Consider a topological n–sphere Sn, where n ≥ 1. By a commutative product I mean a continuous function f: Sn × Sn → Sn such that

Let X and Y denote normed spaces and T:D(T) ⊂ X → Y a linear transformation. It is shown that even in the case where both X and Y are incomplete, the quantity remains constant under both small and compact perturbation, provided that T is relatively open, R(T) is closed, and the perturbation is made in the right ‘direction’. If in addition and N(T) is topologically complemented, the topological complementation of the kernel is also preserved under small perturbations made in the right ‘direction’ and arbitrary compact perturbation. Various counter-examples are exhibited proving these results to be best possible.

We give a simple differential geometric description of the canonical Cartan (or tractor) bundle and connection in CR geometry, thus offering an alternative definition to the usual abstract Lie algebraic approach.

Theorem 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that

A generalization of the Neville–Aitken method is described which allows the construction of interpolating functions, other than polynomials, by means of simple recurrence relations. In particular, simple constructions are given for rational functions and trigonometric series which interpolate prescribed function values at non-equispaced positions of the independent variable.

Restrictions imposed by requiring the interpolating functions to be invariant under linear transformations of the coordinates are discussed, and application of the technique to the problem of inverse interpolation is also considered.

In recent years several authors have treated the fundamental problems of two-dimensional statical elasticity for isotropic and aeolotropic materials by the use of functions of a complex variable; references are given at the end of (7). In this paper Stevenson's notation (8,9) is adopted. Dawoud (2) has expressed the continuity conditions across a curve between two differently loaded regions in terms of the complex potentials and particular integrals for the two regions. A form of the boundary condition defining certain types of boundary constraint, including the rigidly clamped and hinged boundaries, has been introduced by the author and Dawoud (1). The introduction of this boundary condition is of practical importance, since neither rigidly clamped nor simply supported conditions can be realized fully under actual physical conditions and thus any case met in practice must lie somewhere between these two limiting cases.

Let E be a locally convex vector space, and let (At) be a sequence of subsets of E, which cover E. Dieudonné (3) has proved the two following results.

Let $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for $\mathcal{C}$ the class of finite p-groups.

We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups.

This paper is concerned with the notion of independence relating sets of elements in a ring A to a proper ideal a of A. A set of elements a1, …, an ∈A is called a-independent if every form in A[X1, …, Xn] vanishing at a1,…, an has all its coefficients in a. This notion leads to many questions (cf. (2) and (12)), which are of some interest in their own right, several of which are considered here. On the other hand, this independence is related to the structure of the graded ring associated to the ideal generated by the set of elements, hence is often relevant to some problems concerning regular sequences and complex.

The problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.

It is shown that a long-lived radioactive cobalt isotope, which had been previously obtained by Livingood and Seaborg by bombardment of cobalt with slow neutrons, can be produced by irradiation of nickel with fast neutrons.

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:

Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

We prove that the LS category of the symplectic group Sp(n) is bounded above by $(n+1 \choose 2)$. This is achieved by computing the number of critical levels of a height function.

We analyse the steady, fully developed, laminar flows of uniform, electrically conducting, incompressible fluids along arbitrarily shaped ducts of constant cross-section under the action of transverse magnetic fields to show that these flows are uniquely specified by certain parameters, namely: either the pressure drop along the duct per unit length or the volume flow rate and either the electric current leaving the walls of the duct at each point where the duct is connected to an external electrical circuit or the electric potential at these points. The most important, and novel, feature of the analysis is the proof that no secondary flow can exist in these fully developed flows. The analysis includes the special cases where the walls of the duct are non-conducting, and where the fluid is non-conducting (e.g. Poiseuille flow); in the latter case the uniqueness of the flow has not previously been demonstrated, at least in print.

Throughout this paper, let A be a Noetherian local ring with maximal ideal m and M a finitely generated A-module with d = dimAM ≥ 1. Denote by N the set of all positive integers.

Let x = (x1, …, xd) be a system of parameters (s.o.p) for M and let

We consider the following two problems: (i) When is the length of Koszul homology

a polynomial in n for all k = 0, …, d and n1; …, nd sufficiently large (n ≫ 0)?

(ii) Is the length of the generalized fraction in a polynomial in n for n ≫ 0?

A JB-algebra is a real Jordan algebra A which is also a Banach space and whose norm and multiplication satisfy the two following conditions

(i) ∥a2∥ = ∥a∥2,

(ii) ∥a2 − b2∥ ≤ max{∥a2∥, ∥b2∥},

for all elements a and b in A. A JB-algebra which is also a Banach dual space is called a JBW-algebra. The properties of JB-algebras and JBW-algebras can be found in (3), (4), (8) and (15).

The notion of a left (right, double) multiplier may be regarded as a generalization of the concept of a multiplier to a non-commutative Banach algebra. Each of these is a special case of a more general object, namely that of a quasi-multiplier. The idea of a quasi-multiplier was first introduced by Akemann and Pedersen in ([1], §4), where they consider the quasi-multipliers of a C*-algebra. One of the defects of quasi-multipliers is that, at least a priori, there does not appear to be a way of multiplying them together. The general theory of quasi-multipliers of a Banach algebra A with an approximate identity was developed by McKennon in [5], and in particular he showed that the quasi-multipliers of a considerable class of Banach algebras could be multiplied. McKennon also introduced a locally convex topology γ on the space QM(A) of quasi-multipliers of A and derived some of the elementary properties of (QM(A), γ).

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.

In his paper (2), Ripley has shown that some of the basic results of point process theory can be established without making any use of topological assumptions concerning the phase space. For the construction of distributions of point processes he needs pseudo-topological assumptions. In the present paper, another sufficient condition is given for the fact that a distribution of a random measure can be constructed by help of finite-dimensional distributions. This condition is expressed by purely measure-theoretic notions. Roughly speaking, it is supposed that concerning the existence of conditional probabilities the phase space behaves as if it were a Polish space. Measureable spaces with such a property are called ‘full measurable spaces’; they are identical with the measurable spaces of type (B) in (1).

An intrinsic pseudodistance dM is associated to the underlying causal structure of a Lorentzian manifold (M, g). An infinitesimal form for dM is defined on the bundle of null vectors over M and is used to prove that, under certain physically reasonable energy conditions, M is conformally hyperbolic, i.e. that dM is a true distance. As a corollary we obtain sufficient conditions for the conformal hyperbolicity of a globally hyperbolic space-time. Some exact solutions of the Einstein field equations are discussed and possible applications to the study of singularities are outlined.