The classical theory of partition of numbers is applied to the problem of determining the equilibrium profile of a simple cubic crystal. It is concluded that it may be thermo-dynamically profitable for the surface to be ‘saw-toothed’ rather than flat, the extra entropy associated with such an arrangement compensating for the additional surface energy. For both a two- and a three-dimensional ‘saw-tooth’ the extra entropy varies, to a first approximation, in the same way as the surface energy, i.e. is proportional to or respectively, where N is the number of molecules in a ‘tooth’. For the simple cubic lattice, the entropy associated with the formation of a tooth containing N atoms is estimated to be 3.3 It is also possible to estimate the variation of the ‘equilibrium roughness’ of a crystal with temperature, if its surface energy is known.

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

that is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

In this note we shall study random labelled graphs. Denote by

the set of all graphs with n given labelled vertices and M(n) edges. As usual, we turn (M(n)) into a probability space by giving all graphs the same probability. The question we address ourselves to is the following. Given a graph H and a constant p, 0 < p < 1, for what functions M(n) is it true that the probability PM(n)(H ⊂ G) that a graph G∈ (M(n)) contains H tends to p as n∞→? This question was posed by Erdös and Rényi (3), (4), who also proved several beautiful and surprising theorems. In order to state the main general result of Erdös and Rényi in this direction, and for our use later, we introduce some definitions.

We study self-similar sets with overlaps, on the line and in the plane. It is shown that there exist self-similar sets that have non-integral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parameter value. We also explore conditions for the validity of Falconer's formula for the Hausdorff dimension of self- affine sets, and study the dimension of some fractal graphs.

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):

In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

A method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.

For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:

the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which

the step problem: for the set

rearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.

Group algebras and crossed products have always played an important role in the theory of C*-algebras, and there has also been considerable interest in various twisted analogues, where the multiplication is twisted by a two-cocycle. Here we shall discuss a very general family of twisted actions of locally compact groups on C*-algebras, and the corresponding twisted crossed product C*-algebras. We shall then establish some of the basic properties of these algebras, motivated by the requirements of some applications we have in mind [2, 9, 10]. Some of our results will be known to others, at least in principle, but we feel that a coherent account might be useful.

A group is said to have the property T or to be a T-group if every subnormal subgroup is normal. Thus the class of T-groups is just the class of all groups in which normality is a transitive relation. Finite T-groups have been studied by Best and Taussky (l), Gaschütz (4) and Zacher (11). Gaschütz has shown that if G is a finite soluble T-group and G/L is the unique maximal nilpotent quotient group of G, then G/L is Abelian or Hamiltonian and L is an Abelian group of odd order prime to |G:L| ((4), Satz 1). Our aim is to study infinite T-groups and more especially infinite soluble T-groups with a view to extending Gaschütz's results. One of the simplest results on soluble T-groups is THEOREM 2.3.1. Every soluble T-group is metabelian.

It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘A-indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and ‘proofs’ are morphisms over A. We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A.

A detailed account is given of an absorption method, using the β-radiation of radium E as a standard of comparison, by which the maximum energy and some indication of the form of the energy spectrum of the β-particles from any other radioelement may be obtained. The method is applicable alike to weak sources and to sources of finite thickness and is designed to yield the largest amount of information under such limitations. Experimental results with sources of UX2, Na24, Ag106 and P32 are treated by the method described. A modified empirical range-energy relation is given.

Introduction. The effect of a vibrating point force on the surface of a semi-infinite elastic solid was investigated in a classic paper by Lamb ((8)). When the point force is replaced by a vibrating rigid disc the problem is more complicated, because the displacement is specified under the disc and the remainder of the surface is stress-free. This gives rise to a mixed boundary-value problem.

The Rayleigh assumption, which concerns the domain of validity of a representation for the field scattered by a periodic surface under time-harmonic plane-wave excitation, is re-examined. The analysis employs a technique developed to locate singularities of solutions to the Helmholtz equation. When applied to the surface profile v = b cos κu (-∞ < u < ∞) considered originally by Lord Rayleigh, it is found that the Rayleigh assumption is valid if 0 ≤ κb < 0·448 and is not valid if κb > 0·448; this is more precise than an earlier result. It is shown how these findings may be reconciled with the work of others who have suggested, or concluded, that the Rayleigh assumption is never valid.

1. Introduction. The spaces L1 and L2 of unbounded operators associated with a regular gauge space (von Neumann algebra equipped with a faithful normal semi-finite trace) are defined by Segal(5) definitions 3.3, 3.7. The spaces Lp (1 < p < ∞, p ± 2) are defined by Dixmier(2) as the abstract completions of their bounded parts. Dixmier makes use of the Riesz convexity theorem to prove the Hölder inequality, and the uniform convexity, and hence reflexivity, of LLp (2 < p < ∞).

