For any (real or complex) transcendental number ξ and any integer n > 0 let ϑn(ξ) be the least upper bound of the set of all positive numbers σ for which there exist infinitely many polynomials p1(x), p2(x), … of degree n, with integer coefficients, satisfying

where ‖pi‖ denotes the “height” of pi(x), i.e. the maximum modulus of the coefficients. Plainly ϑn(ξ) serves as a measure of how well (or how badly) the number zero can be approximated by values of nth degree integral polynomials at the point ξ. It can be shown by means of the “Schubfachprinzip” that, at worst,

if the transcendental number ξ is real, and

if it is complex, i.e.ϑn(ξ) is never smaller than these bounds. Furthermore, a conjecture of K. Mahler may be interpreted as stating that for almost all real and for almost all complex numbers the equations (2) and (3), respectively, are actually true; in other words, almost all transcendental numbers have the worst possible approximation property for any degree n.

The main object of this paper is to give a self-contained elementary proof of a result (Theorem 1, below), which could be deduced from a theorem of Siegel ([1], Satz 2). It seems worth while to do so, because Siegel's proof is long and difficult, though his result is deeper and more precise than mine.

Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

1. Introduction. Let I0 be a closed rectangle in Euclidean n-space, and let ℬ be the field of Borel subsets of I0. Let ℱ be the space of completely additive set functions F, having a finite real value F(E) for each E of ℬ, and left undefined for sets E not in ℬ. In recent work, we used Hausdorff measures in an attempt to analyze the set functions F of ℱ. If h(t) is a monotonic increasing continuous function of t with h(0) = 0, a measure h-m(E) is generated by the method first defined by Hausdorff [2].

1. Let f = f(x1, …, xn) be an indefinite quadratic form in n variables with discriminant d = d(f) ¹ 0; and let ξ1, …, ξn be real numbers. We consider how closely the inhomogeneous quadratic polynomial

can be made to approximate to a given real number α by choice of suitable integral values of the variables xi. The best that is known seems to be that the inequalities

can always be satisfied if the implied constant is given a suitable value depending only on n. For α ≥ 0 this is a restatement of a result proved by Dr. D. M. E. Foster.

An exact expression in finite terms is found for the small deflexion at any point of an infinitely large plate clamped along an inner curvilinear edge, with outer edge free, and loaded by a concentrated force at an arbitrary point of the plate. The plate can be mapped on the area outside the unit circle by a rational mapping function involving two parameters. By varying these parameters holes having various shapes and several axes of symmetry are obtained. Infinite plates with holes in the forms of regular and approximately rectilinear polygons are included as special cases.

Let G1,… Gn be groups, let *Gi be their free product, and let ´ Gi be their direct product. A homomorphism may be defined by requiring it to be trivial on Gj and the identity on Gi for i ¹ j. Let [G1 … Gn] = Çker pj. P. J. Hilton [2] proves that [G1, …, Gn] is a free group, and, if HiÌGi, i = 1, …, n, that [H1,… Hn] is a free factor of [G1 … Gn]. He asks whether, if Hλi Ì Gi, λ = 1, …, k, i = 1, …, n, and Hλ = [Hλ1, …, Hλn] the group generated by the Hλ is a free factor of [G1, …, Gn].

In recent papers [1, 2] I have given expressions for the stresses and displacement due to elastic distributions in an infinite isotropic elastic solid bounded internally by a spherical hollow, the boundary of which is either stress or displacement free. This paper gives corresponding results for a semi-infinite elastic solid together with expressions for the stresses and displacement due to thermoelastic distributions in such a solid, the boundary of which is maintained at a constant temperature and is either stress or displacement free.

Let N be a large positive integer and let n1, …, nN be any N distinct integers. Let

Hardy and Littlewood proposed the problem: to find a lower bound for

in terms of N, this lower bound to be a function of N which tends to infinity with N. It is easily seen, on examining the case when n1, …, nN art in arithmetic progression, that a lower bound of higher order than log N is impossible.

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.