In an earlier publication [1] we introduced the notion of a Baer assembly and applied it to obtain a coördinatisation theory for semilattices. This was achieved by considering the semigroup of quasi-residuated (i.e. ℴ-preserving and isotone) mappings on a bounded semilattice. In the present paper we consider the semigroup of quasi-residuated ∪-homomorphisms (or hemimorphisms) on a bounded lattice and thus show how a particular type of one-sided Baer assembly can be used to provide a coördinatisation theory for lattices; and in particular for complemented, modular and distributive lattices.

3-Methylfluorene-9-propionic acid (1) with hydrofluoric acid undergoes ring-closure on the substituted ring to give 1,2,3,10b-tetrahydro-5-methylfluoranthen-3-one (II).

Wolff-Kishner reduction of the ketone yielded l,2,3,10b-tetrahydro-5-methylfluoranthene which on dehydrogenation gave 2-methylfluoranthene (III, R=H) identical with a sample prepared according to the method of Tucker (1952) and differing from 8-methylfluoranthene. This proved that ring-closure of (I) had occurred as expected on the methyl-bearing benzene ring. In this instance ring-closure occurs in the position meta to the methyl group and is reminiscent of the similar ring-closure of 2-phenyl-2-p-tolylpropionic acid to give 6-methyl-3-phenylindanone (Pfeiffer and Roos 1941). It thus provided a further example of the limitations of von Braun's statement that Friedel-Crafts ring-closure occurs much less readily at the position meta to a methyl group than on a phenyl ring (von Braun, Manz and Reinsch 1928).

In this paper it is shown that the question of the existence of a classical solution of the first initial-boundary value problem for a non-linear parabolic equation may be reduced to the problem of the derivation of suitable a priori bounds.

This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) and

The extension consists in replacing f′ by M[f] where

choosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the form

where K is a positive number independent of f.

The analysis involves a fourth-order differential equation and the second-order equation associated with M.

A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

A previous discussion (Feather 1969a) of the energetics of the separation of an α-particle in the ternary fission of an (e, e) nucleus is generalised to cover the case of a light particle of arbitrary mass and charge separating from a fissioning nucleus the nucleon-parity type of which may be any one of the four possible types. A ‘profitability index’ is defined in terms of which certain broad differences in the light-particle spectra from different types of fissioning nucleus are predicted, and the results of detailed calculations of the minimum energy expense of separation are given for various light particles emitted in the fission of and, for comparison, for an α-particle emitted in the fission of , an (o, o) nucleus.

In a previous paper (Jack 1970, § 5) the pre-zonal polynomials were defined by the surface integral (1) below. In that paper they were evaluated using a series of differential type operators acting on the sums of powers of the roots of a matrix.

The present paper, although it is a continuation of Jack (1970), is self-contained and consists of three parts. The first part, §§ 3–7, shows how the pre-zonal polynomials may be calculated, certainly for low degree, using only the first of the differential operators. The next part, §§ 8–11, is devoted to establishing various recurrence relations between the pre-zonal polynomials and to conjecturing similar recurrence relations between the zonal polynomials (James 1964, p. 478). In the final part, §§ 12–14 the pre-zonal polynomials are evaluated directly from the integral (1) below, in terms of the augmented monomial symmetric functions.