Recently Wang (5) has given some results concerning characterizations of Poisson, normal, gamma, binomial and negative binomial distributions. We shall show here that these results are incorrect and also attempt to rectify them.

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

For any group G and x ∈ G, and any n ∈ ℕ (the natural numbers) let

i.e. the set of all nth roots of x in G.

In this article we extend the study of ‘plumbing’ initiated in [10] to the case of fibred even-dimensional knots. Plumbing is a geometric operation on the fibre-surfaces of two fibred knots of the same dimension that produces another such knot.

Previous work on the existence and period of radium C′ is discussed with reference to an experiment of Jacobsen which provides evidence that a γ ray transformation of period comparable with that of radium C′ precedes the expulsion of α particles. It is shown that, from Jacobsen's results, part of the γ radiation from a source of recoil atoms should originate in the space surrounding the source.

A rough calculation is made which shows that the γ rays above the source should be detectable by ordinary methods, and a description is given of an ionisation method capable of detecting the effect. The γ rays predicted by Jacobsen's experiment were tested for by using specially prepared sources of radium C. Phenomena associated with α recoil were also investigated for sources of radium (B + C) and thorium (B + C).

No evidence of a γ ray emission from the space above any of the sources was obtained. The negative result indicates that the interval between the departure of the disintegration particle and the emission of the γ ray quantum is considerably less than 10−5 second.

The absorption curves and the photo-ionisation experiments agree to give a value for the molecular photo-electric threshold of λ 2555 ± 20 a.u. If our view of the ionisation process is correct, the energy of dissociation is 0·505 ∓ ·01 volts. If the picture of the ionisation process is wrong, the energy of dissociation may be a little lower (a few hundredths of a volt). The corresponding heat of dissociation is 11400 calories. In the range of temperature (200° C. to 500° C.) generally used in experiments on potassium vapour, the fraction associated varies from 10−4 to 10−2. An equation for calculating the fraction associated at any temperature is given by the writer in an earlier paper.

It has been observed by various workers that, in addition to particles ejected from the various atomic levels of the atoms concerned, some are ejected by rays which correspond to the natural K X-radiation of those atoms. It has also been shown that some radioactive substances emit β particles of comparatively low energy. Therefore it was thought advisable to make an attempt to see whether this “soft” radiation has a line spectrum; for one would expect to find traces of the natural L-radiation, just as one finds evidence of the K-radiation, provided that the effect is sufficiently intense.

An attempt is made in this paper to find the nature of terms that occur in the arc spectrum of antimony. The spectrum is analogous to those of the singly ionized elements of the sixth group. Prof. A. Fowler investigated the spectrum of ionized oxygen (O II). The classification was shown to be in exact accord with the “Hund-Heisenberg theory of complicated spectra” by R. H. Fowler and D. R. Hartree. Recently G. R. Toshniwal has classified the arc spectrum of bismuth.

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution function G, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.

In this paper we study the faithful irreducible representations of certain families of wreath products and show how a particular class of such representations is related to the automorphisms of these groups. Our methods enable us to calculate the orders of the automorphism groups, although they do not deal with their structure. The wreath products that we consider here include most of the Sylow p-subgroups of the classical groups, which are described in (1) and (4).

Throughout this brief note V denotes a fixed, smooth, real-valued function on such that there exists a bounded, open, strictly convex set C with (the boundary of C), V > 0 in C, V = 0 on and V′(0) = 0.

It is shown that, assuming conservation of energy and momentum in individuai transitions, radiation must be regarded as propagated by linearly directed quanta. Evidence is brought forward in favour of the view that these quanta are spatially localised longitudinally as well as laterally. As these conclusions seem unavoidable, an attempt is made to see if there is any possibility of accounting for interference effects in terms of quants possessing periodic properties.

The idea that a single quantum can produce interference effects is untenable. The conception of the phase of a quant is introduced, and it is supposed that atoms may absorb a quant in a given phase provided they are in a “state” corresponding to that phase. Considering radiation emanating from a point, it is shown that a tentative explanation of interference may be obtained if the quants have a constant phase at emission, and if they give atoms at any point an impulse towards the state corresponding to their phase at the point. According as these impulses reinforce or oppose each other, bright or dark parts of a fringe System are obtained. Practically the whole nomenclature and method of the wave theory may be taken over and applied to the quant phase field surrounding an emitting source. Essentially, on this view, an explanation of interference is sought in terms of different quants, and the dark and bright parte of a fringe system are to be explained, not in terms of different numbers of quante arriving at them, but in different phase relations of quante which do arrive.

