The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

We extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

The following problem is the simplest of a group of problems motivated by work of Durran and Truman [1] on planetesimals.

Over the last few years a number of results giving conditions on a derivation of a Banach algebra implying that its range is contained in the radical have been obtained (see survey articles of Mathieu[7] and Murphy [8]). If an algebra is semi-simple, these conditions, of course, imply that a derivation is zero. In this paper we consider inner derivations that are non-zero in general, but their ranges are rather special and ‘small’ in some sense.

Let K be a knotted theta-curve with exterior X, and let ∂_X be one of the two pieces into which ∂X is divided by the meridians of the edges of K. Let X be the universal abelian cover of X. Then is a module over the group ring of H1(X); i.e. over . We call this the Alexander module of K, and denote it by A(K). This, rather than H1(X), seems to be the analogue of the Alexander module of a classical knot; it is a torsion module of deficiency 0. Moreover, it is not an invariant of X alone.

In a stratified wave guiding system, it is often possible to adjust parameters, such as the wave frequency or some dimension of the guide, so that the propagation constants of two modes become equal. These modes are then said to be degenerate. In a tapered guide some of the parameters are slowly varying functions of distance x measured along the guide, and the conditions of degeneracy can be satisfied for some value x0 of x, possibly complex. Here the two modes are said to be critically coupled, and x0 is called a ‘coupling point’ in the complex x-plane. Cases are known where two modes in the earth-ionosphere wave guide could approach critical coupling in the twilight region of the earth, where the guide is tapered. The x dependence of the composite field for the two modes can then be expressed in terms of an Airy Integral function. The theory is closely analogous to that used for studying critical coupling of Ordinary and Extraordinary radio waves within a horizontally stratified ionosphere.

A family of sets {Fd}d is said to be ‘represented by the measure μ’ if, for each d, the set Fd comprises those points at which the local dimension of μ takes some specific value (depending on d). Finding the Hausdorff dimension of these sets may then be thought of as finding the dimension spectrum, or multifractal spectrum, of μ. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approximation, Kleinian groups and hyperbolic dynamical systems.

Throughout this paper the word ‘ring’ will mean ‘commutative Noetherian ring with non-zero identity’, and A will always denote such a ring. It will only be assumed that A is local when this is explicitly stated, but when this is the case m (resp. k) will always be used to denote the maximal ideal (resp. residue field) of A. It is to be understood that ring homomorphisms respect identity elements.

Let the elliptic curve E be defined by the equation

formula here

with a1, …, a6 ∈ ℤ. Define a finite set of places S = {q1, …, qs−1, qs = ∞} of ℚ and put Q = max {q1, …, qs−1}. Let E(ℚ) denote the set of (x, y) ∈ ℚ2 satisfying (1) and the infinite point [Oscr ].

The multiplicative height of a rational point P = (x, y) ∈ E(ℚ) is defined as the following product over all places q of ℚ (including q = ∞):

formula here

where the [mid ]x[mid ]qs are the normalized multiplicative absolute values of ℚ corresponding to the places q.

A group G is said to be complete if the centre of G is trivial and every automorphism of G is inner; this means that G is naturally isomorphic to Aut G, the group of automorphisms of G. In response to a question of Rose (10) we shall describe the construction of an example demonstrating the following result. (Rose has pointed out that the problem was mentioned earlier by Miller (8).)

1. Introduction. The stability of a system of queues in series was investigated in (1) but the results obtained were not quite complete, for the situation in which all component queues are substable and at least one is critical was scarcely touched upon. The only reference to this situation, in fact, was an example on page 513 showing that necessary and sufficient conditions are not obvious. (The ideas and terminology of (1) will be used freely throughout this paper.)

We establish that the set of pairs (α, β) of real numbers such thatwhere ‖ · ‖ denotes the distance to the nearest integer, has full Hausdorff dimension in R2. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems.

We consider singular block operator problems of the type arising in the study of stability of the Ekman boundary layer. The essential spectrum is located, and an analysis of the $L^2$ solutions of a related first order system of differential equations allows the development of a Titchmarsh–Weyl coefficient $M(\lambda)$. This, in turn, permits a rigorous analysis of the convergence of approximations to the spectrum arising from regular problems. Numerical results illustrate the theory.

