Isotropic ionospheric propagation of electromagnetic waves yields refractive index profiles via the standard transcendental equations. Some transformations of these equations have long been regarded as standard. Here, every possible transformation is derived by a systematic analysis, yielding all possible refractive index profiles satisfying certain stated criteria regarding the wave frequency.

Introduction. The object of this note is to give a proof of the following theorem on the solution of an important functional equation.

Our motivation for this paper is to be found in (2) and (3). In (2) Varopoulos considers inductive limits of topological groups, in particular what he calls ‘ℒ∞’. (He calls a topology an ℒ∞-topology when it is the inductive limit of a decreasing sequence of locally compact Hausdorff topologies.) In (2) he proves that much of the classical theory of locally compact Abelian groups also goes through for Abelian ℒ∞-groups, in particular Pontrjagin duality.

Three generalizations of Neumann's integral connecting the two kinds of Legendre function are obtained. They are not subject to restrictions that some parameter be integral, as the original formula and various known extensions of it all appear to be. These known extensions are exhibited as quite particular cases of the chief generalization obtained; and even when this generalization is specialized to the ‘unassociated’ Legendre functions of non-integral order it still seems to be new. Generalizations of Rodrigues's formula and of the orthogonality property are auxiliary results involved.

We are concerned in this work with the following question:

Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal number

We observe two easy results.

The problem discussed is that of solving the integral equation

where g(x) is given, Kv(z) is associated with Bessel functions of purely imaginary argument and f(x) is to be determined.

I prove that, by means of fractional integration, it is possible to reduce this equation to the form of a Laplace transform which can be solved by known methods.

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.

The Navier–Stokes equations governing the motion of viscous compressible fluids are

where ρ is the density, q is the velocity vector, V is the scalar potential of the external force field, I is the idemfactor and Φ is the stress-tensor

Milne-Thomson ((l)) has given a general solution of (1) and (2) in the form

Williams ((3)) has transformed certain series equations in Legendre functions to Bessel integral equations; his main result being that if

then

and vice versa. (Here, n is written for Williams's n + m. The coefficients an are related to Williams's An by a factor. There is an extra factor 2 in equation (7) in Williams's paper which appears to be an error.)

We shall study the following complex Stiefel fibring:

.

In particular we shall study the problem: for what values of n and k does the fibring(1.1) admit a cross-section? A necessary condition for the existence of a cross-section has been found by Atiyah and Todd (8). We shall show (Theorem 1.1 below) that the condition of Atiyah and Todd is sufficient (as well as necessary) for the existence of a cross-section. The problem stated above is therefore completely solved.

The two formulations of quantum mechanics in q-space only are compared by the construction of Schrödinger's operator identity. The examination in detail of the examples of the harmonic oscillator and free particle shows clearly the role of the re-normalization term in both formulations.

A function K(x) is said to be a symmetrical Fourier kernel if the integral equation

remains valid when f(x) and g(x) are interchanged amongst themselves.

The investigation of the isotropic scattering of radiation in an infinite spherically symmetric atmosphere, which was carried out in a previous paper, is extended to a finite atmosphere bounded by a totally absorbing shell.

It is shown that the solution may be expressed as a series which will certainly converge when the radius of the shell is large enough, and whose terms are found by iteration.

Numerical investigation shows that even for quite modest values of the radius, the convergence is so fast that for most practical purposes all but the first term can be neglected. Except at the very ends of the range, a very simple expression then gives an excellent approximation.

Let A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and s is x1, …, xm and t is x2, x3, …, xn. Call a subset S of A a P-block if, for every infinite ascending sequence x1, x2, … of positive integers, there exists an m such that x1, …, xm belongs to S. A quasi-ordered set Q (i.e. a set on which a reflexive and transitive relation ≤ is defined) is better-quasi-ordered if, for every P-block S and every function f:S → Q, there exist s, t ∈ S such that s ⊲ t and f(s) ≤ f(t). It is proved that any set of (finite or infinite) trees is better-quasi-ordered if T1 ≤ T2 means that the tree T1 is homeomorphic to a subtree of the tree T2. This establishes a conjecture of J. B.Kruskal that, if T1, T2, … is an infinite sequence of trees, then there exist i, j such that i < j and Ti ≤ Tj.

