1. Introduction. In (1) we proved that the direct sum of a finite number of unique factorization rings is a unique factorization ring (UFR), and in particular that the direct sum of a finite number of unique factorization domains (UFD's) is a UFR. The converse, however, does not hold i.e. not every UFR can be expressed as a direct sum of UFD's. Here we investigate the structure of UFR's and show that every UFR is a finite direct sum of UFD's and of special UFR's. There is thus a relationship with the structure theorem for principal ideal rings ((2), p. 245).

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequence

is exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphisms

one for each M ∈ ℳ, satisfying the following two conditions:

(i) Exactness. Given a commutative diagram with exact rows,

then tA(f) + tC(h) = tB(g).

In this note we prove a gloss on the usual s-cobordism theorem, which may be of interest, since it holds one dimension lower than the theorem itself. Our result is that every s-cobordism is its own left and right inverse in the groupoid of all h-cobordisms, studied by Stallings in (3).

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:

Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?

Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)

Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Following a program of Reiter, sets of synthesis in Banach algebras with certain Ditkin conditions are discussed. It is shown that the Ditkin Shilov lemma is in a general context best possible. Various methods for the construction of algebras are discussed. These are applied to show that very strong homogeneity conditions are needed to ensure that star sets are of synthesis for regular Ditkin algebras with Rn as carrier space.

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) by

and, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

1. Introduction. The purpose of this paper is to correct a number of points in an earlier paper (1) of the same title, and to prove a number of new results which supplement those given in that paper. The notation and definitions of this paper are those of (1).

1. Introduction. Let (X, C) be a Riesz space (or vector lattice) with positive cone C. A subset B of X is said to be solid if it follows from |x| ≤ |b| with b in B that x is in B (where |x| denotes the supremum of x and − x). The solid hull of B (absolute envelope of B in the terminology of Roberts (2)) is denoted to be the smallest solid set containing B, and is denoted by SB. A locally convex Hausdorff topology on (X, C) is called a locally solid topology if admits a neighbourhood-base of 0 consisting of solid and convex sets in X; and (X, C, ), where is a locally solid topology, is called a locally convex Riesz space.

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

The object of this note is the following theorem together with a table of the (1 + i)n division points for n = 1, 2, 3, 4, 5 on the elliptic curve y2 = x3 − Dx which has complex multiplication by i = √(−1).

1. Introduction. In a recent paper the author (2) has derived a number of integrals with the help of results ((5), p. 343)

where α, β, α′, β′ are parameters such that

and

whereK has the same meaning as above.

1. Introduction. The object of this paper is to obtain some generating functions for Appell functions with the help of fractional derivatives. The results are general in character and include as particular cases some of the results given earlier by Mexiner (3) and the author (4, 5, 6).

0. Our aim in this note is to obtain theorems about measures, perfect sets, and transforms in the group G of p-adic integers (for a prime p), related to recent developments in compact Abelian groups (6, 7) and again to Fourier analysis in Euclidean space (2–5).

1. Introduction. Suppose that we wish to make inferences about an unknown real parameter θ, and that we have a random sample of observations (Xl, X2, …, Xn) from a distribution which depends on θ.

1. Introduction. Various types of multiple-valued dynamical systems have appeared in the literature over the past twenty years (2). The motivations for such generalizations are varied, but among others, control theory plays a significant role. In a recent paper (2), Bushaw has introduced an abstract approach which consolidates a broad class of these dynamical systems. The model is a quasi-ordered set (E, ρ); i.e. Ε is a non-empty set and ρ is a reflexive and transitive relation on E. E represents the ‘event’ space of some real-world system and ρ represents some class of ‘admissible’ inputs on the system; i.e. (e1, e2) ∈ ρ if and only if there is an admissible input function which ‘shifts’ the system from event e1 to event e2. ρ is called the attainability relation on E and the statement ‘ an event e2 is attainable from an event e1 ’ means that (e1, e2) ∈ ρ. Adopting the terminology of (2), we shall refer to the quasi-ordered set (E, ρ) as a polysystem.

A method of deriving the basis functions of the double-valued representations of a point group by the reduction of Kronecker products is described. The method has been used to derive expressions for these bases for cubic point groups for which the results are tabulated.

The preferred-mode method of Roberts (11) is applied to the stability of liquid contained between coaxial cylinders, the inner of which is rotating. Accordingly the velocity field of a steady Taylor vortex is approximated by a truncated Fourier series and the steady state corresponding to a chosen wave number is obtained. The stability of a steady state so obtained is investigated:

(a) With respect to an axisymmetric perturbation associated with a Taylor vortex of a different axial wave number, so that a stable or ‘ preferred ’ steady state is determined. This is done for a narrow gap, η (the ratio of the radii of the cylinders) = 0·95, for Taylor numbers up to 40%, and for a wide gap, η = 0·5, up to 30% beyond the critical values.

(b) For a narrow gap, η = 0·95, with respect to a non-axisymmetric perturbation of a different axial wave number. The solutions obtained for the steady states are found to be stable to such perturbations for a range of Taylor numbers 40% beyond the critical value.

In (1) the velocity and magnetic fields were studied in the stagnation point region of a magnetized blunt body rotating with angular speed Ω. Some familiarity with this paper is assumed here: briefly, the nose section of the body was approximated by a disc of thickness t and conductivity σ′ and a perturbation solution was derived for small values of the diffusivity ratio ε (= ν/λ) and of the magnetic force coefficient N = σB2/4ρa. B is the uniform normal field component at the upperside (z = 0) of the disc, a is the strength of the external flow, p and σ are the density and conductivity of the fluid. The other two governing parameters are ω = Ω/a and β = σ′/σL, where L = (λ/a)½.

The paper deals with the free convection flow along a vertical plate moving arbitrarily in its own plane. The basic equations of the boundary-layer flow and heat transfer are linearized and the first two approximations are considered. The first approximation is the case of steady-state free convection flow while the second approximation is the response of the fluid velocity and temperature fields to the motion of the plate for which limiting solutions are obtained by the Laplace transform technique in two regions; namely, for large times and for small times. The particular case when the plate is given an impulsive start at t = 0 is investigated in detail. It is shown how the skin friction and the rate of heat transfer at the plate respond to the motion of the plate.

The problem of the propagation of time harmonic waves in an isotropic elastic half-space containing a submerged cylindrical cavity is solved analytically. Linear plane strain conditions are assumed. Using an expansion theorem proved in a previous paper (Gregory (3)), the elastic potentials are expanded in a series form which automatically satisfies the governing equations, the conditions of zero stress on the flat surface, and the radiation conditions at infinity. The conditions of prescribed normal and tangential stresses on the cavity walls are shown to lead to an infinite system of equations for the expansion coefficients. This system of equations is shown to be a regular L2-system of the second kind and from its unique l2-solution, the solution to the problem is constructed. The fundamental questions of existence and uniqueness are fully treated and methods are described for constructing the solution.

Three applications of the general theory are presented dealing respectively with the production, amplification and reflexion of Rayleigh waves.