Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered, amongst others, in the fundamental paper of Erdös and Rényi (4). The technique normally used to obtain such approximations in random graph theory is based on showing that the factorial moments of the quantity concerned converge to those of a Poisson distribution as the size of the graph tends to infinity. Since the rth factorial moment is just the expected number of ordered r-tuples of events occurring, it is particularly well suited to evaluation by combinatorial methods. Unfortunately, such a technique becomes very difficult to manage if the mean of the approximating Poisson distribution is itself increasing with the size of the graph, and this limits the scope of the results obtainable.

A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

We prove that every closed, connected contact 3-manifold can be obtained from $S^3$ with its standard contact structure by contact (${\pm}1$)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

1. By expanding the radiation field in a series of spherical waves the complete expressions for the field emitted by an electric and magnetic 2l-pole are obtained (§§ 1, 2).

2. It is shown that the wave emitted by a (electric or magnetic) 2l-pole has an angular momentum about the z-axis Mz = mU/ν (U = total energy), where m, according to the orientation of the 2l-pole in space, can assume the values −l, −l + 1, …, + l (§ 3). The angular momentum is contained in that region of the field in which the product EH decreases as r−3.

3. By quantizing the waves it is shown that the angular momentum of a light quantum emitted by a 2l-pole behaves like the angular momentum of an electron in a central field of force without spin (commutation relations, etc.) (§ 4). The angular momentum of a single light quantum is an integral multiple of ħ.

We present the following sequence of polynomial identities:

is the Gaussian polynomial denned to be zero for m < 0 or m > N, one for m = 0 or N and

A new Möbius-invariant metric δG defined on an open set G⊂[Rmacron]n with at least two boundary points is introduced. This metric coincides with the hyperbolic metric if G is the unit ball. Some inequalities between this and other metrics are proved.

The behaviour of δG under K-quasiconformal maps is also studied. In particular, a generalization of the Schwarz lemma is obtained for K-quasiconformal maps defined on quasiballs.

A new method is developed for evaluating certain integrals occurring in molecular problems, where distances measured from three different centres of force are involved. The convergence of the results is discussed numerically and theoretically.

A method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.

Difficulties associated with the evolution of stars by radiation alone are briefly discussed. It is clear that some other process is also affecting the stars and it is shown that the stars are capable of adding to their mass by the process of accretion of the cosmical cloud. The gravitation of a moving star causes additional collisions of the atoms of the interstellar matter and the motions become randomized to such an extent that the star probably captures all material passing within the distance at which the velocity of the star relative to the cloud is the parabolic velocity. This rate of accretion of mass of a star is 4πγ2ρM2/ν3, and is accordingly of great importance for stars of low velocity. Stars of high velocity are least affected by accretion and therefore in general remain of low mass, while stars of low velocity must attain great mass. The periods of time involved in bringing about appreciable changes in the mass of a star are of the order of 5 × 1010 years and are in agreement with independent estimates of the time scale, as deduced, for example, from the companion of Sirius. The evolution of the components and orbits of binary stars are consequences of the accretion process. The more massive component increases in mass more rapidly than the less massive component in the case of wide pairs, and may therefore in general continue to emit more ergs per gram. The orbit evolves in such a way that the total angular momentum remains constant. For equal masses the separation is proportional to the inverse cube of the mass, and the period to the inverse fifth power, so that great changes of separation and period occur. The evolution of the stars is governed almost entirely by their velocities relative to the cosmical cloud. In the case of double stars the evolution takes the form of decreasing period and decreasing separation. Such features as galactic concentration and the correlation between spectral type and velocity are direct results of accretion.

Harding and Sneddon(3) and Sneddon(6) have shown that the systematic application of Hankel transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch, reduces the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh(8) and Busbridge(1). The same method has also been applied by Sneddon(7) to the problem of the distribution of stress in the neighbourhood of a circular crack in an elastic solid. In both types of problem the results are expressed in quite simple forms independent of Bessel functions and this suggests that there should be an alternative method of solution. It is now possible to supply such a method with the help of a recent paper by Copson(2) on the problem of the electrified disk. In addition, some generalization of the stress systems considered by Harding and Sneddon can be made since they confined their analysis to stresses which were symmetrical about an axis.

