The rings studied in this paper were first constructed in connexion with the problem (Proposition 1) of extending a place from an integral domain to its field of fractions. They provide (Proposition 3) a series of examples of places not so extendable, generalizing an example implicit in the work of Samuel(3); and by the use of ultraproducts they show that such extendability is (in certain precise sense) not an elementary property (Proposition 4). On the other hand they have some unexpected properties, especially in connexion with the relation 1/y0 + … + 1/yr = 1, and they have very small automorphism groups (Proposition 2 and Corollaries); and they are not unique factorization rings (Proposition 5), though they are integrally closed (Proposition 6).

This note exhibits an infinite independent set of laws for monoids (viz. (xpyp)2 = (ypxp)2, p prime) and a variety of groups defined by one law (which is x2y2 = y2x2) such that the smallest containing variety of monoids is not finitely definable.

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).

If M is a group, Z(M) its integral group ring and AM the augmentation ideal, then following Passi we can form the Abelian groups

The main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.

In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.

The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.

The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).

The aim of this paper is to reformulate the work of Massey(6), and Spanier(9, 10), on higher order cohomology operations, in terms of vector bundles in such a way as to produce geometrically some higher order operations in K-theory, which we will call Massey products.

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:

Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.

The conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.

In the general theory of relativity a number of apparently unrelated identities peculiar to a 4-dimensional space are frequently used. However, the proofs usually presented appear to have no common ideas and, furthermore, it is not clear at what stage the dimensionality restriction plays a significant role. In this note a technique is presented which explicitly exhibits the dimensionality dependence and thereby enables the above identities to be generalized for n-dimensional spaces. It is also shown that the above identities are all special cases of a more general identity valid for n = 4.

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.

It has been pointed out to me by Professor A. M. Macbeath that the proof of Theorem 2 which is given in (2) is incorrect, in that it relies on Proposition 3·1 of (2), which is false, the mistake in its proof being the assumption that in an arbitrary compact topological space, every sequence has a convergent subsequence. However, we can still prove Theorem 2 of (2), simply by replacing section 3 of (2) by the corrected section 3 below, and replacing the proof of Proposition 4·2 of (2) by the corrected proof of Proposition 4·2 given below.

Let G be a locally compact, unimodular, topological group with μ Haar measure on G, and μ* the corresponding inner measure. If (G) denotes the Borel subsets of G of finite measure, and V(G) = {μ(E):E∈(G)}, then the Product Set function, or P.S.-function, of the group G, written ΦG, is defined by

We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.

A decomposition of a topological vector space E is a sequence of non-trivial subspaces of E such that each x in E can be expressed uniquely in the form , where yi∈Ei for each i. It follows at once that a basis of E corresponds to the decomposition consisting of the one-dimensional subspaces En = lin{xn}; the theory of bases can therefore be regarded as a special case of the general theory of decompositions, and every property of a decomposition may be naturally denned for a basis.

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

Suppose that λ > − 1 and that

The system of equations here to be analysed is

The method of solution is by way of appeal to the theory of singular integral equations of Cauchy type. Apart from the self-reciprocal series of Titchmarsh(l) no comparable systems appear to have been discussed in the literature, the order of the matrix element falling foul of the restrictions otherwise there imposed.

A result due to Titchmarsh(1) is that corresponding to the series

is the reciprocal formula

By a generalization of the Infeld-Hull Factorization Kernels we find it possible to factorize a wide family of second-order ordinary differential equations at specified points of their spectrum.