Assume is a holomorphic map fixing 0 with derivative λ, where 0 < |λ| ≤ 1. If λ is not a root of unity, there is a formal power series φf(z) = z + (z2) such that φf(λ z) = f(φf(z)). This power series is unique and we denote by Rconv(f) ∈ [0,+∞] its radius of convergence. We denote by Rgeom(f) the largest radius r ∈ [0, Rconv(f)] such that φf(D(0,r)) ⊂ U. In this paper, we present new elementary techniques for studying the maps f ↦ Rconv(f) and f ↦ Rgeom(f). Contrary to previous approaches, our techniques do not involve studying the arithmetical properties of rotation numbers.

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.

This paper is a continuation of (3), in which we proved extensions, to generalized Stieltjes transforms, of some inversion theorems of Widder(l) for ordinary Stieltjes transforms. The numbering of formulae, lemmas and theorems continues on consecutively from that in (3). The last formula in (3) is numbered (32), and the last lemma or theorem is Theorem 18.

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

The disintegrations by which Ac B passes into the inactive AcPb are accompanied by a γ-radiation very weak compared with the intense γ-emission in the case of Ra or Th-active deposit. The analysis of the secondary β-ray spectrum of actinium-active deposit has revealed the existence of at least five γ-rays (1) (see Table I), of which the ray with energy 0·349 × 106 e.V. definitely belongs to the disintegration Ac C—C″ and is associated with the fine-structure of α-particles of Ac C. According to the measurements of Surugue the two rays of 0·4038 × 106 and 0·4257 × 106 e.V. energy are to be attributed to the disintegration AcB—C, whereas the origin of the 0·829 × 106 e.V. ray is less definite. The fit with experiments is best if this ray is assumed to be emitted from Ac B—C; but it may also (within experimental error) be attributed to the disintegration Ac C″—Pb. Experiments on the absorption coefficient of the γ-radiation of RaAc and its disintegration products prove that the 0·829 × 106 e.V. ray is the hardest γ-ray emitted by the active deposit of actinium(2). The value for the absorption coefficient between 4·6 and 10·6 cm. of lead was found to be μ/ρ = 0·76, in good agreement with the value found in previous experiments at smaller absorptions in aluminium (3). There is, on the whole, excellent agreement concerning the energies of the γ-components as measured according to different methods by different observers.

Conflict sets are loci of intersecting wave fronts emanating from $l$ different manifolds. We show that for generic manifolds their conflict sets are projections of smooth conic Lagrangian manifolds. Thus conflict sets locally admit the structure of wave fronts. Simple stable singularities for this problem in $\mathbb{R}^n$ occur when $0\leq n-l \leq 4$. Relations with centre sets, billiards, pedals and orthotomics are also discussed. Throughout canonical relations are used as an essential tool to carry out these geometrical constructions.

The irregular radial Coulomb wave function in repulsive fields is expressed, for any angular momentum value, in series of Bessel functions of arbitrary order. Previous expansions, obtained for angular momenta zero and one, can now be obtained as special cases. The expansion can be most conveniently used for high-energy particles.

Let E be a Banach function lattice such that L∞[0, 1] [rarrhk ] E [rarrhk ] L1[0, 1]. We characterize the strict singularity of the embedding L∞[0, 1] [rarrhk ] E and the strict cosingularity of E [rarrhk ] L1[0, 1] in terms of functionals defined by using characteristic functions.

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.

In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

The knots and links which can arise as the closure of 3-string braids, and their relations to the braids which give rise to them have been studied by Murasugi [5] and others, including Hartley [2] and more recently Przytycki[6]. Three-braids appear to form a rather special class among braids from some points of view [3]; they are also the only group of braids for which Burau's representation is known to be faithful [1]. They are, however, varied enough to provide an interesting range of knots and links on which to test a number of conjectures.

