If A, B are ideals of a commutative ring R, such that B ⊆ A, and, for some positive integer r, Ar = BAr−1 then B is said to be a reduction of A. (This concept was defined and developed by Northcott and Rees in (1).) In this paper, I shall consider commutative Noetherian rings with the property that no non-zero principal ideal is a reduction of an ideal properly containing it.

In this note we give a new, easy proof of the results on invertible powers of ideals that we established in (3). The deduction of the Dade–Taussky–Zassenhaus theorems (1) may be carried out as we did before in (3).

In (13) Wall classified up to diffeomorphism, PL-homeomorphism, topological homeomorphism, and homotopy equivalence all closed, oriented, 1-connected smooth (or PL) 6-manifolds with torsion-free homology and vanishing second Stiefel-Whitney class. This paper extends these classifications to all closed, oriented, 1-connected topological 6-manifolds with torsion-free homology. The method differs from that of (13) but uses two special cases of Wall's classification.

The aim of the paper is to exhibit the structure of products of flows that are suspensions of diffeomorphisms and show their relationship with suspensions of product diffeomorphisms.

The problem is important for applications of the following type. Suppose we have two dynamical systems each with a stable attractor. If the systems are brought into contact, as, for example, in the way a forced oscillation is imposed upon a natural oscillation in synchronization theory, then this is represented by taking the product flow to find the resulting behaviour of the joint system. However, the product of two stable attractors is no longer stable in general, and therefore it is important to analyse why, in order to understand what new stable attractors can arise from perturbations.

Measures, μ which can be realized as an infinite convolution

where each measure μn is a discrete measure, arise naturally in many parts of analysis and number theory (see (15)). The basic property of these measures is ‘purity’; i.e. such a measure μ 1must be absolutely continuous, continuous and singular, or discrete.

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ≤ ∞, was defined(1) as the limit of the sequence {fngn}, provided this sequence is regular on (a, b), where

for n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:

(1) ρ(x) = 0 for |x| ≥ 1,

(2) ρ(x) ≥ 0,

(3) ρ(x) = ρ(−x),

We denote by ω the space of all complex sequences with the topology given by the semi-norms

where δn(x) = xn. An FK-space, E, is a subspace of ω on which there exists a complete metrizable locally convex topology τ, such that the inclusion (E, τ) ⊂ ω is continuous; if τ is given by a single norm then E is a BK-space.

Let ℘ be a Weierstrass elliptic function satisfying the differential equation

and let ζ(z) be the associated Weierstrass ζ-function satisfying (z) = −℘(z). Corresponding to a pair of fundamental periods ω1, ω2 of ℘(z), there is a pair of quasi-periods η1, η2 of ζ(z) defined by

and we have ηi = 2ζ(ωi/2) for i = 1, 2.

Mills(7), Gordon(5), Birnbaum(1), and the author(6) have studied the ratio of the area of the standardized normal curve from x to ∞ and the ordinate at x. Des Raj (4) established the monotonic character of, and obtained lower and upper bounds for this ratio for the standardized type III population. This ratio, as shown by Cohen (2) and Des Raj (3), has to be calculated for several values of x when solving approximately the equations involved in the problem of estimating the para-meters of type III populations from truncated samples. When the areas and ordinates are small, either this ratio cannot be obtained from existing tables prepared by Salvosa(8) or that very few significant digits are available for its calculation. The object of this note is to obtain an upper bound which could satisfactorily locate this ratio over the range x > −1.

Consider a real-valued random walk

which is defined on a Markov chain {Xn: n ≥ 0} with countable state space I. We assume that a matrix Q(.) = (qij(.)) is given such that

This paper is concerned with the steady plane flow of a viscous fluid in symmetrical channels with slowly curving walls. The product of local channel half-width and local wall curvature is bounded by a small parameter ∈. We review the essentials of the formal approximation, in powers of ∈, proposed in (4) and (5); resolve a question, left open there, regarding the existence of the approximate series for the stream function for any value of the Reynolds number and to arbitrary order in ∈ and prove that, under certain restrictions on the Reynolds number and the divergence angle of the channel walls, this formal series is in fact a strict asymptotic expansion (for ∈ → 0) of an exact solution of the Navier–Stokes equations. As a result, the traditional picture of laminar separation, due to Prandtl, emerges as part of the steady flow field predicted by an exact solution that is known explicitly to arbitrary asymptotic order.

An initial-boundary-value problem for the equation

is considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.