The Formanek polynomial gives an explicit expression for the calculation of central elements in a prime ring with polynomial identity, in such a form as to provide a set of linear maps of the ring into its centre. This note makes use of properties of these maps with the object of giving a direct proof of the remarkable theorem of M. Artin on the relationship between Azumaya algebras and rings with polynomial identities. We show that such a central map can be chosen to have some of the crucial properties of the reduced trace in matrices, notably that of non-degeneracy, and our proof of the theorem rests on these properties. The method can also be used for the study of prime PI rings in general; this is illustrated by giving new proofs of theorems of Formanek and Cauchon. The changes of proof essentially amount to the elimination of the need for a change of rings to obtain a split extension in simple algebras. The main theorem is the following.

Let G be a simple adjoint algebraic group over an algebraically closed field K. We are concerned to describe the conjugacy classes of unipotent elements of G. G operates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements of G and the nilpotent elements of g which preserves the G-action, provided that the characteristic of K is either 0 or a ‘good prime’ for G. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjoint G-action.

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): Q → Q; q ↦ q.x and L(x): Q → Q; q ↦ x.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | x ∈ Q}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

Kaplansky (8, 9, 10) introduced the concept of an AW*-algebra and, by purely algebraic methods, succeeded in extending the ‘Type Theory’ of Murray and von Neumann from W*-algebras to these more general algebras. Every W*-algebra is an AW*-algebra but the converse is false, for there exist commutative AW*-algebras which are not W*-algebras.

The work of Coxeter, Longuet-Higgins and Miller (1953) and of Skilling (1975) is extended to give a complete list of uniform compounds of uniform polyhedra. Symmetry relationships between polyhedra and their compounds are discussed.

A polygon is said to be uniform if and only if it is connected and regular. An n-dimensional polytope (n > 2) is called uniform if it has uniform cells and possesses a symmetry group transitive on the vertices. A polytope is either connected or a compound. If it is a compound we also require the components to be isomorphic and to have no cells of dimension ≥ n − 1 in common. The connected 3-dimensional uniform polytopes have been investigated by Coxeter, Longuet-Higgins and Miller (1). The convex 4-dimensional polytopes have been investigated by Conway and Guy (2). Compound polytopes have been less extensively described than connected polytopes, but see, for example, (3), (4). Based on 2- and 3-dimensional experience, one might navely conjecture: ‘given a symmetry group , a uniform polytope P and an integer n, if a uniform compound polytope with symmetry group and consisting of n copies of P exists, then it is either unique or else a member of a one-parameter family’.

In this paper, k is a positive integer, k ≧ 2; K denotes 2k−1, and ε is an arbitrary positive number. X = (x1, …, xs) and U = (u1, …, ur) are integer vectors; is the distance from the real number y to the nearest integer.

H. Herrlich ((2), p. 193) defines, for a given nearness ξ, a property ξ̃ (which by analogy with the case of filters on a proximity space may be called compression) and remarks that ξ is determined by ξ̃. He continues: ‘Consequently, each of the axioms (Ni) can be translated into a condition concerning ξ̃, thus providing an axiomatization of the concept of collections of sets containing arbitrary small members.’ This last remark seems slightly misleading; it might reasonably be taken to mean that the reciprocity between ξ and ξ̃ is a mere set-theoretic tautology, independent of the nearness axioms themselves. (Compare, for example, the relation between the ideas of ‘closure-point of a set’ and ‘neighbourhood of a point’ in the axiomatics of topology.) This is not in fact the case; however, one cannot select from Herrlich's axioms a subset which is necessary and sufficient for ξ to be determined by ξ̃. Moreover, the relation between ξ and ξ̃ appears to be asymmetrical. We shall exhibit, in terms of stacks ((4), p. 36), an elegant set of mutually independent axioms, first for the more general ‘Čech nearness’ discussed by Naimpally in (3) and then for a Herrlich nearness (called in (3) a LO-nearness, with a compatible topology as there defined); these axioms make the reciprocity between nearness and compression, and its relationship with the axioms, explicit and obvious.

Following LeVan (6), we were led in (3) to investigate a purely categorical formulation of shape theory, which had hitherto been confined to certain rather circumscribed areas of topology. However, several authors (6, 7, 8, 9, 10, 11) had noted that, within this topological context, shape theory could be related to the process of expressing objects of the larger category T as filtered limits of objects of the given full subcategory P.

The object of this paper is to study stable operations on the K-cohomology of torsion-free spaces and spectra. Such a study was begun by Novikov ((4), § 8). We hope the present work may among other things serve to clarify some of the obscure points in this part of his work.

By an orthonormal system in a general complex Banach space, we mean a collection {eα: α ∈ } it vectors such that, for each α, there is an hermitian (in the numerical range sense, see (4)) projection Pα whose range is lin (eα) and such that PαPβ = 0, if α ≠ β. This paper is devoted to the study of orthonormal systems in general Banach spaces, and their applications to problems of characterizing isometries and hermitian operators.

0·0. Let G be an infinite compact abelian group with dual X. Parseval's identity shows that if f ∈ C(G) and ω ∈ l∞(X) then . Edwards has shown in (2) that L2(G) here cannot, in general, be replaced by any smaller Lp(G) space. Precisely: there exist f ∈ C(G) and ω: X → {± 1} such that . We strengthen this result by showing much more can be said about the summability of the Fourier series of f than . For example, when G is the circle group, f can be chosen to also satisfy

The functions introduced here and called darts, generalize this type of series condition.

The main theme of this paper is that under wide conditions the product of a large number of square non-negative matrices is close to a positive matrix of rank 1. (A non-negative matrix is one whose elements are non-negative real numbers.)

The aim of the present note is to point to a strange fact, through an example, in connexion with the asymptotic distribution of sums of a random number of independent and identically distributed (i.i.d.) random variables. We shall formulate our conclusion after the introduction of some notations.

That causality implies the Lorentz group was first proved by Zeeman (1). Later, Barruchi (2) and Teppati(3) proved a generalised version of the theorem by using completely different arguments. In this paper, we give a very simple and elementary geometric proof of the same theorem.

We consider in this note two problems which arise naturally in Smale's paper (3).

First of all we show that it is easy to extend Smale's results on the dynamics of LCR circuits with non-linear elements to circuits containing triode valves. It should be noted that transistors and other n-ports (1) can also be easily accommodated within this theory with suitable differentiability assumptions. We then further extend the theory to include elements whose voltage depends explicitly on time.

The initial value problem for the equation

has been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equation

where At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .

The paper continues earlier work on the generalization in n-dimensional space of the differential equations governing the propagation of electromagnetic waves in inhomogeneous anisotropic ionospheric plasma. Here there are investigated certain properties of the equations and their solutions that are independent of the dimension n of the space in which propagation takes place. The isotropic case is discussed, together with a special anisotropic model for which the various planes of circular polarization are geometrically related to each other.

A line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.