The theorem is—

The distance between the circumscribed and the inscribed centres of a triangle it a mean proportional between the circumscribed radius and its excess above the inscribed diameter.

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.

In Kottler's theoretical discussion1 of the diffraction of a plane wave of monochromatic light of wave-length 2π/k by a black halfplane, the function

where (r, θ, z) are cylindrical coordinates, plays an important part. In particular it is necessary to have asymptotic formulae for f (r, θ), valid when r is either very large or very small compared with the wave-length.

1. The subject-matter of this paper is in some sense known; but we will try to organise, explain and reprove it, and to give examples.

We study certain convexity-type properties of homogeneous functions on topological vector lattices, focusing on a concept of 0+-convexity, and using some probabilistic inequalities.

Following the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.

In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.

AMS 2000 Mathematics subject classification: Primary 20F17; 20D35

This paper deals with a few of the simpler specialisations of the intersections of a plane curve and the envelope of the family to which it belongs. It follows the method adopted by Professor Chrystal in dealing with the p-discriminant of a differential equation of the first order. This method is specially applicable to definite problems; in these it is safer to work out the result than to rely on theory.

The principal result of this paper is a characterisation of those commutative semigroups S which have the property that each character of each subsemigroup of S can be extended to a character of S. This work was partially inspired by the discovery that Theorem 5.65 of (1) is incorrect; it is related to that of Hill,who has obtained a different solution of the problem (2). Warne and Williams proved in (4) that any bounded character defined on an inverse subsemigroup of an inverse semigroup can be extended to the semigroup.

Einstein's fundamental equations of the gravitational field are

where Tμν are the components of the energy tensor and λ is the cosmical constant. In empty space these equations become

which may be reduced to

since G = 4λ, by contraction of (2).

The permanent of an m x n matrix A = (aij), m ≤ n, is defined by

where the summation is over all one-to-one functions σ from {1, … , m} to { 1, …, n}. In other words, the permanent of A is the sum of all the diagonal products of A, that is, all the products of m entries of A no two of which lie in the same row or in the same column. Thus the permanent of A may be evaluated by first multiplying all the row sums of A and then subtracting from the product all terms that contain as factors two or more entries from the same column of A. This is the idea behind the formulas of Binet and of Ryser.

This Graduation of the Circumference of a Circle is effected by the aid of an instrument called a trisector which I contrived with the view of trisecting an angle by its assistance, but subsequently perceived that it could help to divide an angle into 5 equal angles, and recently discovered that it could contribute towards dividing the circumference of a circle into 360 equal degrees or arcs.