The method of leading variables is presented as a new method of solving Linear Programming, and is compared with the Simplex and the Dual Simplex methods.

The effect of a monomolecular film on the reflexion of light from a plane surface is considered. The film is treated as a two-dimensional assembly of scattering centres, and the possibility of molecular orientation is allowed for. Experimental results, when available, should indicate whether the scattering properties of groups of molecules in the “two-dimensional” state differ from those of molecules in bulk.

In a recent paper, the solubility of hydrogen in palladium was discussed. The statistical theory employed was essentially the same as that given by Fowler† for critical adsorption, the only modification necessary being to take into account the dissociation of the hydrogen molecules on absorption. In the present paper Peierls' theory‡ is applied to the same problem in order to enable a comparison to be made between the theories. According to both theories we must postulate that the absorbed (adsorbed) particles go into a definite number of potential energy holes and that the heat of absorption increases as the number of holes filled increases. Fowler uses the Bragg and Williams type of approximation to determine the energy of the particles in the holes; while Peierls supposes that there is an interaction energy V for each pair of neighbouring adsorbed particles and that all those arrangements of the particles in the holes having the same number of nearest neighbours have the same energy. The calculation of the dependence of the energy on the arrangements is approximated to by a method due to Bethe.

Self-consistent fields without exchange have been calculated for the ground state of Pr3+ and Tm3+. The spin-orbit coupling constant ζ(4f) is found to have the values 785 and 2742 cm−1 in Pr3+ and Tm3+, respectively. The corresponding values of are 29·4 and 77·5 Å−3. Values of the Slater integrals F2(4f, 4f), F4(4f, 4f) and F6(4f, 4f) are also given for each structure.

Henderson proposed a theory of the stopping of swift α-particles by matter. He treated the electrons in the atoms as free and at rest and ignored all collisions of the α-particle with them except those in which the electron would on that assumption gain sufficient energy to leave the atom. Fowler has shown that Henderson's theory gives stopping-powers only of about 60% of those observed. Fowler also made a calculation of the stopping-power of hydrogen by combining the effects of close collisions treating the electron as free and slight collisions as perturbations of the electron's motion in a circular orbit. He obtained much better agreement. This method is, of course, the natural extension of Bohr's original theory to that model.

In this paper we continue the study of Orlicz sequence spaces initiated by Lindberg (5) and Lindenstrauss and Tzafriri (7), (8) and (9). Our main concern is to investigate features of the theory which occur when the restriction of local convexity is lifted. It is clear that some results will hold with identical proofs, at least when the space is locally bounded. However, we are chiefly interested in the differences which arise. We always assume that the Orlicz function F satisfies the Δ2-condition.

1. Introduction and summary. A chain of N links is allowed to assume a random configuration in space. The extent of the chain in any direction is defined as the shortest distance between a pair of planes perpendicular to that direction, such that the chain is contained entirely between them. In the present paper the probability distribution of the extent is discussed, starting with a chain in one dimension for which formulae are derived for the probability and mean extent for all values of N. The limiting forms for large N are then considered. The results are extended to the case of a chain in three dimensions, and it is shown that the extents in two directions at right angles tend to be independently distributed when N is large. It is assumed that the links are infinitely thin, so that a point in space may be occupied by the chain any number of times.

The notion of a reduction of an ideal I of a noetherian ring R and the related notions of the integral closure I* of I in R and the integral dependence of an element x of R on I have proved useful in many situations in commutative algebra. The purpose of this paper is to extend these notions to modules. To motivate the definitions which follow, it will be useful to consider these notions for ideals before extending them to modules.

Sack (3) and Sneddon (4) have considered the problem of a penny-shaped crack in an infinite medium under tension normal to the face of the crack. The same problem has been considered more recently by Green (1) by a different method. In this note we consider the case of the same shaped crack with the medium subjected to a uniform shearing stress parallel to the plane of the crack. It is found that the solution is analogous to that of the corresponding two-dimensional problem considered by Starr (5).