Some time ago Besicovitch(l) gave an example of a linear Gδ-set E of Besicovitch dimension 1, with the property that E – A also has dimension 1, for each Fσ.-set A contained in E. This shows that an Fσ-set contained in E cannot be a very close approximation to E, but it leaves plenty of scope for Fσ-sets contained in E to be reasonably good approximations to E.

We study processes in which units (particles) associate into clusters, which are then also capable of dissociation. Such processes are discussed generally in section 2, where a stochastic kinetic equation (10) is proposed which bridges the gap between the conventional kinetic equations (8) and the statistical equilibrium concept of the Gibbs distribution.

In section 4 we consider the equilibrium behaviour of a process for which the association rate of two units which are already bound to j and k other units respectively has the form (21). This is very much more general than the equi-reactive bond model usually discussed. The principal results are given in Theorem 1; from a single pair of equations (28) and (29) based on the Hj of (21) one can determine critical points, expected number of bonds, the distribution and moments of cluster size, and most other quantities of interest. This is without reference to any other consideration, such as kinetic or stoichiometric relations.

Some particular cases are worked through in section 5. The classic Flory-Stockmayer results for units with f equi-reactive sites are obtained systematically and economically, with all parameters in terms of physically given quantities. Another type of example seems to indicate the existence of a second critical point.

Corresponding results for the case of several types of unit are stated and illustrated in section 6.

The combined effects of uniform thermal and magnetic fields on the propagation of plane waves in a homogeneous, initially unstressed, electrically conducting elastic medium have been investigated.

When the magnetic field is parallel to the direction of wave propagation, the compression wave is purely thermo-elastic and the shear wave is purely magneto-elastic in nature. For a transverse magnetic field, the shear waves remain elastic whereas the compression wave assumes magneto-thermo-elastic character due to the coupling of all the three fields—mechanical, magnetic and thermal. In the general case, the waves polarized in the plane of the direction of wave propagation and the magnetic field are not only coupled but are also influenced by the thermal field, once again exhibiting the coupling of the three fields. The shear wave polarized transverse to the plane retains its magneto-elastic character.

Notation.

Hi = primary magnetic field components,

ht = induced magnetic field components,

To = initial thermal field,

θ = induced thermal field,

C = compression wave velocity.

S = shear wave velocity,

ui = displacement components,

cv = specific heat at constant volume,

k = thermal conductivity,

η = magnetic diffusivity,

μe = magnetic permeability,

λ, μ = Lamé's constants,

β = ratio of coefficient of volume expansion to isothermal compressibility.

Introduction. When the distribution of a statistical function could not be determined exactly, a suitable form for an approximate formula has sometimes been found by the use of information about the order of smoothness of the distribution and about any singularities it might have (see, for example, Geary ((4))). The object of the present paper is to develop a general theory that will provide information of this kind. For the singularity in the distribution of a function at an end of its range Hotelling ((8)) obtained formulae for cases where the original distribution is uniform over a curved space. However, some special sampling distributions are known to have singularities in the interior of the range, for example, the distribution of √ b1 (see (4)) and those of various serial correlation coefficients (see (6)). In view of this I shall deal below with orders of smoothness and with singularities over the whole range. For simplicity I shall assume that the original distribution has a fairly high order of smoothness: attention will therefore be concentrated on the singularities (if any) that arise from the form of the statistical function.

In this note we shall consider the series

where lim sup |an| = 1 and

Given a totally finite measure space (S, S, μ) and two μ-integrable, non-negative functions f(x) and φ(x) defined in S, such that when

then

we define correlated sampling as the technique of estimating

by sampling an estimator function

where ξ is uniformly distributed in S with respect to μ (i.e. for any T ∈ S, p(T) = μ(T)/μ(S) is the probability that ξ lies in T): and importance sampling as estimating L by sampling the estimator function

where η is distributed in S with probability density φ(x)/Φ

Then, clearly,

It follows that υ(ξ) and ν(η) are both unbiased estimators of L, and that their variances can both be made to approach zero arbitrarily closely by making φ(x) a sufficiently close approximation to f(x).