The following embedding theorem was proved in the paper (7). Let Π be a set of primes and P, H and K periodic Π-groups satisfying

Then there is a period Π-group J generated by isomorphic copies of H and K such that P ≤ J.

Let π denote a finite group. It is well known that every element of the projective class group K0 ℤπ may be realized as Wall obstruction of a finitely dominated complex with fundamental group π (cf. (13)). We will study two subgroups N0ℤπ and Nℤπ of K0ℤπ, which are closely related to the Wall obstruction of nilpotent spaces. If the group π is nilpotent and if S denotes the set of elements x ∈ K0ℤπ which occur as Wall obstructions of nilpotent spaces, then

It turns out that in many instances one has N0,ℤπ = Nℤπ (cf. Section 3) and one obtains hence new information on S. The main theorem (2·4) provides a systematic way of constructing finitely dominated nilpotent (or even simple) spaces with non-vanishing Wall obstructions.

Any ribbon knot K in S3 can be obtained from the unlink of r + 1 components {U0, U1,…, Ur}, for some positive integer r, by the simultaneous performance of r band operations (fusions), using disjoint bands. These bands may be assumed to run from U0 to U1, U2, …, Ur respectively, and labelled b1, b2…, br accordingly. This paper investigates the question of what conditions on these bands guarantee that the knot K so constructed is non-trivial. Besides being of independent interest (see 1·2 (A) of (2), and the example below), this problem is related to the ‘slice implies ribbon’ conjecture (1·33 of (2)), to the triviality question for 2-spheres in the 4-sphere (section 10 of (4)), and other unknotting questions for surfaces in 4-space (see, for example, 4·30 of (2)). The main result proved here is that K is non-trivial if the union of the bands ‘geometrically links’ (in a sense made precise below) every Ui at most once, and at least one U0 exactly once. The linking of the bands with U0 does not affect the argument, so may be completely arbitrary.

Work by L.S.Pontrjagin(18) and M.W.Hirsch(7) allows us to identify the stable n-stem with the bordism group of oriented compact closed smooth n-manifolds immersed in ℝn+1. In a recent paper (11), U. Koschorke discusses invariants thereby defined on by analysing the self-intersections of immersed manifolds. In particular he discusses the homomorphism

defined by assigning to a generic immersion Mn → ℝn+1 number (modulo 2) of its (n+ 1)-fold points.

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π┴ denote its orthogonal complement. If x ∈ Rn we shall write x = x0 + x′ with x0 ∈ Πand x′ ∈ Π ┴. If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x′ )is defined as the integral of f over the affine subspace Π + x′ with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x′ ) regarded as a function of x′ for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x′) with respect to x′ for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.

It is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)′-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for WM-spaces can be easily deduced.

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined as

where c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).

The method of steepest descents for finding the asymptotic expansion of contour integrals of the form ∫ g(z) exp (Nf(z)) dz where N is a real parameter tending to + ∞ is familiar. As is well known, the principal contributions to the asymptotic expansion come from certain critical points; the most important are saddle-points where df/dz = 0. The original contour is deformed into an equivalent contour consisting of paths of steepest descent through certain saddle-points, the relevant saddle-points. The determination of these is a global problem which can be solved explicitly only in simple cases. The function f (z) may also depend on parameters. The position of the saddle-points depends on the parameters and at a certain set of values of the parameters it may happen that two or more saddle-points coincide. The ordinary expansion is then non-uniform, but appropriate uniform expansions have been shown to exist in earlier work.

A recently discovered abstract relation arising in multiparameter spectral theory is used to unify some of the integral relations and equations associated with the ellipsoidal wave equation.

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matrices

satisfying

and such that

Though extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

In this paper we shall generalize a theorem of Bogoyavl'enskii (2) showing the equivalence of two apparently different families of integrals of the ‘higher stationary Korteweg–de Vries (KdV)’ equations. We recall that the KdV equations form a hierarchy of evolution equations

for an unknown function u(x, t). The equations can be written in ‘Lax form’ (7)

where L is the Schrödinger operator (δ2/δx2) + u, and P+ runs through all the (ordinary) differential operators such that the commutator on the right of (1·1) has order zero.

Boundary effects in the classical bifurcation experiments of fluid mechanics have been the subject of much recent investigation, most notably for us the work of Benjamin(2) on bifurcating steady flows in the Taylor problem in a very short annulus. His paper broaches a fundamental conceptual issue: how does the number of cells in the primary flow, an integer, depend on the length L of the annulus, a real parameter? By the primary flow we mean the flow which develops at high Reynolds number R as the rotation speed is increased gradually from rest, L being held constant (say L = L*). The relevant part of Benjamin's data is presented in Fig. 0·1. For the very short annuli of his experiment the primary flow possessed either two of four cells only. (See Section 1 concerning the bias towards an even number of cells.) If L* > L2, L1 < L* < L2 or L* < L1 the primary flow has 4, 2 or 2 cells respectively; however when L1 < L* < L2 the evolution of the flow as R is increased is not smooth but suffers a jump as R passes the middle of the three intersections of the line {L = L*} with the cusped curve Γ in the figure. (It is now widely accepted that the development of cells in the flow occurs over a range of Reynolds numbers; this point is not the issue here.) The curve Γ divides the plane into two regions such that there are one or two stable, experimentally observable flows according as (R, L) lies in region I or region II, respectively. We may refer to the two stable states in region II as 2 cell or 4 cell, although their cellular structure is only partially developed at the moderate values of the Reynolds number occurring in the figure. One or the other of the two flows in region II loses stability across Γ, giving rise to the possibility of jumps when, as above, (R, L) crosses the boundary moving from region II to region I. In general if (R, L) is varied quasi-statically along a path which begins in region I, passes through region II, and re-enters region I, a jump will occur on re-entry if and only if the point H lies between the two points of Γ where the path crosses the boundary (between in the sense of distance along Γ).

The bifurcation problem of an infinite elastoplastic medium surrounding a spherical cavity and subjected to uniform radial tension or compression at infinity is studied. The material is assumed to be incompressible, and its behaviour is modelled by both hypoelastic (flow theory) and hyperelastic (deformation theory) constitutive relations. No bifurcation was found with the flow theory. Surface bifurcation modes were discovered with the deformation theory in both tension and compression. An independent study is also presented of surface bifurcations of a semi-infinite elastoplastic material under equi-biaxial stress. The critical strain for the half-space coincides with the strain at the spherical cavity at the lowest bifurcation.

Bifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force.

At fixed axial extension (a) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; (b) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; (c) an asymmetric or ‘wobble’ mode is always possible before either (a) or (b) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders.

At fixed axial force the results are similar to (a), (b) and (c) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.