A commutant property of ternary algebras, defined as linear subspaces of ℬ(ℋ) (bounded operators on some Hubert space ℋ) closed under three-fold multiplication, is demonstrated through the recovery of an operator from its non- orthogonal restrictions.

We give a new criterion for an order in a commutative split algebra over the quotient field of a discrete valuation ring R to be self-dual. We derive a result on the ideal classes whose order is an extension by R of a self-dual order. This yields a wide generalization of a theorem of Faddeev(1). We can also answer a question arising from a paper of Fröhlich(2).

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

The objective of this note is to complete in one essential way the study undertaken in (2) of the relation between complex bordism and the connective k-homology theory. Specifically, let us denote by MU*( ) the generalized homology theory associated to the Thorn spectrum MU(6), and by k*( ) the generalized homology theory associated to the connective bu spectrum (2, 4). Recall that

Let G be a locally compact Abelian group.

DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that

Definition (Moukoko Priso(2)). A locally convex spaceE[T] is said to have a strict absorbent network of type Σ if there exists in E a familyof absolutely convex absorbent sets such that

(1) if {nk} is a sequence of positive integers andfor each k, then the seriesconverges in E[T]

(2) for each sequence {nk} there is a sequence {λk} of positive real numbers such that, ifand 0 ≤ μk ≤ λkfor each k, then

(i) converges in E[T], and

(ii) for each p.

The following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequences

are uniformly distributed on the circle, for almost all x in G.

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditions

Jones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this case

and the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) is

so that

Using this relation, Churchhouse has shown that

and

where

with

Moreover

and b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,

and

The object of this note is to prove these conjectures.

In this paper we first demonstrate how a certain formula, which expresses (n − 1 )th divided difference in the form of a multiple integral, may be used to obtain the density function of a suitable random variable and then apply this to obtain the density of a useful variate.

A generalization is given of the notion of stochastic monotonicity. This is used in setting up, and showing how to obtain approximate solutions for, an integral equation formulation of GI/G/l queues, when access to the waiting-line is subject to periodic interruptions.

We consider the equation:

where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,

f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.

A metric is constructed, which represents the gravitational field of two spherical static bodies of masses +b and −b, together with a massless source of angular momentum between them. When the separation of the two masses is allowed to tend to infinity the N.U.T. metric can be obtained from the limiting process. This supports the view that the N.U.T. solution represents a static Schwarzschild body together with a rotating semi-infinite spike of pure angular momentum.

1. The proofs of many results in the theory of stability and boundedness basically depend on dividing the vicinity of some kind of invariant set (or other convenient set) into suitable subsets and then trying either to prove that solutions cannot leave such sets or to estimate the escape time. This observation makes it possible to give some global results in terms of arbitrary sets which can be employed as tools in dealing with various problems of stability and boundedness. In applications, these tools enlarge the class of useful Lyapunov like functions and also offer more flexibility.

A formal asymptotic theory, valid at high frequencies, is developed for the propagation of time harmonic Rayleigh surface waves over the general smooth free surface Σ of a homogeneous elastic solid. It is shown that on Σ these Rayleigh waves can be described by a system of surface rays, which are shown to be geodesics of Σ. The amplitude of the waves on Σ is shown to vary in such a way that the energy propagated along a strip of surface rays is constant. The waves are also shown to be dispersive and an explicit first-order dispersion formula is derived.

The flow of a viscous incompressible fluid past a paraboloid of revolution is described by matching a series of boundary-layer approximations valid far downstream to a series of potential flow solutions valid far from the solid surface. The development parallels that of Goldstein (3) and Murray (10) for the flat plate, becoming identical for infinitely large Reynolds number; it is found that logarithmic terms must be introduced at the third stage of the matching, and that these produce constants in the downstream solution which remain indeterminate. These terms result from interaction between inner and outer solutions, rather than from appearance of a complementary function of the inner equation.

Expansions obtained from classical subsonic thin-aerofoil theory break down in the neighbourhood of the aerofoil edges. At sharp edges the method of matched asymptotic expansions fails to remedy this. Here this failure is explained, and in the case of incompressible flow past a symmetric aerofoil at zero incidence it is shown that by proper choice of the dependent variable an expansion may be obtained which is uniformly asymptotic. Finally, the case of a circular-arc aerofoil is considered in more detail.

A method of solution is given to the problem of internal wave motion in a non-homogeneous viscous fluid of variable depth. The approach is based on the inviscid theory of Keller and Mow(l) and on boundary-layer analysis. For internal progressive waves in uniform depths, it reduces essentially to the theory given by Dore(2). The present results are also applicable to surface waves when the fluid is homogeneous.

In this paper, the general theory of a Cosserat surface given by Green, Naghdi and Wainwright(1), has been applied to the problem of a thermo-elastic Cosserat plate containing a circular hole of radius a. We assume that the major surfaces of the plate and the boundary of the hole are thermally insulated and that a uniform temperature gradient τ exists at infinity. In the limiting case, when h/a → 0, where h is the thickness of the plate, the thermal stresses at the circular hole reduce to those obtained by Florence and Goodier (4), by means of the classical plate theory. Results for the other limiting case h/a → ∞ are also given.