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

The classification of all finite primitive distance-transitive graphs is basically divided into two cases. In the one case, known as the almost simple case, we have an almost simple group acting primitively as a group of automorphisms on the graph. In the other case, known as the affine case, the vertices of the graph can be identified with the vectors of a finite-dimensional vector space over some finite field. In this case the automorphism group G of the graph Γ contains a normal p-subgroup N which is elementary Abelian and acts regularly on the set of vertices of Γ. Let G0 be the subgroup of G that stabilizes a vertex. Identifying the vertices of Γ with G0-cosets in G, one obtains a vector space V on which N acts as a group of translations, G0, stabilizes 0 and, as Γ is primitive, G0 acts irreducibly on V.

The aim of this note is to construct generators for II*(MU). Generators of a similar type were constructed by various authors; Hazewinkel (2), Kozma (3) and Liulevicius (4). The construction in (4) is in fact for II*(BP) at the prime 2.

This paper presents the asymptotic expansion, with respect to the large real parameter λ, of the singular Fourier integral

formula here

where 0 < b [les ] + ∞, the symbol fp designates an integration in the finite part sense of Hadamard and complex function K(x, u) belongs to a specific set of pseudo-functions which may present a singular behaviour of logarithmic nature at the endpoints.

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

Throughout this paper we let A = (aij) be a non-zero n × n matrix-we study real matrices only–with row sums R1,…Rn and eigenvalues λ1,…,λn, ordered λ1≥…≥λn if they are real. We denote E = (1,…,1)T and su A = ΣiΣjaij = ETAE. If A is symmetric, a simple property of the Rayleigh quotient is that

satisfies

and

Wilson's cloud method, provides a means of measuring the ranges of β-rays which is free from any uncertainties arising from the scattering of the rays, and for this reason it is an invaluable method for obtaining the true rate of loss of energy of β-rays as they traverse matter. Thetrue rate of loss of energy or ‘stopping power’ is defined as the quantity , where is the mean energy lost by β-rays of energy T in travelling a distance dx measured along their paths. The cloud method of course enables one to observe only the total ranges of β-particles, and in deducing the stopping power from the mean range it is necessary to bear in mind that owing to ‘straggling,’ which in the case of β-rays is large, the differential coefficient , where is the mean range of particles of energy T, is not exactly the same as the stopping power . is however the quantity which the cloud method enables us to observe. It is the purpose of this note to consider the exact relation between these two quantities so that the information provided by the cloud method may be rigorously interpreted.

In the light of the hypothesis that the helium atom can exist in a hydrogen-like form in which one electron is relatively far removed from the nucleus with respect to the other, a series of experiments have been carried out with a view to determine whether or not helium could be obtained in the form of a compound similar to certain known hydrides. The mass of evidence points to a positive result. A gaseous compound was shown to form when excited helium was passed over a strong source of the active deposit of radium. Compound formation was proved by counting scintillations caused by the transportation of the radioactive material from the discharge tube into a bulb containing a zinc sulphide screen. That the effect observed in the bulb was due to the presence of a gaseous radioactive helide and not to radium emanation or solid particles in suspension was definitely proved; that it was due to hydrogen impurities in the helium was shown to be very unlikely. The compound in hydrogen was more efficiently condensed by a carbon dioxide-ether cooling mixture than that in helium. The catalytic effects of tap grease and mercury vapour on the reaction were noted.

In conclusion the writer wishes to thank Professor Sir Ernest Rutherford, O.M., P.R.S., for his interest in the work; Dr J. Chadwick, F.R.S., for advice and helpful suggestions; Mr J. A. Ratcliffe, for the wireless set used in producing the electrodeless discharge; Mr Crowe, for the preparation of the radioactive sources; and Mr Brown, for help in counting scintillations.

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

Let X be a (based) space of the homotopy type of a CW-complex. Let H(X) denote the classical (ungraded) cohomology ring Πi≥0Hi (X;Z/2). In [1] Atiyah and Hirzebruch described the group of natural ring automorphisms of H(X) (‘cohomology automorphisms’) with group operation given by composition. They showed that is isomorphic to the group of formal power series of the form with group operation given by ‘substitution’ of one power series into another. In particular the most famous ‘cohomology automorphism’, the total Steenrod Square, corresponds to x + x2.