Mr. President, ladies and gentlemen, I have had great pleasure in accepting the invitation to deliver this tenth Jeffery-Williams lecture. As one of that generation who were strongly influenced by Ralph Jeffery and Lloyd Williams, I also appreciate the challenge of maintaining the high standard that these lectures named in their honour have established. May I also say that the lecture earlier this morning by Professor Dieudonnè was certainly fortunate for us.

Lie algebras, in the form of algebras of observables, play an essential role in the formulation of classical and quantum mechanics. We discuss whether lie groups play a similar role in general relativity through the holonomy group. We also explore what interrelations these ideas provide between classical physics, relativity and quantum physics.

The following theorem is a widely used corollary of the Thorn-Riesz convexity theorem ([1], pp. 526).

Theorem. Let (Ω, a, μ) be a positive measure space, and let

be a linear operator with ‖T‖1≤1, ‖T‖∞≤1. Then ‖T‖p≤1, 1≤p≤∞

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

Sufficient conditions are given for the error norm and coefficients of best rational Chebyshev approximation on a domain to depend continuously on the domain. Examples of discontinuity are given.

Recently Bruck initiated the study of the structure of the fixed-point set of a nonexpansive selfmap T of a Banach space, where T satisfies a conditional fixed point property. We generalize many of his results to a Hausdorff locally convex space X. Also, we generalize a result of Holmes and Narayanaswami and use it, along with a procedure of Kiang, to obtain a fixed point theorem for families of asymptotically nonexpansive mappings in X.

Local criteria are given which imply the lipschitzian character of mappings defined in complete and convex metric spaces.

In the study of quasi-analytic classes (see [3], pp. 372-379), a class C{Mn} is shown to have the following properties:

1. (a) If M0= 1 and (i.e. {Mn} is log convex), C{Mn} forms an algebra.

2. (b) C{Mn} is invariant under affine transformations.

3. (c) C{Mn} is quasi-analytic iff it contains non non-trivial function with compact support.

The construction of images of morphisms in an elementary topos E has hithero required the use of colimits. For example, in [1], Freyd constructs the image of a morphism by taking the equalizer of the cokernel pair of the morphism. In particular, the construction of the direct image functor, or, as it is sometimes referred to, existential quantification along a morphism, has required the use of images, and hence colimits. However, Mikkelsen has defined existential quantification using only limits.

The notion of entire function of Bounded Index is by now well established. It may be stated as follows.

An entire function f(z) is said to be of Bounded Index if for some fixed s

for all n and all z. (See [1], [2].)

Recall that an automorphism ϕ of an algebra A is called regular if, for all x∊A, ϕ(x) = x implies x = 0. For various types of algebras it is known that the existence of a regular automorphism implies that the algebra is radical. Here, for associative algebras, we relax the restrictions on the fixed points so that all automorphisms of commutative algebras immediately satisfy the hypothesis on the fixed points.

In 1957, M. A. Subhankulov established the following identity

where ; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).

In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.

It is shown that R is a biregular near-ring if and only if it is isomorphic with the near-ring of sections of a sheaf of reduced near-rings over a Boolean space. Also, some ideal properties of a biregular near-ring are proved. These are considered as generalizations of some works of R. Pierce on biregular rings.

As a contribution to various investigations [1-11] about packing of convex bodies with certain conditions imposed on the number of neighbours of each body, V. Chvátal [12] recently proved the following theorem: If in a packing of translates of a square each square has at least six neighbours then the density of the packing is at least 11/15.

One of the interesting mathematical problems is whether the system of four Diophantine equations

1

2

has a solution in x, y, z, I, m, n, w. To this day the problem has not been shown to be impossible, nor has it been solved.

An (r, λ)-system is an arrangement of v objects (or varieties) into subsets (or blocks) such that each variety appears in exactly r blocks and each pair of distinct varieties appears in exactly λ blocks. To avoid trivial designs, we assume that 1≤λ<r.

It is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

The dual object T of a compact group is called a local central A(p) set if there is a constant K such that ‖X‖P < K ‖X‖1 for all irreducible characters X of G. For each γ∊Γ, Dr is an irreducible representation of G of dimension dγ. Several authors [1, 2, 3, 4] have observed that Γ is a local central Λ(p) set for p<l provided sup{dγ:γ∊Γ}>∞, and some of them [2, 3] conjectured the converse. Cecchini [1] showed that Γ is not a local central Λ(4) set if G is a compact Lie group.