functions.

Let K = {K1, K2, …, Kp} be a system of p bounded closed convex sets in affine space An of n dimensions. If λ1, λ2,…, λp are any p real numbers, we use λ1K1+λ2K2+…λpKp to denote the bounded closed convex set consisting of all the points

The n-dimensional content or volume of this set is a homogeneous polynomial of degree n in the parameters λi, that is

where the summation is over all sets of suffixes 1 £ij, £p, for 1 £j £ n. Further the coefficients may be chosen to be positive and symmetric in their arguments These coefficients, which are in number, are called the mixed volumes of the convex sets.

A metric is defined for a plane affine geometry with coordinates in GF(2n) and is applied to the geometry of a triangle.

The purpose of this note is to settlo a question that was left incompletely solved in the author's paper [1].

Let E be a compact Hausdorff space, and C(E) the class of all continuous real-valued functions on E. A subset F of C(E) is called an upper semi-lattice (or, more briefly, is said to admit ں) if ƒ ں g ε F whenever ƒ, g ε F. A subset F of C(E) is said to be upper semi-equicontinuous (u.s.e.c.) if for every ε > 0 and s ε E there exists a neighbourhood U of s such that

We say that F is locally u.s.e.c. if every uniformly bounded subset of F is u.s.e.c.

Previous workers in this field have only considered plane waves. In this paper the Fourier integral method recently devised by Lighthill is used to estimate displacements at large distances from a harmonic point source in an isotropic elastic medium with infinite electrical conductivity subject to a uniform magnetic field. The effect of the applied field is to introduce anisotropy, and the method used gives a complete geometrical description of wave and energy propagation.

A Room square is an arrangement of the k(2k−1) unordered pairs (ar, as), with r≠s, formed from 2k symbols a0, a1 …, a2k−1 in a square of 2k−1 rows and columns such that in each row and column there appear k pairs (and k−1 blanks) which among them contain all 2k symbols.

The incidence matrices for the finite projective planes, for the so-called λ-planes (where two lines meet in λ points instead of the usual 1) and for some other configurations, including those of Pappus and Desargues, are all cases of what are defined below as (v, k, t, λ)-matrices. These are shown here to possess arithmetical properties which reduce, in the case of cyclic projective planes, to properties remarked on by Marshall Hall [3]. And if a certain group hypothesis (which suggests itself in a natural way) is satisfied by a matrix of this type, the matrix is shown to be equivalent (by rearranging the rows and columns) to a direct sum of incidence matrices for λ-planes, each of which satisfies the same group hypothesis.

The main object of this paper is to show that an indefinite nonsingular quadratic form which is incommensurable (that is, is not a constant multiple of a form with integral coefficients) takes infinitely many distinct small values, for a suitable interpretation of the word small. This proves a conjecture made by Dr. Chalk in conversation with the writer. I believe that the theorems proved are new and of interest, though they are easy deductions from known results.

In [1] and [2] the flow of a conducting fluid past a magnetized sphere was considered. The magnetic distribution was an arbitrary axially symmetric one. The magnetic field, velocity, vorticity and drag were evaluated for the particular case of a dipole field when the dipole was in the free stream flow direction. Astronomically the more interesting case is that in which the dipole is perpendicular to the free stream flow. The axially symmetric nature of the problem is thereby lost.

In previous papers [1, 2], the authors have solved the Stokes flow problem for certain axially symmetric bodies, with the velocity at infinity uniform and parallel to the axis of symmetry. Each of the bodies considered possessed the property that the meridional section intercepted a segment of the axis of symmetry. In the present paper this assumption is removed; in addition, we shall consider the particular case of the Stokes flow about a torus.

Let Κ be a finite number field and let ο be the ring of algebraic integers in Κ. The algebraic integers in a finite extension field Λ of Κ form a ring . We shall be concerned here with the structure of such rings , viewed as modules over ο. It will be useful to begin with a brief discussion of a new concept of the discriminant of Λ/Κ, introduced in a preceding paper [1], which will be our principal tool (see also [2]).

Let Λ′ be a lattice in three dimensional space. Let G be any point and denote by Λ the “displaced lattice”, consisting of all points X + G where X ε Λ′. Suppose that a sphere of radius 1 is centred at each point of Λ and a sphere of radius R−l, where 1 < R ≤ 2, is centred at each point of Λ′. If the lattice Λ′ and the point G are such that no two spheres of the system overlap we shall call Λ a mixed packing lattice, and shall denote its determinant by d(Λ) = d(Λ′). Let Δ be the lower bound of d(Λ) taken over all mixed packing lattices Λ. It can easily be proved that this is an attained lower bound, and any mixed packing lattice Λ, having d(Λ) = Δ, will be called a critical mixed packing lattice. We prove that

and shall describe, in Lemmas 1–3, the critical mixed packing lattices.

The stability of a non-rotating column of liquid was discussed by Rayleigh [1] who showed that axisymmetric disturbances of wave-lengths greater than the circumference of the column decreased the surface area and the consequent release of surface energy enabled the disturbance to grow. Rayleigh [2] also considered the effect of viscosity on the same problem and found that for high viscosity the range of stability was unaltered. For disturbances which are the same in all planes perpendicular to the axis of the column, it is clear that any disturbance increases the length of the boundary of a plane section of the column and hence increases the surface energy. Thus the column is stable for these disturbances.

The membrane theory for calculating stresses in symmetrically loaded elastic shells of revolution was introduced in 1828 by Lamé and Clapeyron [1] who assumed that a thin shell is incapable of resisting bending. In 1892 Love [2] gave the general equations of equilibrium for an element of an elastic shell taking bending into consideration and obtained expressions for the strains in terms of the displacements as well as the stress-strain relations. Since then the problem of the elastic shell has been the subject of numerous researches. The spherical shell has, however, drawn the attention of many investigators due to its importance in structural and mechanical engineering, e.g. roof and boiler constructions.

An ordered triangle in the plane S2 is defined as a sextuple (PlP2, P3; l1, l2, l3) consisting of three points Pi and three lines lj restricted by the relations of incidence Pi Ì lj (i ¹ j) If we map unrestricted sextuples by the points of a V12—Segre product of six planes—we obtain an image-manifold Ω6 for ordered triangles as the appropriate subvariety of V12. The variety Ω6 possesses an ordinary double threefold ɸ3 whose points map the totally degenerate triangles (i.e. those for which P1 ═ P2 ═ P3 and l1 ═ l2 ═ l3); Ω6 is therefore unsuitable as a basis for the construction of an enumerative calculus for triangles, for equivalence theory is as yet developed satisfactorily only on non-singular varieties.

In the preceding paper of the same title (cf. [1]) I defined the notion of the principal genus GK of a finite number field K as the least ideal group, which contains the group IK of totally positive principal ideals and is characterized rationally. The quotient group of the group AK of ideals in K modulo GK is the genus group, its order (Ak: GK) = gK is the genus number, which is thus a factor of the class number hK (in the narrow sense). Associated with the genus group is the genus-field, of K, which is defined as the maximal non-ramified extension of K composed of K and of some absolutely Abelian field.