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].

S. T. Tsou and A. G. Walker have defined the I-extension of a given Lie algebra as a certain Lie algebra on the Cartesian product of the given algebra and one of its ideals (Tsou 1955). I-extensions have been studied also in connection with metrisable Lie groups and metrisable Lie algebras. The definition can be applied immediately to any anti-commutative algebra, and in this paper properties of such I-extensions are established. A list of all proper I-extensions of dimension not greater than four over a field of characteristic zero is also given together with a set of characters.

The synthesis of 7-bromo- and 7-chloro-fluoranthene from the readily accessible 1, 2, 3, 1ob-tetrahydro-3-oxofluoranthene is described. Oxidation of the chloro-compound yields 1-chlorofluorene-9-one-8-carboxylic acid and thus opens up a method for the preparation of 1, 8-disubstituted fluorenes and fluorenones.

Thanks are expressed to the Department of Scientific and Industrial Research for the award of a Maintenance Grant to one of us (D. A. C).

In order to estimate the number of partitions of a multi-partite number, the components of which are all large and of approximately the same order of magnitude, it is necessary to evaluate for ℛ(z1) > o(l=1, 2, …,j — 1) the integral

where

for o < u < 2π min (1, |z1|−1, …, |zj−1|−1) and Asymptotic expansions are obtained for I when the z1 are small. Simple expressions give an approximate value of I when every zl is real and exact formulæ are derived when every Zl is real and rational.

An asymptotic formula is given for the number r(s, P; N) of representations of an integer N as the sum of s non-negative squares, where each square does not exceed P2. The numbers s, P and N are large and are subject to certain conditions, one of which is that N is approximately ⅓sP2.

Sets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n=3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

The roots of these equations are of importance in several theories and various authors have studied certain of their properties. Here we solve the equations in the sense that we define two numbers z(1), z(0), and a sequence {zn} which include the roots of both equations. Except for a small, finite number of values of n, we find a rapidly convergent series for zn whose terms are alternately real and purely imaginary. We give a number of expansions and a variety of practical methods which enable us to calculate the small number of remaining roots to any required degree of accuracy.

The main results of this article have been announced without proof or details in Wright 1960.

The preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.

Algebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.

A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

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.