We consider the motion of a particle of mass m and electrical charge e, moving in a constant magnetic field Bk, where k is a unit vector, and acted upon by a force mf(t). The position vector r(t) of this particle is governed by the differential equation where .

In a recent paper, Maher [2] proved that for any algebraic number field K of degree n and discriminant d there exists a constant C depending only on n and d such that for any ceiling λP of K there exists a basis α1 …, αn of the corresponding ideal αλ such that .

It is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.

In the development of the rotor calculus presented in a previous paper space-time was taken to be flat. This work is now extended to the case of curved space-times, which, in the first instance, is taken to be Riemannian. (The calculus bears at times a strong formal resemblance to the spinor analysis of Infeld and van der Waerden, but it is in fact developed quite independently of this.) Owing to the fact that all general relations had earlier already been written in a generally covariant form they may be taken over unchanged into the present context. In particular,

now serves as a defining relation for the connecting quantities τAkl. A linear rotor connection is introduced, and the covariant derivative of a rotor defined. The covariant constancy of the τAkl establishes the relation between the linear connections in w-space and in r-space. A rotor curvature tensor is considered alongside a number of other curvature objects. Next, conformal transformations are dealt with, of which duality rotations may be considered as a special case. This leads naturally to a gauge-covariant generalization of the whole calculus. A so-called rotor-derivative is defined, and some general relations involving such derivatives investigated. The relation of the rotor curvature tensor to the spin-curvature tensor is touched upon, after which the introduction of “geodesic frames” is considered. After this general theory some points concerning the Maxwell field are dealt with, which is followed by some work concerning basic quadratic and cubic invariants. Finally, a certain basic symmetric rotor is re-considered in the context of the classification of Weyl tensors, and the idea of a canonical representation suggested.

W. Feit [1], N. Itô [2] and M. Suzuki [3] have determined all doubly transitive groups with the property that only the identity fixes three symbols. It is of interest to the theory of projective planes to determine whether any of these groups contain a sharply doubly transitive subset (see Definition 1). It is found that if such a group G contains such a subset R then R is a normal subgroup of G, i.e. R is a doubly transitive normal subgroup of G in which only the identity fixes two symbols.

A polygon is said to be rational if all its sides and diagonals have rational lengths. I. J. Schoenberg has posed the interesting problem, “Can any polygon be approximated as closely as we like by a rational polygon?” Besicovitch [2] proved that right-angled triangles and parallelograms can be approximated in Schoenberg's sense, the proofs were improved by Daykin [5]. Mordell [7] proved that any quadrilateral can be approximated by a rational quadrilateral. By adapting Mordell's proof, Almering [1] generalised Mordell's result by showing that, if A, B, C are three distinct points with the distances AB, BC, CA all rational, then the set of points P for which PA, PB, PC are rational is everywhere dense in the plane that contains ABC. Daykin [4] extended the results of Besicovitch [3] and Mordell [7] by adding the requirement that the approximating quadrilaterals have rational area.

Let Mbe a σ-field of subsets of a space X. A partition of Xmeans a countable partition Π of Xinto sets belonging to M; the set of partitions is directed by refinement. A. Kolmogoroff in 1930 [1] discussed an integral (Moore-Smith limit as Π gets finer) for set-functions F defined on M. When it exists, IF is σ-additive, and if by chance F is already σ-additive, then IF = F

We say that a subgroup H is an n-th maximal subgroup of G if there exists a chain of subgroups G = G0 > G1 > … > Gn = H such that each Gi is a maximal subgroup of Gi-1, i = 1, 2, …, n. The purpose of this note is to classify all finite simple groups with the property that every third maximal subgroup is nilpotent.

In this paper we obtain the sum of some infinite series involving hypergeometric functions of one or more variables.

Let (X, B, m) be a measure space and let f(x) be a real-valued or complex-valued measurable function on X. A non-negative measurable function s(x) will be said to dominate f(x) provided |f(x)| ≦ s(x) for almost all x in X. The function s(x) will be said to dominate the sequence {f(x)}n∈N, N = {1, 2,…}, provided it dominates each fn(x) in the sequence. Unless otherwise specified, each integral will be over X with respect to m.

Let G be a finite group of odd order with an automorphism ω of order 2. The Feit-Thompson theorem implies that G is soluble and this is assumed throughout the paper. Let Gω denote the subgroup of G consisting of those elements fixed by ω. If F(G) denotes the Fitting subgroup of G then the upper Fitting series of G is defined by F1(G) = F(G) and Fr+1(G) = the inverse image in G of F(G/Fr(G)). G(r) denotes the rth derived group of G. The principal result of this paper may now be stated as follows: THEOREM 1. Let G be a group of odd order with an automorphism ω of order 2. Suppose that Gω is nilpotent, and that G(r)ω = 1. Then G(r) is nilpotent and G = F3(G).

A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

Journal of the Australian Mathematical Society 4 (1964), 452–453

The second paragraph should be deleted. The alleged commutator identity (3) is false and is certainly not due to Philip Hall. The correct form is

as Dr. N. D. Gupta of Canberra has pointed out to me. According to Professor B. H. Neumann, this identity appeared in his (Professor Neumann's) thesis.

Nevertheless the theorem is valid and the proof given is correct.