The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean that

for all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

In an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f′′′ + ff″ + β(1 − f′2) = 0, in addition to those found by Hartree, is investigated.

It is found that when β < − 0·1988 there are no solutions satisfying |f′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f′(0) and δ1 are displayed. The impact of these solutions on the theory of separation is discussed.

It is generally believed that isolation has played an important part in evolution. If an organism is to evolve so as to adapt itself to a special type of environment, e.g. a cave or a desert, it must not be swamped in each generation by migrants from the original habitat.

The paper examines the structure obtained by abstracting from the conventional (manifold) representation of relativistic space-time the concept of an event-set equipped with two partial orderings, whose counterparts are the notions ‘causally precedes’ and ‘chronologically precedes in the history of some observer’.

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,

or, in an alternative notation,

Let X1, …, Xn be independent Bernoulli random variables, and let pi = P[Xi = 1], λ = Σi=1n pi and Σi=1n Xi. Successively improved estimates of the total variation distance between the distribution ℒ(W) of W and a Poisson distribution Pλ with mean λ have been obtained by Prohorov[5], Le Cam [4], Kerstan[3], Vervaat[8], Chen [2], Serfling[7] and Romanowska[6]. Prohorov, Vervaat and Romanowska discussed only the case of identically distributed Xi's, whereas Chen and Serfling were primarily interested in more general, dependent sequences. Under the present hypotheses, the following inequalities, here expressed in terms of the total variation distance

were established respectively by Le Cam, Kerstan and Chen:

(Kerstan's published estimate of ([3], p.174, equation (1)) is a misprint for , the constant 2·1 appearing twice on p. 175 of his paper.) Here, we use Chen's [2] elegant adaptation of Stein's method to improve hte estimates given in (1·1), and we complement these estimates with a reverse inequality expressed in similar terms. Second order estimates, and the case of more general non-negative integer valued X's, are also discussed.

Suppose that

is an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Then

where a, b, c, d are integers such that ad − bc = 1.

The function c(p) is defined for positive integers p ≥ 4 by

where > denotes contraction. Random graph examples show

In 1968 Mader showed that c(p) ≤ 8(p − 2) [log2 (p − 2)] and more recently Kostochka showed that p√(log p) is the correct order for c(p). We present a simple argument showing c(p) ≤ 2.68p √(log2p)(l + ο(l)).

1. It has been shown, in a well-known paper by Born and Oppenheimer, that in evaluating the energy levels of a molecule, the nuclei may be treated as fixed, or slowly vibrating, centres of force. If M be the average mass of the nuclei, and m the mass of an electron, the error involved in doing this is of order of magnitude (m/M)½. The possibility therefore of treating the nuclei as fixed depends on their great mass, compared with that of the electron.

The quantum theory of the electron allows states of negative kinetic energy as well as the usual states of positive kinetic energy and also allows transitions from one kind of state to the other. Now particles in states of negative kinetic energy are never observed in practice. We can get over this discrepancy between theory and observation by assuming that, in the world as we know it, nearly all the states of negative kinetic energy are occupied, with one electron in each state in accordance with Pauli's exclusion principle, and that the distribution of negative-energy electrons is unobservable to us on account of its uniformity. Any unoccupied negative-energy states would be observable to us, as holes in the distribution of negative-energy electrons, but these holes would appear as particles with positive kinetic energy and thus not as things foreign to all our experience. It seems reasonable and in agreement with all the facts known at present to identify these holes with the recently discovered positrons and thus to obtain a theory of the positron.

The methods used to measure separately the electronic and lattice heat conductivities κe and κg in a metal are reviewed, and it is pointed out that care is necessary in interpreting the results in view of the underlying assumptions. The equations given by Wilson for κe and for the electrical conductivity σ are used to plot the theoretical values of the electronic Lorentz ratio Le = κe/σT as a function of T, both for the monovalent metals and for a model metal with 1·8 × 10−2 conduction electrons per atom, which is taken to represent bismuth sufficiently accurately for this purpose. Curves for κe and κg as functions of T are given in both cases, and these, together with a comparison of the observed Lorentz ratio and Le, show that in the monovalent metals κg is unimportant at any temperature, but in bismuth it plays a major part at low temperatures, in agreement with experimental conclusions. Quantitatively the agreement is good for copper and, as far as the calculations go, reasonable for bismuth.

Scattering of lattice waves at the boundaries of single crystals (including insulators) at temperatures of a few degrees absolute is shown to be consistent with the experiments of de Haas and Biermasz on KCl and to be responsible for the rise in thermal resistance in this region as suggested by Peierls.

The assumption in the theory of electronic heat conduction that the lattice energy distribution function has its thermal equilibrium value is examined in an appendix. The conclusion is that it should be satisfactory, though the proof of this given by Bethe is seen to be inadequate.

A study is made of the actual trajectories of fluid particles in certain motions of classical hydrodynamics. When a solid body moves through an incompressible fluid, it induces a drift in the fluid, such that the final positions of the particles are further on than those from which they started. The drift-volume enclosed between the initial and final positions is equal to the volume corresponding to hydrodynamic mass, that is, the mass of fluid to be added to that of the solid in calculating its kinetic energy. This result is proved quite generally. The work involves integrals which are not absolutely convergent, and these are discussed in relation to the general mechanics of fluids. When the trajectories are considered of the fluid surrounding a rotating body, it is shown that the fluid particles slowly drift round the body, even though the motion is irrotational and without circulation. There seems to be in some respects a closer resemblance between the behaviour of the idealized hydrodynamic fluid and a real fluid than might be expected from the well-known discrepancies between them.

This paper is an attempt to apply the methods of the theory of algebras to the more general problem of the structure of semi-groups, i.e. systems with one composition, satisfying the associative law.

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the form

and require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.

As we have stressed in a previous paper (9), the main concern in Monte Carlo work is to achieve without inordinate labour a respectably small standard error in the final result. Mere replication of the Monte Carlo results is unrewarding; for, to reduce a standard error by a factor k, the labour must be increased k2-fold, and this will be beyond the resources of even electronic computers when k = 1000, say. The remedy lies in a skilful choice of sampling technique and the substitution of analytical methods for random processes wherever possible. The efficiency of a Monte Carlo process may be taken as inversely proportional to the product of the sampling variance of the final estimate and the amount of labour expended in obtaining this estimate; and it is profitable to allow some increase in the labour if that produces an overwhelming decrease in the variance. For instance, in the last example quoted below (Table 2), we reduce the variance by a factor of four million at the expense of only multiplying the labour sixteenfold, thereby attaining a 250,000-fold gain of efficiency.

In this paper we first of all consider the dual integral equations

where f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.

The theory of holes in liquids, suggested in a previous paper, is developed by means of classical statistical mechanics, and it is shown that the principal thermodynamic properties of the liquid state can be derived in this way and that they are in numerical agreement with the experiments.

This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.