Characterization and uniqueness results are given for uniform approximation which extend to suns previously known results for convex sets.

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.

Let X be a complex Banach space, and let and denote respectively the algebras of bounded and compact operators on X. The quotient algebra is called the Calkin algebra associated with X. It is known that both and are complex Banach algebras with unit e. For such unital Banach algebras B, set

and define the numerical range of x ∈ B as

x is said to be hermitian if W(x)⊆R. It is known that

Fact 1. ([4 vol. I, p. 46]) x is hermitian if and only if ‖eiαx‖ = (or ≦)1 for all α ∈ R, where ex is defined by

A congruence ρ on a semigroup is said to be idempotent-separating if each ρ-class contains at most one idempotent. For any idempotent e of a semigroup S the set eSe is a subsemigroup of S with identity e and group of units He, the maximal subgroup of S containing e. The purpose of the present note is to show that if S is a regular O-bisimple semigroup and e is a non-zero idempotent of 5 then there is a one-to-one correspondence between the idempotentseparating congruences on 5 and the subgroups N of He with the property that aN ⊆ Na for all right units a of eSe and Nb ⊆ bN for all left units b of eSe. Some special cases of this result are discussed and, in the final section, an application is made to the principal factors of the full transformation semigroup on a set X.

In this note we prove some cyclic inequalities which are generalisations of known results. We shall assume throughout that ai+n = ai ≧ 0 for all i, that no denominator in the statement of a result vanishes and finally that p, m and q are positive integers. We shall also use A(i, m) to denote with the convention that A(i, 0) = 0. The most interesting of our results is probably Theorem 2 since, in the special case p = 1, m = 2, r = 0, it gives a lower bound of ⅓n for the Shapiro sum . Although it is by no means best possible, see (2), our method implicitly gives a really simple way of obtaining this lower bound which, incidentally, is an improvement on Rankin's original result (5).

The theorem which I propose to establish first attracted my attention while I was turning over the pages of a volume of Cayley's Collected Mathematical Papers (Cayley, 1). The enunciation of the theorem (with no attempt towards a proof) had been published earlier by Kirkman (3) in a lengthy paper on combinatorial analysis (one of the three-score papers of which Kirkman was the author); among the topics discussed in this paper was the enumeration of the total number of different ways D(r, k) in which a (convex) polygon of r sides can be dissected into k+l parts by drawing k non-intersecting diagonals (i.e., no two diagonals may cross each other except at a vertex or outside the polygon).

When P is joined to four points A, B, C, D coplanar with P, a pencil of four lines is formed whose cross ratio is constant if ABCD are collinear. If A, B, C, D are not in a line the cross ratio P(ABCD) has a value which in general varies with the position of P, but which should be known when P is given in position and also A, B, C, D. A simple expression for the cross ratio is given and its utility in locus problems is illustrated by a variety of simple examples, which in several cases furnish methods for constructing a general cubic curve, with or without double point, a trinodal quartic, etc.

In reducing some experiments, I noticed that the logarithm of 237 is about 2.37 …. Hence it occurred to me to find in what cases the figures of a number and of its common logarithm are identical:—i.e., to solve the equation

where m is any positive integer.

The possibilities under rearrangement of terms in a complex series were discussed by Levy and by Steinitz. A reference the Steinitz paper as first disposing of the questions raised made by Bieberbach. It is the purpose of the present paper give an independent treatment by methods somewhat resembling those of the Levy paper.

The following method of establishing the existence and properties of the Focal Circles of a Circular Cubic is, as far as I know, new, and it has the advantage of dispensing, almost entirely, explicitly with analysis, while many of the properties can be proved without using the complicated method of generating the curve given in Salmon. The results which I have arrived at in connexion with the Nodal Cubic and Cuspidal Cubic are not given in Salmon.

This paper is an attempt (I) to deduce from first principles the number of conditions required to determine a plane polygon of n sides; (II) thence to deduce the numbers for special cases; and (III) to discuss the effects of a redundancy and a deficiency in the number of conditions. An investigation of this kind should form an important as well as interesting accompaniment to the ordinary study of elementary geometry.