Relations are derived between the dose in r./min. delivered by the γ-ray beam from a betasynchrotron and the mean electron current which is incident on the internal target. Comparison with experiment shows that multiple passage of electrons through the target probably plays an important part in production of the γ-ray beam. Some possible effects connected with the multiple passage of electrons are discussed.

For each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

A new type of product of field operators is considered and the equations that it satisfies obtained.

Notation and results. The main results of this paper are (a) a new proof of the canonical Schoenflies theorem and (b) a non-compact version of the canonical Schoen-flies theorem.

The result (a) is already known(5); the existing proof is based on the proof by Brown of the ordinary Schoenflies theorem, whereas ours is based on that of Mazur and Morse.

In Feynman's ‘space-time’ formulation of quantum mechanics the Green's function for the Schrödinger equation is defined by an integral over all histories of the system. By integrating over one-parameter sets of functions, one gets the same Green's function as by integrating over a Fourier series, in simple cases. The method may be useful for estimating the result in cases when the integration over all histories cannot be performed exactly.

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.

It seems worthwhile to point out that a few of the theorems in (2) can, as indeed one would expect in a paper of such length and scope, be somewhat generalized or simplified. This note uses the notation of (2) and all references are to that paper.

Let N be a compact normal subgroup of a locally compact group G. One of our goals here is to determine when and how a given compactification Y of G/N can be realized as a quotient of the analogous compactification (ψ, X) of G by Nψ = ψ(N) ⊂ X; this is achieved in a number of cases for which we can establish that μNψ ⊂ Nψ μ for all μ ∈ X A question arises naturally, ‘Can the latter containment be proper?’ With an example, we give a positive answer to this question.

The group G is an extension of N by GN and can be identified algebraically with Nx GN when this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically with N x GN. When GN is discrete and X is the compactification of G coming from the left uniformly continuous functions, we are able to show that X is an extension of N by (GN)(X≅N x (G/N)) even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.

The automorphism group of a p-group acts naturally on the mod-p cohomology of the group. This action need not be faithful, but we show that it contains every simple module as a composition factor.

The double cusp is the real, compact, unimodal singularity

see (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T > 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T > 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equations

can be defined, following Perron ((12)), as

where ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in a previous paper by the author, of the fibrewise notion of closure for sublocales of locales over a base. We establish the implications which hold between these axioms and the traditional separation axioms for locales, and give counter-examples to show that some of these implications are irreversible.

The asymptotic values of the variances and covariances for the distribution of the transition numbers AA, AB, BA and BB for a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for the k-state chain under the restriction that the first and the last observations belong to the rth and the sth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the order nr (p11 – p21)n−r where r is the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for the k-state also. This is done by differentiating directly the determinantal characteristic equation.

We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a Λ-tree X into a group acting freely, without inversions, and transitively on a Λ-tree in such a way that X embeds into by means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an ℝ-tree X into (H) for some suitable group H, again in such a way that X embeds G-equivariantly into the ℝ-tree XH associated with (H). The group (H) referred to here belongs to a class of groups introduced and studied by the present authors in [3]. As a consequence of these two theorems, we find that -groups and their associated ℝ-trees are in fact universal for free ℝ-tree actions. Moreover, our first embedding theorem throws light on the question, arising from the results of [7], whether a group endowed with a Lyndon length function L can always be embedded in a length-preserving way into a group with a regular Lyndon length function; modulo an obvious necessary restriction we show that this is indeed the case if L is free.

This paper is devoted to the study of unions of ideals in commutative rings. The starting point is the prime avoidance lemma and an accompanying but diverse body of results on coverings of ideals by unions of ideals, which is described in Section 1. In Sections 4 and 5 these known facts, about finite and infinite unions, are combined and generalized, two distinct but overlapping cases emerging. All that is proved in Sections 4 and 5 turns on the crucial Lemma 2·1 in Section 2, which shows that a cover of an arbitrary ideal by cosets can be lifted to a cover of the entire ring. In Section 3 we introduce and define α-sieves, which provide a concise framework for the expression of applications. In Sections 6 to 9 various applications of Sections 4 and 5 are investigated.

We extend the notion of topological degree to a suitable class of continuous mappings of a locally convex Hausdorff topological vector space into itself which are not completely continuous. This extension and the resulting existence theorems for non-linear equations are further extended to product spaces. Our extension is developed with application to quantum field theory in mind; this application is made elsewhere.

A Petrie matrix (Kendall (1), (2)) is a matrix of O's and l's such that the l's in each column occur consecutively. An example follows.

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S•(V) and the exterior algebra ∧•(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S•(V) ⊗ ∧•(V), twisted by an arbitrary determinant power.

The following ‘partial’ integral equation arises in the quantum theory of fields:

where f(x1x2) = f(x2x1), K(x1x2, y) = K(x2x1, y), and a symmetric solution of the equation is desired. The solution of (1) will be obtained as a special case of the solution of the following equation:

The Peter-Weyl theorem states that for a compact topological group G, the set of finite dimensional (almost invariant) functions on G is uniformly dense in the set of continuous functions on G. In this paper, we consider the question: which subalgebras of C(G) inherit this property?