From measurement of tracks formed in an expansion chamber range distribution curves are obtained relative to 2134 α particles from thorium C′ and 729 α particles from radium C′. These curves exhibit marked differences in the proportion of short range particles which each contains. The origin of these differences is discussed. Estimates are made of the linear straggling coefficients relative to the absorption of the two groups of particles in air (in the experimental arrangement other media entered as well). These results very definitely confirm the conclusions reached by Briggs from measurements of the straggling of velocity of the particles. The sources of error which enter when the expansion chamber is used for the measurement of straggling coefficients are briefly reviewed.

If Q is a finitely generated abelian group, k a field, V a finitely generated kQ-module Bieri and Groves have proved that [otimes ]mV is a finitely generated as a kQ-module with diagonal Q-action if and only if ∧iV is finitely generated as a kQ-module for all 1[les ]i[les ]m. We generalize this result by showing that if the mth exterior power of V or the mth symmetric power of V is finitely generated as a kQ-module so is the mth tensor power of V. Further we show the equivalence between the finite generation of symmetric and tensor powers in the case when the ground ring is a PID of characteristic 0.

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.

It was remarked in (1) that many of the results valid for compact (this term is taken to imply Hausdorff) semigroups with jointly continuous multiplication also hold if the multiplication is only separately continuous. It is the purpose of this note to show that this is true of the structure theorem for compact simple semigroups. Any unsubstantiated facts about semigroups which we use may be found in the second section of (1) (the results given there do not depend on the assumption that a semigroup should contain an identity).

In a previous paper [6] the Bergman projection in D = A ∪ B, the union of domains A, B ⊂ CN, was described by alternating projections procedure. It was shown that when N = 1, and A, B are two halfplanes, the procedure can be carried out by explicit analytic calculations. In the present paper we show that the same is true for A = {|Re z| < π} and B = {Re z > 0}. In this case the calculations are more involved, and an essential use is made of L2-theorems of Paley–Wiener type due to T. Genchev, M. Dzrbasjan and W. Martirosjan. Of central importance is an operator W which maps f(z) onto . We show that for every g ∈ L2H(B) the sequence Wng (n = 1, 2, …) converges to zero locally uniformly in B.

The tensile deformation of single crystals of aluminium was examined by Taylor and Elam, using test-pieces composed of one crystal only. Miss Elam kindly supplied the author with a test-piece containing three large crystals, and an investigation of the effect of the constraint at the crystal boundary was carried out upon it as described below.

The theory of the evaporation of adsorbed atoms and ions of the alkali metals from a hot metal surface has been developed from somewhat different viewpoints by Langmuir and Kingdon and by Becker. These two treatments, which yield essentially the same results, are applicable only when equilibrium conditions obtain on the surface, so that the surface concentration is not changing. It is the purpose of the present note to extend Becker's treatment to take account of the rate of attainment of equilibrium on the surface, in order to lead to an interpretation of some experiments on the rate of evaporation of the alkali metals adsorbed on hot tungsten.

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).

This third and last part of the paper is concerned with the interpretation of the Schläfli symbol {k1, k2, …, km−1} when the k's are unrestricted. It is shown that, whenever the k's are integers (greater than 2), the symbol represents a polytope in a generalized space having time-like as well as space-like dimensions.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

1. The purpose of this paper is to examine in detail the types of radiation emitted by a quantum mechanical system (representing a nucleus) when the radiating particle changes its azimuthal quantum number l by two, one, or nought. We assume that the radiating particle is spinless* and that it moves in a field possessing central symmetry, so that the particle may be described by a wave function ψn,l,m which factorizes into radial and angular parts,

We assume secondly that the nucleus has a finite radius aN of the order of 10−12 cm., that is to say that π(r) = 0 for r > aN.

This note is intended to correct the proof of a lemma in the article ‘Hit polynomials and conjugation in the dual Steenrod algebra’, which appeared in 1998 in Volume 123 of these Proceedings.

It is shown that P, depending on x, u and derivatives of u, satisfies Div P ≥ 0 for all such x, u, if and only if Div p = Φ(x) ≥ 0 where Φ is independent of u. Applications to theories of continuum thermomechanics are discussed.

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose that

the series being boundedly convergent for 0 ≤ x ≤ 1.

Let p be an odd prime number, and let M2 be the extra-special p-group of order p5 and of exponent p2. For every p-group K, we denote by H*(K) the mod p cohomology of K. The purpose of this paper is to calculate the mod p cohomology groups of M2 and of Cp2* M2 (the central product of the cyclic group Cp2 of order p2 and M2).

The generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

Let M1 denote the set of Borel probability measures on the real line and M1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M1 and M1 is in fact a homeomorphism, provided M1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, if

where μn ∈ M1 (and so φn ∈ M1), then φn(λ)→φ0(λ) uniformly on all compact sets if and only if μn ⇒ μ0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φn(λ)→φ0 pointwise (for all λ), the conclusion that μn ⇒ μ0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative, even in a much more general setting.

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.