The purpose of this paper is to give two characterizations of scalar-type spectral operators.

A radiating point source is abruptly switched on and, thereupon, constantly maintained while immersed within a multi-dimensional, non-dispersive, originally uniform, unbounded, propagative medium with transverse isotropy, equivalent to an intrinsic axisymmetry. The number of spatial dimensions concerned is odd. The pertinent radiation problem is posed as a partial differential equation coupled with a preactivation zero condition. This problem is, first, resolved by a Fourier integral, which is then ‘frozen’ onto (i.e. evaluated at points on) the axis of symmetry. Postulating appropriate hyperbolic-type as well as elliptic-type conditions, a systematic reduction of the ‘frozen’ integral is carried out, enabling an analytic exact solution to be constructed with reference to certain phase curves in a phase plane. Away from the source, this solution is valid throughout the symmetry axis except at certain discrete, moving singularities. These fall into three different categories. The behaviour near each typical singularity is estimated. Three significant physical features are demonstrated (as corollaries); namely, the implicit compliance with a radiation principle, the linear interception of the symmetry axis by a Petrow sky's lacuna (associated with the instantaneous point impulse), and, the eventual attainment of either a steady or quiet state after an infinite period. Interpretations are provided in terms of the group velocity. To assist in probing into these various aspects, two appendices are separately included. An application is made to consider magnetically aligned, radiation reception within a field permeated, electrically conducting gas.

The foregoing paper by Dr Atkinson relates to the new formula for the rate of accretion of mass by a star that has recently been proposed by us. The derivation of this formula was made by means of a dynamical argument, certain physical aspects having to be neglected.

In a recent paper [7], Erdmann has calculated Ext1G between Weyl modules for SL2. In this paper we generalize this result to solve the corresponding problem for quantum GL2 as defined by Dipper and Donkin in [2]. We also show how our result also holds for the Manin quantization. To apply the methods of [7], it is necessary to determine the block structure of quantum GL2, so the first main result of this paper is a description of this, derived from the analysis of the subcomodule structure of the symmetric powers in [10].

After an initial section of generalities, the next section consists of the determination of the block structure. We also need a quantum analogue of two short exact sequences from [11], which we give in the following section. With these results, the argument now follows much as in [7]; we consider the infinitesimal case, and then use the Lyndon–Hochschild–Serre spectral sequence to obtain the desired result. Finally we show how the result also holds for the Manin quantization.

It should be noted that the result here uses the classical case, so is not independent of that in [7]. The only real difference in the arguments used occurs in Lemma 4·8 where the original methods do not generalize, so we use a more direct argument. There is also an unfortunate typographical error in the statement of the main result in [7].

1. Prof. Fekete, in a letter to the author, wrote: “…Satz von Myrberg über den Zusammenhang zwischen dem transfiniten Durchmesser und dem logarithmischen Mass zugehörig zum Exponenten 1 + ε, ε > 0 (vgl. Satz 10 in seiner Acta Arbeit).…Ich würde einen elementaren Beweis für den letzteren Satz sehr begrüssen.” The object of this note is to supply such a proof but, since the concepts involved may not all be matters of common knowledge, I shall begin by outlining the necessary definitions, the relevant known results, the nature of the theorem and its relation to what is known.

For a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.

Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG(M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.

In the situation when all types in SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.

In developing the theory of adsorption taking into account the interaction between adsorbed particles it has been usual to use a physical model in which it is assumed that there is a fixed interaction energy between particles adsorbed on neighbouring sites on the surface. In this paper the differences between the behaviour of this model and that of actual surfaces are discussed by considering a one-dimensional film in which the potential energy of a single adsorbed particle varies continuously and periodically with its position on the surface and in which there is a repulsive force between adsorbed particles which varies with the distance between them according to an inverse power law. For such a physical model the variation of the heat of adsorption with the fraction of the sites occupied is considered in detail and it is shown in particular that there is much less difference between the behaviour of mobile and immobile films than is indicated by the earlier model in which a fixed interaction energy is assumed. These results are considered in connexion with the interpretation of experiments on the adsorption of hydrogen on tungsten.

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into the stream, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary so that a method lately developed by N. Muschelišvili can be used in solving the biharmonic equation; a simple expression in finite terms is found for the stream function ψ1. Fig. 1 shows the section ABC of the fixed boundary of the liquid, the equation of the curve ABC is given in § 2, and ψ1 in § 3.

1. In this paper we consider the conditions at the surface of a semi-infinite elastic body due to the action of an external force applied, in a direction at right angles to the surface, to the interior of the body.

In 1914 Whittaker(12) conjectured that the Heun differential equation is the simplest equation of Fuchsian type whose solution cannot be represented by a contour integral; instead the nearest approach to such a solution is to find a homogeneous integral equation satisfied by a solution of the differential equation. In this paper we reconsider Whittaker's conjecture and show that in fact solutions of Heun's equation can be represented in terms of contour integrals, similar to those of Barnes for the hypergeometric equation. The integrands of these integrals are of a rather complicated nature and cannot be said to involve known or simpler functions although they do provide expressions for the analytic continuation of Heun functions analogous to those for the hypergeometric functions.

