1. Introduction. The wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p (Baumslag (1)). When A is an Abelian p-group of exponent pk, and B is the direct product of cyclic groups of orders pβ1, …, pβn and β1 ≥ β2 ≥ …, ≥ βn, then Liebeck has shown that the nilpotency class c of A wr B is

Fröhlich(1) has obtained certain invariants for modules over commutative separable orders over dedekind domains, whose values are ideals in the dedekind domain. In two particular applications he has shown that these invariants supply a criterion for a module over a given order to be projective ((1), Theorem 4), and another one for a fractional ideal to be invertible ((1), Theorem 5).

1. Introduction. Let A[λ] be the irreducible invariant matrix of a general matrix of order n × n, corresponding to a partition (λ) = (λ1, λ2, …, λr) of some integer m. The problem to be discussed here is that of determining the canonical form of A[λ] when that of A is known.

We introduce and compare four procedures for defining the order-continuity of a function from one topological ordered space into another, where each reduces to the usual conception when the orderings of the two spaces are trivial. Chiefly for the purposes of this comparison, we use the idea of an ‘order-connected’ space, and in the course of investigating under which types of order-continuous functions this property is preserved, we are helped in assessing their relative importance.

1. Introduction. Let Ω be a non-singular quadric in an n-dimensional projective space PG(n, K) whose coordinate field is K. With respect to Ω the linear subspaces of PG(n, K) fall into various types: the subspaces of a given type each have the same dimension and the same geometrical kind of quadric section with Ω. Each element of the collineation group Γ preserving Ω takes a subspace into one of the same type, but Γ may divide the subspaces of a given type into several transitivity classes or orbits.

0. Introduction. An immersion f of one space X in another Y is a continuous map which is locally an embedding; that is, for any x ∈ X there is a neighbourhood N(x) of x such that f|N(x) is an embedding. Thus the image of an immersion may intersect itself, but it can contain no worse singularities. For example, the well-known model of the Klein Bottle is the image of an immersion into ordinary 3-space; the locus of points of self intersection is a circle, where two distinct circles in the Klein Bottle have their images. This paper obtains a sufficient condition for a map of PL manifolds to be deformable into a PL immersion.

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

1. Let f(t) be a perodic function with period 2π and integrable in the sense of Lebesgue over (–π, π), then its Fourier series is

Introduction. We will study here for normal T1 spaces the space of cuts—defined and analysed by Hurewicz (3) for separable metrizable spaces.

1. Introduction. This note is a sequel to (1), the notation of which will be used without further explanation. In (2), certain deductions from the theorems proved in (1) were made in the case where K(x, t) is derived from a function K(x), a.c. in [0, 1], by the equation

In (2) the author determined a formula for computing, by means of the Eilenberg-Moore spectral sequence, the action of the group in a principal G-bundle, under the hypothesis that H*(G) be universally transgressively generated. This rather severe restriction was due to the method of proof, which used the theory of Guy Hirsch. Since Hirsch's work can be derived as a special case of Brown's twisted tensor products (3), one might hope to obtain the formula for the group action in the general case from the latter theory, and this we shall do (Theorem 1·2 below).

1. Introduction. In his papers (2), (3) and (4), and in his book (5), Husain studied the locally convex, Hausdorff, linear topological spaces (hereafter abbreviated to l.c. spaces) which he called B() spaces and Br() spaces and which satisfy weakened forms of the B completeness and Br completeness conditions of Ptak (8). L.c. spaces satisfying another weakening of the Br completeness condition were studied by Persson in (7) and were called by him, weakly t-polar. We consider some examples in connexion with these spaces.

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

1. Introduction. In the study of diophantine equations in two variables, it is often necessary to consider rational functions on a curve with prescribed zeros and poles. Although it is well known that such functions can, in principle, always be effectively constructed, the detailed proof does not appear to have been given. The purpose of the present paper is to give the complete proof of such a construction. Our method, and the statement of our results, are motivated by the applications to diophantine equations which we have in mind. In particular, our results will play an important role in a subsequent paper (1), in which explicit bounds will be established for the integer points on any curve of genus 1.

In a paper (1) published in 1922 in these Proceedings, I conjectured that a curve of genus greater than one contained only a finite number of rational points, but this has never been proved. A particular case is given by the equation of genus three,

and so any results for such equations may be of interest.

1. Introduction: The well-known result that every function f(x) ∈ L2(–∞, ∞) can be expanded in L2(– ∞,∞) by where and was recently followed by an expansion theorem for generalized functions. In papers by Wildlund(8), Giertz(4) and Zemanian(9) it was shown that a tempered distribution f∈S′ can be expanded by Hermite functions that is

for all ψ∈S (the space of rapidly decreasing test functions).

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

1. Introduction: Consider the system of ordinary differential equations

where the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

1. Introduction: In what follows, X will be a real inner-product linear space (inner product (x, y), norm |x| = (x, x)½); En will be n-dimensional real Euclidean space. We shall be dealing with sets {as: 1 ≤ s ≤ rk; r ≥ 1, k ≥ 2} of elements of X; we write a for the vector (a1; …, ark), and put . We suppose that |as| ≤ ∈ (< 0) for every s.

1. Introduction. A Markov Renewal Process (MRP) with m(<∞) states is one which records at each time t, the number of times a system visits each of the m states up to time t, if the system moves from state to state according to a Markov chain with transition probability matrix P0 = [pij] and if the time required for each successive move is a random variable whose distribution function (d.f.) depends on the two states between which the move is made. Thus, if the system moves from state i to state j, the holding time in the state i has Fij(x) as its d.f. (i, j = 1,2, …, m).