§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

Par un point fixe A d'une circonférence donnée on mène deux cordes AB et AC dont le produit a une valeur constante m2, puis on joint BC. Trouver 1° le lieu du pied D de la bissectrice de l'angle A du triangle ABC; 2° le lieu des centres des cercles inscrits et exinscrits à ce triangle.

In this paper we continue our study of the tensor product of distributive lattices which was begun in (2). We obtain a representation of the tensor product as a ring of sets and we describe a simple way to construct the tensor product of finite distributive lattices.

In the Proceedings of the London Mathematical Society, Ser. 2, Vol. 20 (1921), pp. 465–489, Professor H. W. Turnbull has studied the projective invariant theory of three quadrics. The following paper is based on this work and develops one definite section of the theory. From the geometrical point of view the linear complex is now seen to be fundamental in the study of three arbitrary quadrics; particularly when their (2, 2, 2) invariant φ123 vanishes.

This paper is an attempt to collect and arrange some of the propositions regarding the so-called Simson line, contained in various Mathematical Treatises and Journals. Proofs have been altered, or new ones substituted to suit the arrangement.

Maximal left ideals in matrix rings were studied by Stone [10]. Similar results are not necessarily valid in the general near-ring case and one of the objectives of this paper is to study these differences. Furthermore, although much is known about 2-primitivity in general matrix near-rings (Van der Walt [11]), quite the opposite is true for 0-primitivity and the other objective of this paper is to present some results on 0-primitivity in matrix near-rings in certain restricted cases.

Let S be a compact semigroup (with jointly continuous multiplication) and let P(S) denote the probability measures on S, i.e. the positive regular Borel measures on S with total mass one. Then P(S) is a compact semigroup with convolution multiplication and the weak* topology. Let II(P(S)) denote the set of primitive (or minimal) idempotents in P(S). Collins (2) and Pym (5) respectively have given complete descriptions of II(P(S)) when S is a group and when K(S), the kernel of S, is not a group. Choy (1) has given some characterizations of II(P(S)) for the general case. In this paper we present some detailed and intrinsic characterizations of II((P(S)) for various classes of compact semigroups that are not covered by the results of Collins and Pym.

On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma has

and points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integral

which was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.

Def. The arithmetic mean of n quantities a, b, c, d …

Def. The geometric mean of n quantities a, b, c, d …

The notation employed in the following pages is that recommended in a paper of mine on “The Triangle and its Six Scribed Circles”* printed in the first volume of the Proceedings of the Edinburgh Mathematical Society. It may be convenient to repeat all that is necessary for the present purpose.

R will denote a Dedekind domain and Pone of its prime ideals. A P-primary module will be an R-module all of whose non-zero elements have annihilators that are powers of the prime P. In all that follows E is such a module.

The height of 0 ≠ x ∈ E will be max{n : x ∈ PnE}. It is denoted by h(x). If this maximum does not exist we will say h(x)∞.

Clearly the condition is equivalent to E having no non-zero elements of infinite height. Adopting the terminology of (2, Ch. XI) where such modules over the ring of integers are studied, we will call these modules separable and reduced.

We use Priestley's duality to characterize, via their dual space, the distributive double p-algebras on which all congruences are principal.

In 1982, the first exotic ℝ4 was discovered—a smooth manifold homeomorphic to ℝ4, but not diffeomorphic to it. The object shocked topologists by its open defiance of the rules of high-dimensional smoothing theory. The exotic ℝ4 was constructed by connecting the two powerful machines of Freedman [4] and Donaldson [2] to earlier work of Casson [1].