Let G be a group of order pn where p is some prime. Denote by nr (G) the number of subgroups of G of order pr. If H is the elementary abelian group of order pn, then.

In his paper on sets of primes with intermediate density Golomb1 proved the following theorem: Let 2 < p1 < P2 < … be any sequence of primes for whichfor every i and j. Denote by A (x) the number of P's not exceeding x. Then.

A similarity method is used to develop a solution of the wave equation within a sector with mixed boundary conditions. In this manner the field which results from the diffraction of an incident pulse of step function time dependence is found.

It is known that the theory of Cauchy's problem for differential equations with two independent variables is réducible to the corresponding problem for systems of quasi-linear equations. The reduction is carried further, by means of the theory of characteristics, to the case of systems of equations of the special form first considered by H. Lewy [1]. The simplest case is that of the pair of equationswhere the aii depend on z1 and z2. The problem to be considered is that of finding functions z1(x, y), z2(x, y) which satisfy (1) and which take prescribed values on x + y = 0.

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Coxeter [1] has studied groups defined by the relations

and gives lists of finite groups known to be completely defined by such sets of relations. In a later paper [2] he shows that G3, n, p is finite if n, p are both even and satisfy

and expresses the conjecture that the restriction to even values may be removed. The only case satisfying this inequality and not already known to be finite is G3, 7, 16. In this note we show that G3, 7, 16 is indeed finite, being of order 21504 = 210.3.7, by showing that its subgroup

of index 2 is finite and of order 10752. Thus we add one entry to each of the lists of finite groups in Coxeter [1].

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.

In the study of connected partially ordered spaces a problem of fundamental interest is to determine sufficient conditions to ensure the existence of chains (i.e., simply ordered subsets) which are connected. Recently [5] R. J. Koch proved that, if X is a compact Hausdorff space with continuous partial order (i.e., the partial order has a closed graph), if L(x) = {y: y ≦ x} is connected for each x ∈ X, and if X has a zero (i.e., an element 0 such that 0 ≦ x for all x ∈ X), then each element of X lies in a connected chain containing zero. It is easy to find simple examples which show that this result is false if X is assumed only to be locally compact. However, if it is assumed that the partial order is that of a topological lattice then the existence of such chains can be shown by elementary methods. This solves a problem which was proposed in [3].

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).

MacRobert [2] in 1937 defined the E-function as

where the symbol denotes that to the expression following it, a similar expression with α and β interchanged is to be added. For (1) he also gave the integral representation

where Re β > 0, | arg z | < π.

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.

If p ≧ q + 1, [1, p. 353]

where, if p = q + 1, | z | < 1. The dash in the product sign indicates that the factor for which t = αr – αr + 1 is omitted, while the asterisk indicates that the parameter αr−αr + 1 is omitted.

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.

In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equations

for the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitution

or

It is the purpose of this note to show that the general case can be dealt with just as simply by putting

The analysis is formal: no attempt is made to supply details of rigour.