The invitation, made to me, to write the introduction to this volume of the Proceedings of the Royal Society of Edinburgh, which is dedicated to Norman Feather, suggested that I should place on record some of my reminiscences which might convey a picture of his character and personality. I feel, however, that first I should make some reference, albeit brief, to the distinction of his contributions to science over the 45 years which have elapsed since he first started to research in the Cavendish Laboratory at Cambridge.

The asymptotic behaviour of double integrals over a portion of the plane is investigated when the integrand contains an exponential factor with a large parameter. The exponent can have a stationary point which may or may not be close to the boundary of the domain of integration. Results are first derived for a rectangular region with a particularly simple exponent in the integrand and shown to be uniformly valid under certain conditions. In some circumstances the asymptotic terms can be evaluated by means of a universal function. The theory is then generalised to cover more complicated exponents and arbitrary wedge-shaped domains; it is found that the asymptotic behaviour can still be expressed in terms of the same universal function.

It is a great pleasure to join in this tribute to Professor Norman Feather: My recollections of him as teacher, friend and colleague extend back nearly forty years. I recall attending, as a Cambridge undergraduate, his lectures on ‘Properties of Matter’ (what an archaic and nostalgic flavour that title has now!); and hearing him describe his pioneering experiments on the properties of the neutron. He was my doctoral thesis examiner; and later when he was Editor of the Cambridge monographs, it was at his suggestion, and with his help and encouragement, that I made my first essay as an author. I was privileged to succeed my former teachers, Professor C. D. Ellis and Professor Feather, at Trinity College; and in the 1950s I was regularly and warmly welcomed at Edinburgh, the ambivalent benevolence of my role as ‘External Examiner’ notwithstanding.

This paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problem

With suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.

The curve Γ common to two quadric surfaces has 24 principal chords; they are the sides of six skew quadrilaterals each of which has for its two diagonals a pair of opposite edges of that tetrahedron S which is self-polar for both quadrics. These three pairs of quadrilaterals serve to identify the three pairs of quadrics through Γ that are mutually apolar.

The vertices of the quadrilaterals lie four on each edge of S. Both nodes of the plane projection of Γ from such a vertex are biflecnodes.

Let P be a simplicial d-polytope, and, for – 1 ≤ j < d, let fj(P) denote the number of j-faces of P (with f_1 (P) = 1). For k = 0, ..., [½d] – 1, we define

and conjecture that

gk(d + 1)(P) ≥ 0,

with equality in the k-th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P. This conjecture is compared with the usual lower-bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj(P) is a consequence of the linear inequalities given above.

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x(p, q) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula

and

The following problem was posed by Paul Gauthier: if f(z) and g(z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f(z) and g(z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D. It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

In [5] Knowles proved the following.

Every compact Hausdorff space with no isolated points admits a non-atomic measure.

This note is concerned with the converse problem in a more general set up. Here we deal with certain properties of the family of completely regular spaces admitting no continuous measures. In §3 it is shown that this family contains spaces with no isolated points, thus theorem (1.1) does not generalize to completely regular spaces. In §4 a canonical decomposition of the compact members of the above family into discrete subspaces is obtained, and it is shown that these spaces are metrizable whenever they satisfy the first axiom of countability.

The method of the large sieve has played a very important role in number theory. It turns out that estimates of exponential sums are of basic importance for large sieve inequalities. Let

be an exponential sum with complex coefficients c(n). It follows from Theorem 1 of Bombieri and Davenport [1] that

A section of the non-negative orthant by an affine subspace is a polyhedral set. A technique, analogous to that of Gale diagrams, is described which enables one to determine the facial structure of such a polyhedral set.

Both S. Bergman [1] and I. N. Vekua [13] have constructed integral operators which map ordered pairs of analytic functions of one complex variable onto solutions of fourth order elliptic equations in two independent variables. Such operators play an important role in the investigation of the analytic properties of solutions to higher order elliptic equations and in the approximation of solutions to boundary value problems associated with these equations. Unfortunately, little progress has been made in developing an analogous theory for elliptic equations in more than two independent variables. Recently, however, Colton and Gilbert [7] constructed integral operators for a class of fourth order elliptic equations with spherically symmetric coefficients in p + 2 (p ≥ 0) independent variables, and at present Dean Kukral [11], a student of R. P. Gilbert, is in the process of trying to extend some recent results of Colton [3, 4, 5] for second order equations in three independent variables to the fourth order case.

The concepts of bounded paracompactness and bounded metacompactness were introduced and studied in [3]. In this paper we define bounded full normality and show that this concept is equivalent to bounded paracompactness.

It is known that countable paracompactness and countable metacompactness are equivalent properties in a normal space. We show that bounded countable metacompactness is equivalent to bounded countable paracompactness in a normal space and that bounded countable paracompactness is equivalent to bounded paracompactness in a hereditarily paracompact space.

The properties of the solution of the differential equation governing the evolution of localised line-centred disturbances to a marginally unstable plane parallel flow were described by Hocking and Stewartson (1972). A corresponding study of the properties when the initial disturbance is point-centred is presented here. A localised burst at a finite time can be produced, for certain values of the coefficients which can be determined analytically. When the equation permits solutions with circular symmetry, two kinds of bursting solutions, as well as solutions which remain finite, are possible, but the boundary between bursting and finite solutions could not be determined analytically.

Let ˜ be an equivalence relation on a topological space X. A point x ε X i s stable with respect to ˜ if it is in the interior of an equivalence class. We may also add, if ambiguity arises, that x is stable under perturbations in X. Let E be a Banach space, and let L(E) be the Banach space of continuous linear endomorphisms of E, with norm given by |T| = sup{ |T(x) | : |x| = 1}. In this paper we discuss stability of elements of L(E) with respect to some natural equivalence relations.

Maxwell's equations within a dielectric and/or a magnetic medium were first developed macroscopically and must be complemented by constitutive relations to obtain solutions. These relations connect D with E (and with B in optically active material) and H with B (and again with E in optically active material). The atomic and molecular theories of quantum mechanics allow a microscopic approach to derive these constitutive relations where the macroscopic electric and magnetic fields are averages of the microscopic fields e and b. In classical electromagnetic theory Lorentz [1] originally showed how to derive the Maxwell's macroscopic equations from electron theory using microscopic fields obeying the Maxwell's equations in vacuo but coupled to electronic and ionic sources. There are two distinct steps in this procedure. The first introduces microscopic polarization fields, both electric amd magnetic, p and m from which microscopic electric displacement vector field d and auxiliary magnetic field h are simply constructed. The resulting equations for the microscopic fields e, b, d, and h are called the atomic field equations. The second step is the statistical one where the macroscopic fields E, B, D, and H are defined as averages of the microscopic fields and these macroscopic fields are then shown to obey the phenomenological macroscopic Maxwell's equations. A historical appraisal may be found in the recent book by de Groot [2].

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.