A complete system of combinantal forms (generalised and ordinary) of a pencil of quadrics fλ ≡ λ1f1 + λ2f2 can be chosen such that the coefficients of the various power products of λ1, λ2 in the former give a complete irreducible system of concomitants of the two quadrics f1, f2, and conversely. This result was proved by Todd (1), who used it in conjunction with Schur function analysis (2) (3) to derive the complete irreducible system of concomitants of two quaternary quadratics (4).

Let R ⊊ T be an extension of commutative rings having the same identity. A. Wadsworth (10) studies the situation when R and T are integral domains, and all rings between R and T are Noetherian. In this case (R, T) is called a Noetherian pair. In a similar vein, E. Davis (4) studies normal pairs and I. Papick (8) shows when coherent pairs are Noetherian pairs.

Let K be a field of characteristic zero and let Δ ={δ1,…,δn} be a set of commuting K-derivations of the commutative Noetherian K-algebra R. Let S = R[X1,…,Xn] be the corresponding ring of differential operators, so [Xi, r] = Xir − rXi=δi(r and [Xi, Xj]=0, for 1≦i, j≦n. Let M be a maximal ideal of R with R/M of finite dimension over K. The purpose of this note is to describe the groups

A class of dual integral equations involving Bessel functions is solved by formal application of Mellin transforms.

Throughout this paper S will denote a given monoid and R a given ring with unity. A set A is a right S-system if there is a map φ:A × S→A satisfying

and

for any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem, an S-homomorphism and a congruence on an S-system. The reader ispresumed to be familiar with the basic definitions concerning right R-modules over R. As with S-systems we will refer to right R-modules just as R-modules.

Conditions for the existence of solutions of a class of elliptic problems with nonconvex constraints are given in the general framework of pseudo-monotone operators. Applications are considered in unilateral problems of free boundary type, yielding the solvability of a Reynold's lubrication model and of a biological population problem with nonlocal terms and global constraints.

Prof. W. P. Milne has cited (Proc. Edin. Math. Soc., Vol. XXXI., p. 90) pairs of nonagons inscribed in a plane cubic curve, and in nonuple perspective. A part from this, and the familiar case of triangles, little attention appears to have been given to polygons in multiple perspective.

In what follows the term C*-algebra will mean a complex C*-algebra with identity. We denote the identity element by 1. We shall also use the notation and terminology of Dixmier (3) without comment.

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then

(1)

This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

In a paper under the above title in Vol. XXIV. of the Proceedings it is shown that in a certain system of co-ordinates, the equation of the first degree represents a circle orthogonal to a fixed circle. It follows that any purely graphical theorem regarding right lines in a plane can be extended to orthogonals to a circle. This may be seen otherwise by projecting the figure of right lines on a sphere, the right lines thus becoming circles orthogonal to a circle on the sphere; and then inverting the sphere into the original plane. The geometrical method shows that the extension may also be applied to theorems involving one circle as well as right lines, the circle remaining unchanged, while the lines become orthogonals to a circle; the Pole and Polar Theorem, Pascal's and Brianchon's Theorems are examples. But plane figures involving more than one circle cannot in general be transformed in this way. We cannot, for instance, deduce the construction for a circle touching three great circles on a sphere from the known construction for a circle touching three lines in a plane; nor the Gergonne construction for circles on a sphere from the corresponding method in a plane.

The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.

Perhaps the simplest elementary proof of the prime number theorem, see Erdös (2) and Selberg (5), is Wright's modification (8), (3, p. 362) of Selberg's original proof (5). Another variant is due to V. Nevanlinna (4). Wright's proof uses Selberg's idea of smoothing the weighting process which occurs in the Selberg inequality, (1.2) below, by iterating this inequality. Here it will be shown that the proof requires less ingenuity if use is made of a further smoothing operation, namely first integrating the Selberg inequality itself. Integration has been used on a related inequality by Breusch (1 to obtain a remainder term. This method also makes proof by contradiction unnecessary.