Quante are characterised by a magnetic and electric vector, whose magnitudes and directions define the state of polarisation and the phase. Electron orbite may also be regarded as characterised by two vectors in terms of which it seems as if the “state” may be specified.

The relation between transition character and type of radiation emitted is finally considered and it is shown that the evidence so far available renders plausible an explanation of interference of the kind put forward. Much further experimental work, particularly in connection with the Stark and Zeeman effects, is, however, necessary, before sufficient precision can be given to the views put forward to make them satisfying. On the other hand, it seems premature to suppose that only a formal treatment is possible of the physical phenomena involved.

In the course of a long series of experiments on the scattering of positive rays I have had occasion to take some hundreds of photographs in which positive rays diverging from an approximately linear source pass through a metal slit about ·5 mm. wide parallel to the source and then strike the sensitive surface of the plate. The change caused by the actual impact of the rays can be developed like the ordinary latent image produced by light. The pattern thus formed on the negative is a rectangular strip about 1·8 mm. × 20 mm. answering in shape to the shape of the slit, and bordered by a much less intense edge (of the order 10%) due to the scattered rays whose investigation was the object of the experiment. It was found that, in a majority of the photographs, the rectangular strip, which should have been approximately uniform in blackness, at least across, had its longer edges apparently much blacker than its centre. So marked is this effect that it was supposed to be due to some cause affecting the distribution of the rays. However, careful measurements with a microphotometer showed that the effect was unreal, the edges always being less black than the centre. The effect of the positive rays is thus merely to produce a certain distribution of blackening on the plate, and the increased blackness seen at the edges is purely subjective.

1. Simple linear clustering. One of the simplest clustering problems is the following: if n points are taken independently and uniformly at random on the interval I = [0, L], what is the probability Q = Q(n, a, L) that no two points are closer than a? The well-known answer

if (n − 1) a ≤ L, and Q = 0 otherwise, can be obtained in a variety of ways. Perhaps the simplest one is the original method of E.C.Molina(5). If x1,…,xn are the abscissas of the n points, then there are n! equiprobable orderings of the xi's consistent with the condition that no two points are closer than a. The probability of each one is

Let yi = xi − (i − 1) a for i = 1,…,n, then the above becomes

which is

A recently discovered abstract relation arising in multiparameter spectral theory is used to unify some of the integral relations and equations associated with the ellipsoidal wave equation.

A few years ago, the authors considered briefly Banach algebras whose duals could be identified ‘naturally’ with their multiplier algebras [17]. In this context, naturalness can be interpreted as meaning that, for each element b of the algebra B and each pair of elements u, v of the dual B′,

where 〈, 〉 denotes the dual pairing and the products are of elements of B′ regarded as left or right multipliers on B. In the present paper we return to the same circle of ideas but begin with a more general situation. We assume only that the algebra B is injectively embedded in its algebra of left, and in its algebra of right, multipliers and that its dual B′ can be injectively embedded in the algebra M(B) of double multipliers on B (definition below) in such a way that the above relation holds. From these assumptions we shall prove that there is a normed algebra A such that M(B) is the dual of A and is the algebra of continuous left multipliers on A (or, equally, right multipliers).

In [1] Marc Feighn proves the following result: a proper, C2, codimension-1 immersion in a manifold with no first homology separates the ambient space. In [4] Michelangelo Vaccaro proves a related result: the C1-immersed image of a compact n-manifold with image a (curved) polyhedron has non-zero Hn with ℤ2 coefficients. (In Vaccaro's terminology f(M) is a curved polyhedron if f is smooth and f(M) is the (non-PL) embedded image of a simplicial complex.) Using ideas similar to Feighn's we prove here the following result.

According to the views which we expressed recently (Proc. Roy. Soc. 1929, 124 A, 322) the solid condensed, liquid condensed, expanded and vaporous types of unimolecular films on the surface of aqueous solutions are the two-dimensional analogues of the solid, liquid crystal or smectic, liquid and vapour states of bulk matter. The previously-unsuspected observation that the films show triple point phenomena supports these conclusions.

T. Hori has recently carried out a careful study of the ultra-violet bands of the hydrogen molecule and deduced numerical values for the energy levels involved in their production. Mr R. H. Fowler pointed out to the writer the desirability of re-calculating the specific heat of H2 by using Hori's empirical values of the energy. The present note gives the result of the calculation, which is of course similar to the investigation of Kemble and van Vleck.