We introduce bow varieties and construct some ALF spaces as bow varieties.

In this paper, the Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included are the two-relator groups introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology computations and a classification of finite subgroups. The methods are geometric; technical tools include the original theorems of Magnus and Lyndon, as well as an amalgamation technique due to J. H. C. Whitehead.

Although the existence of an electromotive force in apparently homogeneous circuits under asymmetrical temperature distributions has been questioned by only a minority of workers in this field, all attempts at explaining the effect have been little more than speculations. These speculations may be divided roughly into two types, first(2), those which agree with the contention of Benedicks that the E.M.F. is an essentially new effect not explicable in terms of the classical theory of Kelvin, and secondly(3), those which contend that the effect is essentially spurious, to be explained away as due either to hidden sources of error, or to unsuspected heterogeneities in the circuit. The somewhat fantastic claim of Benedicks' school that the effect was an inverse of the Thomson effect, which proved entirely false upon examination in the light of Sommerfeld's work(4), did much to increase scepticism. Benedicks' later claim to have found yet another inverse of his original effect(5), the “electro-thermal” effect, has also been refuted(6), and this does not inspire confidence in the genuine nature of the original effect.

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

In ([7], p. 181, problem 7) Maury and Raynaud pose the following question: ‘Let J be a ring and G a group such that the group ring J[G] is an order in a quotient ring Q; when is J[G] a maximal order in Q?’ The question is interesting for two reasons. In the first place, the analogous question for universal enveloping algebras of finite-dimensional Lie algebras has been settled very satisfactorily by Chamarie([2], corollaire 2·3·2). Secondly, it has been pointed out by several authors that maximal orders have some very desirable properties – see for example [3], [4], [6], [7] and [12].

We extend a number of known criteria for normality of analytic and harmonic functions to the setting of solutions to elliptic partial differential equations. Some of the results hold for monotone Sobolev functions. We also discuss the boundary behaviour of monotone Sobolev functions.

Schauder bases in Banach spaces are studied in [5].

In ordered Banach spaces a special type of Schauder bases, the O.P. Schauder bases, are studied because then the properties of ordered spaces can be used.

We study here the connections between the well known Puppe-Verdier notion of triangulated category and an abstract setting for homotopical algebra, based on homotopy kernels and cokernels, which was expounded by the author in [11, 13[.

We show that a right-homotopical category A (having well-behaved homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated structure with regard to the suspension endofunctor σ, called σ-homotopical category. If A is homotopical and h-stable (in a sense related to the suspension-loop adjunction), this structure is also h-stable, i.e. satisfies ‘up to homotopy’ the axioms of Verdier[29[ for a triangulated category, excepting the octahedral one which depends on some further elementary conditions on the cone endofunctor of A. Every σ-homotopical category can be stabilized, by two universal procedures, respectively initial and terminal.

In this paper every space will be 2-local, for example BO will mean the 2-localization of the space usually denoted BO.

Let f1, …, fs: ℝn+1→ℝ be homogeneous polynomials, and let

The Euler characteristic χ(X) is an important topological and geometric invariant of X.

In [2], chapter S, and also in the ATLAS ([1], p. 131), we give two presentations for 3 · Suz: 2, the normalizer of a certain 3-element in Conway's largest sporadic simple group Co1. These presentations extend naturally to presentations for groups having Co1 × 2 as image, leading to conjectured presentations for Co1 × 2 and Co1 (see [1], p. 183, although the form of one of the presentations there is slightly different from what we give here).

The Born partial wave integrals of Seaton [3] are considered. It is shown that the treatment of Whelan [5] may be further simplified and that these integrals may be expressed as a finite sum of associated Legendre polynomials of the second kind. This provides a convenient method for their numerical evaluation and allows the relationship between the Born and Bethe reactance matrices to be given in explicit analytic form.

Let G be a hyperfinite locally soluble group and let A be a noetherian ℤZ;G-module. In [2], we proved that A is the direct sum of a ℤZ;G-submodule Af each of whose irreducible ℤZ;G-module sections is finite and a ℤZ;G-submodule each of whose irreducible ℤZ;G-module sections is infinite. In this paper we study the structure of the ℤZ;G-submodules Af and . Our main result gives a complete description of Af.

There are two effects involved in the sucking action of the diffusion pump. One is the penetration of gas molecules into the mercury vapour jet, and the second is the existence of a gradient of the partial pressure of the gas in the direction of the jet, so that the pressure at the origin of the jet is much lower than at the end of it.

There are many constructions of the Suzuki groups in the literature (see for example Suzuki's original paper [3], as well as [1, 2, 4]), and one needs to make a strong case to justify publishing another. Yet I believe the construction below is sufficiently new and sufficiently elementary that in time it will come to be regarded as the standard construction.

The motion of a solid in a viscous conducting fluid permeated by a magnetic field has been the subject of a number of papers. Using the Stokes approximation Chester (3) first derived an expansion in powers of the Hartmann number for the drag on a sphere translating uniformly through a fluid, parallel to the applied magnetic field. In a subsequent paper (4) the same author treated the high Hartmann number limit. Later, Chang (2) considered, in the low Hartmann number regime, the motion of an axisymmetric body translating slowly along the axis of a fluid filled tube, and was able to calculate a drag formula which includes the lowest order wall effect term. An alternative treatment of this class of problems has recently been given by Williams (13), using an intergral equation formulation.

In area theory, Hausdorff measure has been mainly used in the form of Hausdorff ‘spherical’ measure (l) in spite of the fact that work on the geometry of sets of points has mainly used Hausdorff ‘convex’ measure (2,4). The reason for this lies in the inequality This inequality is known (1,3) when the measures are interpreted aslspherical’. But as has for some time been conjectured and this paper shows, with the proper normalization it is false for the convex case.

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.

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.