An elementary theory of the diffraction of light by supersonic waves was put forward by Sir C. V. Raman and myself (4) to explain many of the important features of the phenomenon observed by Bär and others. Recently Mueller (2) pointed out the essential idea that the optical diffraction effects in solids may be interpreted on the basis of this theory if due account is taken of the photoelastic effects arising from the periodic strains caused by the sound waves. He was thus able to give an explanation of the results of Schaefer and Bergmann (5, 6). Mueller's idea, however, required a still further refinement, since it did not consider the phase relationships between the differently polarized components of the diffraction orders. This refinement has, however, been recently incorporated by Mueller and myself (3). This led us to offer an explanation of the diffraction effects observed by Hiedemann and Hoesch (1) in glass blocks.

1. The set ℰ (X) of homotopy classes of self-homotopy equivalences of a space X forms a group under composition. In this note I complete the calculation of ℰ (X) when X is a torsion free rank 2 h-space, and give ℰ (X) up to extension for the remaining principal S3 bundles over S7.

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

The characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.

There are spectra P−k constructed from stunted real projective spaces as in [1] such that H*(P−k) is the span in ℤ/2[x, x−1] of those xi with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse system

which is similar to those of Lin ([15], p. 451). It is a corollary of Lin's work that there is an equivalence of spectra

where holim is the homotopy inverse limit ([3], ch. 5) and Ŝ–1 the 2-adic completion of a sphere spectrum. One may denote by this holim (P–κ), although one must constantly keep in mind that , but rather

The method of steepest descents is widely used to find the asymptotic expansion of contour integrals of the form

where N is a large positive parameter, and f(z) and g(z) are analytic functions of z. (For the sake of simplicity it will here be supposed that f(z) and g(z) are regular except at ∞, but our arguments can be extended to include singularities at other points.) In the method of steepest descents the path of integration is deformed into a new equivalent path consisting of the paths of steepest descent through a and b, together with paths of steepest descent through certain saddle points, the relevant saddle points. (Paths of descent, etc., are defined in §1 below.) Watson's Lemma then shows that the complete asymptotic expansion depends on the local behaviour of f(z) and g(z) at the end-points and at the relevant saddle-points. On the other hand to determine which saddle-points are relevant is a global (non-local) problem and requires either the tracing of paths of steepest descent all the way from saddle-points and endpoints to ∞, or some other global process. It has often been noted that in applications of Watson's Lemma the paths of steepest descent can be replaced by a wider class of paths of descent. In this way we are led to global problems like the following: Given 2points z = α and z = β where Nf(a) > Nf(β), can apath of descent be drawn from α to β? If also Ff(α) = Ff(β), can a path of steepest descent be drawn from α to β?

By measuring the resistance of a phosphor bronze wire in thermal equilibrium via various substances with crystals of iron ammonium alum it is shown that the time for thermal equilibrium between the ionic magnets of the salt and its lattice vibrations is less than 0·5 sec. for all temperatures above 0·025° T*. When liquid helium or a german silver tube forms part of the cooled portion of the apparatus, the time for equilibrium is increased to a few seconds for temperatures below 0·4° K. It appears possible that the thermal conductivity of german silver is less than 10−8 cal. cm.−1 sec.−1 degree−1 below 0·05° T*, and there are indications that the thermal conductivity of liquid helium at temperatures below 0·3° K. is small compared with its value at 2° K.

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.

A central role in the theory of smoothing combinatorial manifolds is played by the Cairns–Hirsch Theorem, which may be expressed (in a weak form) as follows:

If M is a combinatorial manifold and if M × R has a differentiable structure a, compatible with its combinatorial structure then M has a differentiable structure λ, such that (M × R)α is diffeomorphic with Mλ × R.

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number c4(K). In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.

