Let (S, ℳ) be a measurable space (that is, a set S in which is defined a σ-algebra ℳ of subsets) and X a locally convex space. A map M from ℳ to the family of all non-empty subsets of X is called a multimeasure iff for every sequence of disjoint sets An ɛ ℳ (n=1,2,… )with the series converges (in the sense of (6), p. 3) to M(A).

Let be any given infinite series with sn as its n-th partial sum.

We write

and

where

The problems associated with finding solutions of Laplace's equation subject to mixed boundary conditions have attracted much attention and, as a consequence, a variety of analytical techniques have been developed for the solution of such problems. Sneddon (1) has given a comprehensive account of these techniques. The object of this note is to draw attention to some simple orthogonal polynomial solutions to the most basic mixed boundary-value problems in two and threedimensional potential theory. These solutions have the advantage that most quantities of physical interest are easily evaluated in terms of known functions. Two-dimensional problems are considered in §2 and axially-symmetric three-dimensional problems in §3.

Consider a sufficiently smooth simple closed convex plane curve enclosing the origin, expanding linearly with time. The root mean square of the discrepancy (number of lattice points minus area) from time t = M to t = M + 1 is almost as small as the root mean square discrepancy from time t = 0 to t = M, so the discrepancy has no memory.

Sturm theory is extended to the equation

for 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditions

and

Oscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

The aim of the scientific teacher is to teach the pupil how to think along scientific lines. By a suitable presentation of the facts of experience he should lead the mind of the learner to form almost intuitively the scientific law or generalisation which embraces them all. We may of course start with the law or formula, and develope it mathematically into all its ramifications. But that reduces itself to mere analytical skill. If carried out faithfully in the elementary teaching of science, it would tend to give the learner an erroneous conception of the whole method of scientific investigation and the meaning of scientific law.

This note extends to categories Fountain's theorem (2) that for a perfect monoid S, every flat S-set is projective. (The converse is known (4).)

A correspondence is established between a class of coverings of an inverse semigroup S and a class of embeddings of S, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.

By the death of Dr Pinkerton on 22nd November 1930, at the comparatively early age of 60, we lost a fellow member who in times past had given varied and valuable service to our Society, and who, as Rector of the High School of Glasgow, occupied a distinguished position among Scottish teachers.

Peter Pinkerton was born in Kilmarnock on 8th June, 1870. He received his early education in the Academy there, and was Medallist in Classics and Mathematics. In 1886 he entered Glasgow University. In 1890 he graduated M.A. with First-class Honours in Mathematics and Natural Philosophy, and was awarded the Breadalbane and John Clark Scholarships in those subjects. For two years thereafter he attended the Royal College of Science, Dublin. He took first places in Mathematics, Mechanics, Physics and Chemistry, and second place in Botany.

Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker, viz

In this paper, we solve the following dual integral equations

where δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.

With every graph G (finite and undirected with no loops or multiple lines) there is associated a graph L(G), called the line-graph of G, whose points correspond in a one-to-one manner with the lines of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of Gare adjacent. This concept was originated by Whitney (3). In a similar way one can associate with G another graph which we call its total graph and denote by T(G). This new graph has the property that a one-to-one correspondence can be established between its points and the elements (the set of points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent (if both elements are points or both are lines) or theyare incident

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.