In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.

In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

In this note, I show that, if Q is a local ring and s an integer, 1 < s < codim Q, then there is an ideal As of Q, generated by three or fewer elements of Q, such that

and that, if Q is not regular, then there is an ideal B of Q, generated by three or fewer elements and of infinite homological dimension over Q.

Both Hirsch (cf. (5), (3)) and Eilenberg and Moore (cf. (7)) have described filtered complexes for the cohomology of a fibre space, under different hypotheses. Provided both complexes are defined, we shall give an explicit filtration-preserving map of the first into the second which induces homology isomorphisms. In section 3 we shall use this to transfer the action of the group in a principal bundle from the Hirsch complex (as computed by the author in (1)) to that of Eilenberg and Moore. The latter complex has certain advantages, for example, it is natural, and admits a cup-product structure.

If Σn is a piecewise linear n-sphere and Qn is a piecewise linear n-ball which is a subpolyhedron of Σn thenis a piecewise linear n-ball.

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.

The spectrum of the hydrogen energy operator

(Δ is the Laplacian and r is the distance from the origin) consists of the non-negative real axis and a sequence of negative eigenvalues of finite multiplicities converging to O. In the present study we are interested in finding sufficient conditions on a potential q(x) such that the spectrum of the operator

in En has a ‘hydrogen-like’ spectrum, i.e. a spectrum consisting of

(a) the non-negative real axis,

(b) at most a denumerable set of negative eigenvalues of finite multiplicities having zero as its only possible limit point.

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).

In this paper we study partially ordered vector spaces X whose positive cone K possesses a base which defines a norm in X. A positive decomposition x = y − z of the element x is said to be minimal if ‖x‖ = ‖y‖ + ‖z‖. We proved in (6) that the property that every element of X has a unique minimal decomposition is equivalent to an intersection property for homothetic translates of the base. Section 2 of the present paper analyses this intersection property in much more detail and discusses possible generalizations of it.

Let G be a locally compact Hausdorff topological group, with µ the left Haar measure on G, and µ* the corresponding inner measure. If R denotes the real numbers, ℬ(G) denotes the Borel† subsets of G of finite measure, and VG = {µ(E):E∈ℬ(G)}, then, following Macbeath(4), we define ΦG: VG × VG→ R by

where AB denotes the product set of A and B in G.

This paper continues the investigation begun in (6), and uses the definitions and notation of that paper.

We denote by M(G) the space of bounded regular borel measures on a non-discrete locally compact Abelian topological group G. Under the convolution product and norm of total mass M(G) becomes a complex commutative banach algebra. We denote by Φ the space of m.l.f.s (multiplicative linear functionals) on M(G) and by σ the subset of Φ consisting of functionals symmetric in the sense that

where * denotes the usual involution on M(G). These matters are discussed more fully in Williamson (1).

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying

for every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the function

which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.

Let f(θ) be integrable L in (−π, π) and periodic with period 2π and let

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are established

where the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

Slater ((5), p. 27) and Bose ((1), p. 202) have obtained certain recurrence relations about Whittaker and hypergeometric functions. In this note an attempt has been made to obtain generalizations of these recurrence relations.

Cases in which heat is produced in solids are becoming increasingly important in technical applications ((3), pp. 11–12).

In a long series of papers published some years ago both here and elsewhere, Dr L. J. Slater has derived a large number of valuable results concerned with the theory of generalized hypergeometric functions.

The long open question regarding the almost everywhere convergence of the Fourier series of an L2-function having been recently settled by Carleson (1), an interesting problem still awaits solution. It is to investigate whether the Fourier series of a function of class Lp (1 < p < 2), converges (or diverges) almost everywhere; for it is already known that the Fourier series of an integrable function may diverge almost everywhere. See Kolmogorov (3) and Marcinkiewicz (4) or ((8), 1, p. 305), see also Kahane(2). The object of this paper is simply to show that a suitable restriction imposed on the modulus of smoothness of an integrable function guarantees the almost everywhere convergence of its Fourier series. This will establish, in a more general form, the author's conjecture made in (5). Our proof is based on Carleson's theorem according to which the Fourier series of an L2-function converges almost everywhere.