In the “special” or “restricted” theory of relativity, for which the line-element ds of the “world” of space-time is given by , the geodesics of the world are straight lines, and the null geodesics (i.e. the geodesics for which ds vanishes) are the tracks of rays of light. When Einstein discovered the “general“ theory of relativity, in which the effects of gravitation are taken into account, he carried over this principle by analogy, and asserted its truth for gravitational fields; it was, in fact, the basis of his famous calculation of the deviation of light at the sun. The law was, however, not proved at the time: and indeed there is the obvious difficulty in proving it, that strictly speaking there are no “rays” of light—that is, electromagnetic disturbances which are filiform, or drawn out like a thread—except in the limit when the frequency of the light is infinitely great: in all other cases, diffraction causes the “ray” to spread out over a three-dimensional region.

Let X denote a Banach space over the complex field ℂ and let B(X) be the Banach algebra of all bounded linear operators on X. If T ε B(X), we write n(T) = dim ker (T) and d(T) = codim T(X). Suppose that Y is a subspace invariant under T; then TY will denote the restriction of T to Y and Y the operator on X/Y defined by

Y: x/Y →(Tx)/Y

The diffraction of a plane harmonic electromagnetic wave by two parallel, perfectly conducting planes of infinite width and finite length is considered. The wave is assumed to be polarized perpendicular to the edges of the plane, and thus the problem can also be regarded as that of diffraction of a plane sound wave. Two integral equations are obtained, and the equations are solved approximately when the length of the planes is large compared with the wave-length. The resonance of the system is also considered and an equation obtained for the resonant lengths. This equation is solved by an approximation depending on the assumption that twice the product of the wave number (k) and the resonant length (l) is always large.

Erdös has asked whether the plane ℝ2, or more generally n-dimensional Euclidean space ℝn, can be partitioned into countably many sets none of which contains the vertices of an isosceles triangle. Assuming the Continuum Hypothesis (CH), Davies[2] (for n = 2) and Kunen[10] (for arbitrary n) proved that such partitions exist. Assuming Martin's Axiom, Erdös and Komjáth proved in [5] that such partitions exist for n = 2. We will prove here, without additional set-theoretic hypotheses, that there are such partitions in all dimensions.

Let ‖x‖ denote the usual Euclidean norm of a point x ∈ ℝn, so that ‖x − y‖ is the distance between x and y.

In this paper, we consider the problem of diffraction of two-dimensional sound pulses by a homogeneous fluid circular cylinder contained in another homogeneous fluid. The line source is situated outside the cylinder and is parallel to its axis. It is supposed that the velocity of sound inside the cylinder is less than the velocity of sound in the surrounding medium. We investigate the problem by the method of dual transformation as developed by Friedlander. The pulse propagation modes both inside and outside the cylinder are obtained, We interpret the modes as diffracted pulses in terms of Keller's Geometrical Theory of Diffraction. The results agree with Friedlander's conjecture.

The dispersion equation describing the character of the equilibrium of two viscous incompressible superposed fluids under the action of gravity is considered in detail. Approximations for extreme wave-numbers are obtained and exact numerical results for particular cases are given. The existence of certain secondary modes is also considered.

In the spirit of generalized coordinates in Lagrangian mechanics, the finite deformation of a material element is considered to be specified by arbitrary geometric variables, and the associated generalized stresses are generated by work conjugacy. The comparative quantitative influences of such coordinate choices are investigated for the main state variables in anisotropic thermoelasticity, namely the moduli, compliances, thermal coefficients of expansion, pressure coefficients, and specific heats. Transformation formulae are obtained to connect the respective values of the state variables for different choices. Special attention is given to the identification of coordinate-invariant magnitudes and relationships. The entire analysis is carried farther for coordinates in a class of tensor measures of finite strain; such measures are used widely in current theories of the mechanical response of solid materials, but apparently not yet in regard to thermal phenomena.

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U and

It is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).

The study of electrostatic effects on the motion of fluids goes back to the work of Rayleigh (l) who discussed the effect of surface charges on the vibration of spherical drops, and Bassett (2) who studied the effect of a radial electric field on the stability of a jet. The subject has recently been taken up again by Melcher (3) and Crowley (4), and the author became interested in the subject through the monograph of the former author in which a wide variety of wave problems are described.

The main purpose of this note is to show some connexions between Legendre polynomials and random walks. More specifically we shall be concerned with what may be called ‘trinomial’ random walks i.e. walks on the points 0, ± 1, ± 2, …, such that the probabilities of ‘currently’ taking steps of – 1, 0, + 1 are p–1, p0, p1.