A new and fruitful theory of turbulent motion was published in 1941 by A. N. Kolmogoroff. It does not seem to be as widely known outside the U.S.S.R. as its importance warrants, and the present paper therefore describes the theory in some detail before presenting a number of extensions and making a comparison of experimental results with some of the theoretical predictions.

Kolmogoroff's basic notion is that at high Reynolds number all kinds of turbulent motion, of arbitrary mean-flow characteristics, show a similar structure if attention is confined to the smallest eddies. The motion due to these eddies of limited size is conceived to be isotropic and statistically steady. Within this range of eddies we recognize two limiting processes. The influence of viscosity on the larger eddies of the range is negligible if the Reynolds number is large enough, so that their motion is determined entirely by the amount of energy which they are continually passing on to smaller eddies. This quantity of energy is the local mean energy dissipation due to turbulence. On the other hand, the smaller eddies of the range dissipate through the action of viscosity a considerable proportion of the energy which they receive, and the motion of the very smallest eddies is entirely laminar. The analytical expression of this physical picture is that the motion due to eddies less than a certain limiting size in an arbitrary field of turbulence is determined uniquely by two quantities, the viscosity and the local mean energy dissipation per unit mass of the fluid.

The mathematical method of describing the motion due to eddies of a particular size is to construct correlations between the differences of parallel-velocity components at two points at an appropriate distance apart. Kinematical results analogous to those for turbulence which is isotropic in the ordinary sense are obtained, and then the scalar functions occurring in the expressions for the correlations are determined by dimensional analysis. The consequences of the theory in the case of turbulence which possesses ordinary isotropy are analysed and various predictions are made. One of these, namely that dimensionless ratios of moments of the probability distribution of the rate of extension of the fluid in any direction are universal constants, is confirmed by recent experiments, so far as the second and third moments are concerned. In several other cases it can be said that relations predicted by the theory have the correct form, but further experiments at Reynolds numbers higher than those hitherto used will be required before the theory can be regarded as fully confirmed. If valid, Kolmogoroff's theory of locally isotropic turbulence will provide a powerful tool for the analysis of problems of non-uniform turbulent flow, and for the determination of statistical characteristics of space and time derivatives of quantities influenced by the turbulence.

The choice of a numerical method for the solution of ordinary differential equations depends on the associated boundary conditions. When all the boundary conditions are specified at one end of the range of integration, one of the well-known step-by-step methods will generally be used, while the method of relaxation is reserved for the case in which boundary conditions are specified at more than one point (1). In the latter, simple but inaccurate finite-difference formulae are used to provide a first approximation to the required solution; this approximation is then used to give an estimate of the errors involved in the use of the inaccurate formulae, and successive corrections are obtained until the full, accurate finite-difference equations are satisfied (1). The same principle is followed in this paper with regard to step-by-step methods, the main difference being the way in which approximate solutions are obtained. In relaxation methods simultaneous equations are solved, while in the use of the step-by-step methods suggested here successive pivotal values are built up by the use of recurrence relations. All the methods of this paper follow this principle, differing only in the method of obtaining a recurrence relation, and consequently in the form of the correction terms.

There are many types of problem in the study of molecular structure in which integrals involving two or more centres of force arise. We may mention the calculation of molecular energy levels, the Stark effect in molecules, the van der Waals forces between molecules, and, more recently, the internal arrangement of neutrons and protons in nuclei. During the last few years the writer has been accumulating a table of these integrals, and the explicit forms for many of them are given in this paper.

In this paper there is considered the irrotational motion of an infinite fluid when the normal velocity across a plane is specified as a stationary random function of position in the plane, and a solution is obtained in terms of the specified boundary conditions. It is shown that the mean square velocity normal to the plane is equal to the sum of the mean squares of the velocities in the other two orthogonal directions. The asymptotic variations with distance normal to the plane are found for functions representing the important properties of the motion, and, in particular, the energy of the fluctuations is shown to be inversely proportional to the fourth power of the distance from the plane. The conditions postulated are shown to correspond closely to the motion outside a free turbulent boundary, and good agreement is found between the predictions of the theory and the available experimental results.

Two sets of conditions are found on the Q-matrix of an irreducible Markov process on a countably infinite state space which ensure that Q is regular; the first set also implies that the Markov process is ergodic, the second that it is recurrent. If the process is not irreducible, the conditions still imply regularity, and then either non-dissipativity or ‘ultimate recurrence’ (reducible analogues of ergodicity and recurrence) of the process. Conditions sufficient for ergodicity or recurrence of the process in the non-regular and regular case are also given. The results parallel (and use) results for discrete time Markov chains, and the known discrete time recurrence condition is extended to the reducible case. The conditions are illustrated by a competition process example.