In the first part of this paper there are found the numbers of points, lines, etc., in a finite projective geometry of n dimensions. The substance of this has already been worked out by O. Veblen and W. H. Bussey. The second part is concerned with the arrangements of the numbers representing the points in a finite projective plane desarguesian geometry.

The radius of convergence of the power series is 2π.

The function is regular within a circle whose centre is the origin of the z plane and radius 2π, and can be expanded in a Taylor's series converging at all points within the circle.

This note gives a proof of the result:

A necessary and sufficient condition that a trigonometrical series T (x) be the Fourier series of a function is that σn – σm = O(n-n) uniformly in [0, 2π] for all m≤n, where σn is the nth (C, l) mean of T (x).

This paper considers the radial and nontangential growth of a function f given by

where α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.

Composition algebras in which the subalgebra generated by any element has dimension at most two are classified over fields of characteristic ≠2,3. They include, besides the classical unital composition algebras, some closely related algebras and all the composition algebras with invariant quadratic norm.

When a given frequency distribution is to be graduated, it is customary to express the constants of the fitted curve in terms of the moments of the frequency distribution. The rth moment of the distribution, in which the relative frequency of a measure x is φ(x) or, in the case of a continuous variable, the differential of frequency is φ(x)dx, is defined in the respective cases by

the summation or integration being taken over the whole range of possible values of x. In the present paper we make use of another kind of moment, the factorial moment, which has already been considered by several writers, and which is specially suited to the case when the frequencies of the distribution are given for discrete, equidistant values of the variable. The (r + 1)th factorial moment, for the case where x, measured from some arbitrary origin, can increase by increments h, 2h, 3h,.…, will be defined to be

where the summation extends over all possible values of x; it will be denoted by m(r+1). By a suitable choice of scale the increments of x may be taken as equal to unity in any given case.

This note exhibits some old results in what is possibly a new form.

The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x by

where b is a positive constant, has the equation of motion

The solution, appropriate to the initial conditions at t = 0 is conveniently written

A problem of considerable interest in combinatorial analysis is that of determining the number of ways in which a connected figure can be constructed in the plane by assembling n regular hexagons in such a way that two hexagons abut on each other, if at all, along the whole of a common edge. Examples of these constructions can be seen in the various figures in this paper.

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined by

For example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.

In this paper we shall examine the relationship between the numerical range of aninner derivation, and that of its implementing element.

Much of this paper is taken from the author's doctoral thesis (5) written at theUniversity of Newcastle upon Tyne with the helpful guidance of Professor B. E.Johnson. The author also acknowledges the financial support of the Science ResearchCouncil during the period of this research.