It was pointed out by Copson(1) in 1947 that the solution of the integral equation for the electrostatic problem for a circular disc could be reduced to the solution of Abel integral equations and hence a solution obtained in a fairly elementary manner. This result was obtained independently by Lebedev(2), who also obtained a similar result for the electrostatic problem for the spherical cap. The solution for the spherical cap was also obtained independently by Collins (3). In view of the relative simplicity of the approach it seems to be of interest to examine whether there exist other integral equations which can be treated in a similar fashion. In the present paper one such class of integral equations will be considered.

The stability of a rotating compressed rod is studied as a two-parameter bifurcation problem. Imperfections in shape and loading are assumed to be present. A number of solutions and their local behaviour is analysed.

In the first part of this paper I investigated the nature of the isolated singular points which can appear on an algebraic surface without affecting the conditions of adjunction and the arithmetic genus of the surface. It appeared that such points have neighbourhoods analysable into connected chains or trees of curves, each rational and of virtual grade − 2, and arranged either in a single chain (giving a binode, or a conic node if there is only one curve) or in three chains of, say, n, p, q curves respectively, one end curve of each chain meeting one which we may call the central curve of the tree. The values of n, p, q are not arbitrary, but satisfy either p = q = 1, or n ≤ 4, p = 2, q = 1; the unodes given by the former case we call U, those given by the latter case U, with a suffix indicating the reduction in class in each case. I pointed out the similarity between these results and Coxeter's enumeration of groups generated by reflexions, there being a one-one correspondence between these singularities and Coxeter's groups, with the restriction that the only groups with which we are concerned are those in which any two primes of symmetry are inclined at either π/2 or π/3. In fact, the curves which make up a complete neighbourhood correspond to the bounding primes of a fundamental region, two curves which do not meet corresponding to mutually perpendicular, primes, and two which do to primes inclined at π/3.

1. Introduction. Among the more notable attempts to derive a generalization of Einstein's gravitational theory is the recent one of Einstein and Schrodinger ((1)–(8)). This was formulated by dropping the symmetry of the fundamental tensor gμν and the components of the affine connexion. The most serious defect of these non-symmetric theories is that the field equations, in their original form, do not determine the motion of electrically charged particles in an electromagnetic field, as has been proved by Infeld(9), Callaway (10) and Bonnor (n). Together with the lack of an energy-momentum tensor and a geometric description of the paths of charged particles, this seems to indicate that the concept of motion is missing in this type of theory. It is clear that one of the most important results which should follow from a generalization of Einstein's gravitational theory is the correct equations of motion of charged particles in an electromagnetic field.

Experiments on the deformation of soap bubbles in an electric field show that until the bubble begins rapid vibrations there is no difference between positive and negative bubbles.

The positive bubble always bursts very shortly after beginning to vibrate even when resting upon a large supply of soap solution.

The negative bubble, however, when resting on ample soap solution, continues vibrating until a much higher field is reached and usually does not burst until some protuberance on the positive plate allows a spark to pass, which may then strike and destroy the bubble.

The values of the field for which a particular bubble begins rapid vibrations, so that in continuous light it appears to be conical, are found to be the same within experimental error for both positive and negative bubbles. The fields for bubbles of different sizes are found to vary inversely as the square root of the radius.

Photographs are given illustrating the shape of bubbles in various fields.

These experiments were carried out at the suggestion of Professor C. T. R. Wilson, and I am greatly indebted to him for his continued advice and interest throughout their progress.

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.

Wald(4) proved that under rather weak conditions, maximum-likelihood estimators are consistent. In an earlier paper (3) he had promised to publish conditions under which they are uniformly consistent but he did not do so. As the uniformity of consistence of maximum-likelihood estimators is important in studying the asymptotic power of certain tests, particularly when the true value of the parameter lies on the boundary of the parameter space, the purpose of the present note is to fill this gap. The method of proof follows that of Wald(4) very closely.