In an intersting article, ‘Bezout rings and their subrings’ (2) P. M. Cohn has introduced a construction which may also be applied in a slightly different situation, i.e. the imbedding of a Krull ring in a Dedekind ring.

The importance of principal ideal domains (PIDs), both in algebra itself and elsewhere in mathematics is undisputed. By contrast, Bezout rings, ‡ although they represent a natural generalization of PIDs, play a much smaller role and are far less well known. It is true that many of the properties of PIDs are shared by Bezout rings, but the practical value of this observation is questioned by many on the grounds that most of the Bezout rings occuring naturally are in fact PIDs. However, there are several fairly natural methods of constructing Bezout rings from other rings, leading to wide classes of Bezout rings which are not PIDs, and it is the object of this paper to discuss some of these methods.

One of the most widely studied concepts in graph theory is that of the colouring of graphs. Much of the research in this area has dealt with the chromatic number χ(G) of a graph G, defined as the minimum number of colours which can be assigned to the points of G so that adjacent points are coloured differently. One can obtain a natural generalization of this concept as follows: we define the n-chromatic number χn (G) to be the smallest number of colours which one can associate with the points of G so that not all points on any path of length n are coloured the same. The 1-chromatic number is then simply the chromatic number.

Let Q be a quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined. If, for every finite sequence q1, q2, … of elements of Q, there exist i and j such that i < j and qi ≤ qj then we call Q well-quasi-ordered. For any ordinal number α the set of all ordinal numbers less than α is called an initial set. A function from an initial set into Q is called a transfinite sequence on Q. If ƒ: I1 → Q, g: I2 → Q are transfinite sequences on Q, the statement ƒ ≤ g means that there is a one-to-one order-preserving function ø:I1 → I2 such that f(α) ≤ g(ø(α)) for every α ∈ I1. Milner has conjectured in (3) that, if Q is well ordered, then any set of transfinite sequences on Q is well-quasi-ordered under the quasi-ordering just defined. In this paper, we define so-called ‘better-quasi-ordered sets’, which are well-quasi-ordered sets of a particularly ‘well-behaved’ kind, and we prove that any set of transfinite sequences on a better-quasi-ordered set is better-quasi-ordered. Milner's conjecture follows a fortiori, since every well ordered set is better-quasi-ordered and every better-quasi-ordered set is well-quasi-ordered.

Theorem. A simple closed curve J in the plane E2separates E2.

We give a relatively simple proof of the theorem on the spectral sequence of a fibre-space in the sense of Kan. The method consists in the reduction to locally trivial fibre-spaces by minimal fibrings and in the use of a very simple acyclic model theory (without degeneracies).

A group G of homeomorphisms of a topological space X will be called discontinuous if

(1) the stabilizer of each point of X is finite, and

(2) each point x ∈ X; has a neighbourhood U such that any element of G not in the stabilizer of x maps U outside itself (i.e. if gx ≠ x then U ∩ gU is empty). The purpose of this note is to prove the following result.

The purpose of this paper is to give a complete list of those nilpotent groups which can be the fundamental groups of connected, closed, compact but possibly non-orientable 3-manifolds. The starting point is the following theorem of Reidmeister, which is given a neat proof in (1).

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).

A paranormed space X = (X, g) is a topological linear space in which the topology is given by paranorm g—a real subadditive function on X such that g(θ) = 0, g(x) = g(−x) and such that multiplication is continuous. In the above, θ is the zero in the complex linear space X and continuity of multiplication means that λn → λ, xn → x(i.e. g(xn − x) → 0) imply λnxn → λx, for scalars λ and vectors x. We shall use the term semimetric function to describe a real subadditive function g on X such that g(θ) = 0, g(x) = g(−x). Two familiar paranormed sequence spaces, which have been extensively studied (3), are l(p) and m(p). For a given sequence p = (gk) of strictly positive numbers, l;(p) is the set of all complex sequences x = (xk) such that and m(p) is the set of x such that sup Throughout, sums and suprema without limits are taken from 1 to ∞. Simons (3) considered only the case in which 0 < pk ≤ 1 so that natural paranorms would seem to be in m(p). In fact Simons showed that g1 was a paranorm for l(p), but that g2 did not satisfy the continuity of multiplication axiom.

Given a locally convex Hausdorff linear topological space, we construct and examine the following topologies in the space:

the finest locally convex topology with the same convergent sequences as the initial topology, and the finest locally convex topology with the same precompact sets as the initial topology.

We obtain a simple new proof of Bonsall's characterization theorem for type 1 semi-algebras of continuous functions. An extension of our methods produces several new results on stability properties of wedges.

Suppose that λ > − 1 and that

It is easy to show that

With Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the series

is convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

1. For the familiar definition of (R, λn, κ), (R*, λn, κ) and (N, p) means and their notations, see, for example (3). If {fn} is any arbitrary sequence, we adopt the convention throughout that f−1 = 0. A method of absolute summability |A| is said to be ineffective if it is absolutely regular and sums only absolutely convergent sequences. The main object of this note is to prove absolute analogues of Kuttner's results in (3). And for this we need to prove a theorem (Theorem 1) on ineffectiveness for |N, p| method.

We let h1/k1,h2/k2…hN(n)/kN(n)represent the Farey series of order n. It is well known that

Recently, Gupta (l) proved that

and Hans and Chander (2) proved that

The authors, while attempting to give a new generalization of the Laplace transform, have come across a particular function in the form of Meijer's G-function

which, when taken as a nucleus of the transformation K(x) in

defines a new transform. This transform besides serving as a generalization of the Laplace transform and most of its generalizations existing in the literature bears the characteristic property of generalizing the Hankel transform (and hence some of its generalizations) as well. In this paper the authors after having defined the transform have given an inversion theorem which is supported by means of two examples.

The Stieltjes transform is a consequence of the iteration of two Laplace transforms, namely if

then is referred to as the Stieltjes transform of g(t).

The main object of this paper is to derive an expansion formula for a generalized hypergeometric function of two variables in a series of products of generalized hypergeometric functions of two variables and a Meijer's G-function. The result established in this paper is the extension of the results recently given by Srivastava (5) and Verma (6). It is interesting to note that some interesting expansions can be derived from the result by specializing the parameters.