For distributive double p-algebras, a close connection is established between being congruence coherent and congruence regular. Every congruence regular distributive double p-algebra is congruence coherent. Even though every congruence coherent distributive double p-algebra that has either a non-empty core or finite range is congruence regular, an example is given that is congruence coherent but not congruence regular.

In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

This is a note on the theory of continued fractions, in which the chief feature is the use made of the successive remainders or divisors which occur in the reduction of any given ratio to a continued fraction.

The treatment of the Pellian equation also differs from that which is generally given.

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, then

where E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

Let S be a nontrivial monoid with zero and let F be a field. A sufficient condition, on the 0-simple principal factors of S and the characteristic of F, is given for the contracted monoid algebra of S over F to be directly finite.

1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence

for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)

In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B

If G is a hyperfinite locally soluble group and A an artinian ZG-module then Zaĭcev proved that A has an f-decomposition. For G being a hyper-(cyclic or finite) locally soluble group, Z. Y. Duan has shown that any periodic artinian ZG-module A has an f-decomposition. Here we prove that: if G is a hyper-(cyclic or finite) group, then any artinian ZG-module A has an f-decomposition.

We obtain a formula relating the Conway potential functions of links in S3 which are connected by a framed surgery operation. Using this formula we extend the theory of Conway potential functions to links in all oriented ℚ-homology 3-spheres.

The transformation of Continued Fractions into one another is a subject in which very little work has so far been done. Beyond the simple transformations given in works on elementary algebra, few transformation-theorems are known ; the best known being those connected with the “contraction” and “extension” of Continued Fractions, and the transformations of Euler, Bauer and Muir.

When the curvature of a plane curve continuously increases or diminishes (as is the case with a logarithmic spiral, for instance) no two of its circles of curvature can intersect one another.

If G is a group and N a ring, the elements of the group ring NG can be thought of either as formal sums or as functions Φ:G→Nwith finite support. If N is a nearring, problems arise in trying to construct a group near-ring either way. In the first case, Meldrum [7] was abl to exploit properties of distributively generated near-rings (N, S) to build free (N,S)-products and hence a near-ring analogue of a group ring. For the latter case, Heatherly and Ligh [3] observed that the set of functions could be made into a near-ring under multiplication given by provided N satisfies

for all ai,bin∈N and k∈Z+. Such near-rings are called pseudo-distributive. In fact these are precisely the conditions under which the set Nk of k x k matrices over N is also a near-ring and then both NG and Nk are pseudo-distributive.

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functions

fulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.