Let A be a closed subset of a topological space X and f a continuous mapping of A into X with the following two properties:

1.1. f {Fr (A)} & f {Int (A)} are disjoint.

1.2. The mapping f* = f | Fr (A) is 1–1.

It is proved in [5], that if X is the euclidean n–sphere Sn = {x; x e Rn+1 and ― x ― = 1}, then

1.3. f {Fr(A)} = Fr {f(A)}.

[ Hence f{Int (A)} = Int {f(A)}].

Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.

Let

be the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved that

and Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many n

and Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.

(i) What is the condition on a given series for every rearrangement to converge?

(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?

The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

Darling [3] in 1932 and Bailey [2] in 1933 gave certain theorems on products of hypergeometric series. Again in 1948 Sears [4] used the relation which expresses the series in terms of M other series of the same type to derive transformations between products of both basic and ordinary hypergeometric series. In this paper I give certain general theorems on products of bilateral hypergeometric series together with some of their interesting special cases.

The formula to be established is

where | amp z < π, z ≠ 0, R(α + β) > R(ς) > R(alpha;) > 0, R(ς - β) > 0.

In proving (1) use will be m ade of the two following formulae.

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

In § 2 a number of integrals in which the integrand contains a product of a hypergeometric function and an E-function will be evaluated. The following formulae will be employed in the proofs.

If ρ +σ = α + β + γ + 1, and if α, β or γ is zero or a negative integer,

this is Sallschütz's theorem [1].

If R(γ - ½α - ½β)< - ½,

This theorem was given by Wastson [2] for negative integral values of α and later by Whipple [3] for general values α.

R(γ) > 0,

This formula was given by Whipple [3]

l is a positive integer,

In his fundamental paper, “On the structure of semigroups” [6], J. A. Green has examined certain important minimal conditions which may be satisfied bya semigroup S.We say that S satisfies the minimal condition on principal left ideals if every set of principal left ideals of S contains a minimal member with respect to inclusion:this condition is denoted by ℳ1. The corresponding conditions on principal rightideals and principal two-sided ideals are denoted by ℳr and ℳ1 respectively. The purpose of the present paper is to give some further results concerning these three conditions.Extensive use is made of the work of A. H. Clifford ([3] and [4]) onminimal ideals.

In the equation

dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by

and, in particular, all positive integer solutions are given by

In some recent work on uniformization [2], I found it necessary to consider a regular branched covering Riemann surface Ȓ of a given Riemann surface Rf, where Rf is an unlimited branched, but not necessarily regular, covering surface of a portion Rz of the extended complex z-plane Z(2-sphere). The branching of Ȓ over Rf had to be chosen so that Ȓ was regular over Rz, since the uniformization of the functions on Rf is then simpler; in particular, the Schwarzian derivative is then a single-valued function of z.