Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, let

where p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example with

where Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:

where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).

Introduction. The embeddability of l1(Γ), for uncountable sets Γ, into subspaces of Banach spaces of the form C(Ω) was investigated first by Hagler in (6) and subsequently by Haydon in (7), (8) and Argyros and Negrepontis in (1). An important role in the development of the above subject is played by a lemma of Rosenthal (12) that translates the functional analytic problem of finding a family {fξ: ξ Γ} of elements of C(Ω) equivalent to the usual basis of l1(Γ) into the problem of the existence of an independent family {(Aξ, Bξ,): ξ є Γ} of closed subsets.

We consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

Let K be a knot in the 3-sphere S3; let m be a meridian and 1 a longitude of K. In [1] we gave a detailed study of the ‘π-orbifold group’ O(K): π1(S3\K)/〈m2〉 of the knot, where 〈m2〉 is the subgroup normally generated by the square of the meridian. In particular, we found out to what extent the π-orbifold group classifies knots (and links) and determines the symmetry group of a knot. In the present paper, we consider the groups Gn(K): = π1(S3\K)/〈ln〉. Our main results are the following.

Radiologists have often had cause to note certain apparent anomalies in the behaviour of their apparatus. It is quite well known that different X-ray tubes, excited by different kinds of high tension apparatus, yield X-radiation of markedly varying quality and quantity even under conditions which, as measured by spark gap and milliammeter (the usual measuring instruments of the radiologist), are apparently identical. These anomalies seemed to offer an interesting field for investigation, and one which might not be without value on the practical side.

1. The conductivity of liquid sulphur is found to vary with the temperature in a similar manner to that in which the viscosity varies.

2. The conduction is electrolytic in character.

3. There is a definite E.M.F. of polarisation.

In a paper recently published I found it necessary to extend those investigations of Lord Rayleigh which deal with the instability of the surface common to two streams of fluid. The Rayleigh investigation is equivalent to a first order approximation to the flow and concludes that the initial form of the disturbance of the common surface is maintained but that its amplitude increases exponentially with the time, the form of the surface at time t being, with particular values inserted,

where a is the initial amplitude, 2π/k the wave-length, U the velocity of the upper stream, – U the velocity of the lower stream, and ρ the density of both fluids. A second order approximation to the flow for the above particular case shows that the original disturbance does not grow symmetrically, the ordinate of the common surface being given by

(A slightly amended form of this solution is given byequation III(8).) The solution suggests the initial stages in the formation of vortices. The problem was further discussed by a numerical step-by-step method and showed the formation of vortices quite clearly.

1. The following proof of Bateman's expansion may be of interest since it avoids the use of Appell's hypergeometric functions of two variables, though it suffers from the defect that it presupposes a knowledge of the expansion.

The relation between the reaction velocity of catalytic dehydrogenation and the rate of passage of alcohol vapour over the catalyst has been found by experiment. At low rates of flow the concentration of the aldehyde and hydrogen in the effluent gas, and therefore over the catalyst, is comparable with that of the reactant. The whole of the surface is thus not covered with the reactant, and the adsorbed reaction products cause marked decrease in the reaction velocity.

When the concentration of the reaction products becomes sufficiently small, that is, when the rate of passage of the alcohol vapour exceeds some limit, the velocity is approximately independent of the rate of flow of reactant. With very high speeds the reaction velocity tends to fall again, but this is probably due to the difficulty of keeping the temperature of the catalyst constant.

The formula previously deduced connecting the partial pressure of the reactant in a mixture with the fractional reaction velocity has been verified for mixtures of alcohol vapour with water, acetone and benzene; and the composition of the adsorbed gas film with the fractional partial pressure of the gases for mixtures of carbon monoxide with oxygen on platinum.

The effect of mixture of vapours on the temperature coefficient of reaction is considered, and equations deduced to show the alteration produced. If the heat of desorption of the reactant is greater than that of the diluent, then the temperature coefficient is diminished, if they are equal there is no alteration, and if less the temperature coefficient is increased. Experiments show that water present in alcohol has little effect on the temperature coefficient of the reaction.

If it is assumed that the series electron of an atom polarises the core, then it has been shown by Born and Heisenberg that the polarisability α of the core in a given state may be calculated from the corresponding term value by means of the approximate formulae, where q is the quantum defect, δν is the difference between the term and the corresponding hydrogen term, R is the Rydberg constant in cm.−1, α1 is the radius of first hydrogen orbit, n is the principal quantum number, k is the azimuthal quantum number, For terms with small quantum defect either of the formulae (1) and may be used, but for terms with large quantum defect (1) gives a higher degree of accuracy.

The view which the author has expressed regarding catalytic activity as due to frozen groups of atoms with strong specific external fields is further tested by a study of the stability of preparations of copper at the instant of formation, and during the subsequent life of the catalyst.

1. Let ℱ be a family of functions X (t) defined in (0, 2π). The functionals to be considered are of a particular class, namely those which define a correspondence between each function X of ℱ and a number which we denote by F(X). It is convenient to refer to F(X) as the functional.

For countable admissible α, one can add a new infinitary propositional connective to so that the extended language obeys the Barwise compactness theorem, and the set of valid sentences is complete α-r.e.

Aside from obeying the compactness theorem and a completeness theorem, ordinary finitary predicate calculus is also truth-functionally complete.

In (1), Barwise shows that for countable admissible A, provides a fragment of which obeys a compactness theorem and a completeness theorem. However, we of course lose truth-functional completeness, with respect to infinitary propositional connectives that operate on infinite sequences of propositional variables. This raises the question of studying extensions of the language obtained by adding infinitary propositional connectives, in connexion with the Barwise compactness and completeness theorems, and other metatheorems, proved for Some aspects of this project are proposed in (3). It is the purpose of this paper to answer a few of the more basic questions which arise in this connexion.

We have not attempted to study the preservation of interpolation or implicit definability. This could be quite interesting if done systematically.

We prove that the Poincaré conjecture implies that the 4-dimensional version of the realization theorem for Whitehead torsion is false. We show that infinitely many examples may be constructed to demonstrate this.

The first examples of Bohr almost periodic functions that are not almost periodic were given by T. -S. Wu. Later, the present author showed that Bohr almost periodic functions could be distal (and not almost periodic) and even merely minimal. Here it is proved that all Bohr almost periodic functions are minimal. The proof yields an unusual feature about the orbit of a Bohr almost periodic function, one which does not characterize Bohr almost periodic functions, but can be used to show that a Bohr almost periodic function f that is point distal must be distal or, if f is almost automorphic, it must be almost periodic. Some pathologies of Bohr almost periodic functions are discussed.

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximize

where κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

In this paper [1], two corrections must be made.

(1) In the second member of (27), m is to be replaced by m + 1; this agrees with (18), and gives the correct result.

(2) Theorem 2 is mis-stated.

A k-3 cap in a three-dimensional Galois space, S3,q, is a set of k points, of which some 3, but no 4 are collinear. It is shown in (2) that for q ≥ 4.