We investigate, for a locally compact group $G$, the operator amenability of the Fourier–Stieltjes algebra $B(G)$ and of the reduced Fourier–Stieltjes algebra $B_r(G)$. The natural conjecture is that any of these algebras is operator amenable if and only if $G$ is compact. We partially prove this conjecture with mere operator amenability replaced by operator $C$-amenability for some constant $C\,{<}\, 5$. In the process, we obtain a new decomposition of $B(G)$, which can be interpreted as the non-commutative counterpart of the decomposition of $M(G)$ into the discrete and the continuous measures. We further introduce a variant of operator amenability – called operator Connes-amenability – which also takes the dual space structure on $B(G)$ and $B_r(G)$ into account. We show that $B_r(G)$ is operator Connes-amenable if and only if G is amenable. Surprisingly, $B({\mathbb F}_2)$ is operator Connes-amenable although ${\mathbb F}_2$, the free group in two generators, fails to be amenable.

If Γ is a discrete subgroup of PSL(3, ℂ), it is determined the equicontinuity region Eq(Γ) of the natural action of Γ on ℙ2ℂ. It is also proved that the action restricted to Eq(Γ) is discontinuous, and Eq(Γ) agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of Γ in the sense of Kulkarni, Λ(Γ), contains at least three complex lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which Γ acts discontinuously. Moreover, if Λ(Γ) contains at least four complex lines and Γ acts on ℙ2ℂ without fixed points nor invariant complex lines, then each connected component of Eq(Γ) is a holomorphy domain and a complete Kobayashi hyperbolic space.

Measurements have been made of the positive ion currents emitted by tungsten at temperatures between 3000°K. and 3200°K. The results are in general accordance with the values calculated on the basis of the Saha equation from the rate of evaporation of neutral atoms, the electron work-function and the ionisation potential of tungsten. The “work-function” associated with the ionic evaporation appears to lie between 10 and 11 electron-volts.

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measure

and which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided that

Here the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

We study equilibrium states in relativistic galactic dynamics which are described by stationary solutions of the Einstein–Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses.

If X is a uniformizable space any uniform structure compatible with its topology will give a uniform structure on the collection H(X) of all non-empty closed sets of X ((1), Section 2, Exercises 7, 8, definition 1·6): for each entourage U ∈ we define an entourage U on H(X) by putting U = {(A, B); A ⊂ U(B) and B ⊂ U(A)}. It has been conjectured ((6), p. 35, Exercise 17) that if 1 and 2 are distinct uniform structures on X, 1 and 2 induce indifferent topologies on H(X). Unfortunately the method proposed in (6) fails, even for metrizable structures, as we shall show by an example. We shall, however, establish a number of partial results tending to support the conjecture; in particular we shall establish the result when both the uniform structures are metrizable, or when one is precompact.

Introduction. The object of this paper is to prove some formulae for Jacobi polynomials. The formulae (6), (7), (9), (10) and (13) are believed to be new.

Keller and Rubinow(l) have considered the force on a spinning sphere which is moving through an incompressible viscous fluid by employing the method of matched asymptotic expansions to describe the asymmetric flow. Childress(2) has investigated the motion of a sphere moving through a rotating fluid and calculated a correction to the drag coefficient. Brenner(3) has also obtained some general results for the drag and couple on an obstacle which is moving through the fluid. The present paper is concerned with a similar problem, namely the axially symmetric flow past a rotating sphere due to a uniform stream of infinity. It is shown that leading terms for the flow consist of a linear superposition of a primary Stokes flow past a non-rotating sphere together with an antisymmetric secondary flow in the azimuthal plane induced by the spinning sphere. For a3n2 > 6Uv, where n is the angular velocity of the sphere, U the speed of the uniform stream, and a the radius of the sphere, there is in the azimuthal plane a region of reversed flow attached to the rear portion of the sphere. The structure of the vortex is described and is shown to be confined to the rear portion of the sphere. A similar phenomenon occurs for a sphere rotating about an axis oblique to the direction of the uniform stream but the analysis will be given in a separate paper.

We are using the term ‘bivariate Poisson process’ to describe a bivariate point process (N1(.), N2(.)) whose components (or, marginal processes) are Poisson processes. In this we are following Milne (2) who amongst his examples cites the case where N1(.) and N2(.) refer to the input and output processes respectively of the M/G/∈ queueing system. Such a bivariate point process is infinitely divisible. We shall now show that in a stationary M/M/1 queueing system (i.e. Poisson arrivals at rate λ, exponential service at rate µ > λ, single-server) a similar identification of (N1(.), N2(.)) yields a bivariate Poisson process that is not infinitely divisible.