Since the appearance of the article [15] to which this is a sequel, considerable progress has been made in the study of the groups F(r, n) of the title. It is therefore our intention to give a brief account of these developments before proceeding to our main theme, which is to apply the elegant and powerful methods of small cancellation theory to these groups. This has a variety of consequences, perhaps the most important of which is the generalisation to arbitrary r of Lyndon's proof that F(2, n) is infinite for n ≧ 11.

Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.

Integral operators are used to solve the direct and inverse problems of the scattering of acoustic waves by a spherically stratified inhomogeneous medium of compact support. The results are valid for all values of the wave number and an arbitrarily large index of refraction. In the limiting case of small wave number or small inhomogeneities the results are in agreement with those of Rorres and Born.

It is shown that Ф = | grad u |2–uΔu, where u is a solution of Δ2u+pf(u) = 0 in D, assumes its maximum value on the boundary of D. This principle leads one to a lower bound on the first eigenvalue in the non-linear Dirichlet eigenvalue problem and to the non-existence of solutions to this non-linear partial differential equation subject to certain zero boundaryconditions.

This paper is concerned with quasilinear hyperbolic systems of conservation type and establishes two main results. The first is that when a general system is considered in one space dimension and time, then the exceptional nature of a characteristic field implies the coincidence of a shock with one of the characteristics of that field. The second result involves the demonstration by example that quasilinear hyperbolic systems of conservation type may possess solutions that become unbounded after only a finite time, even though they are exceptional.

A new formula is obtained for the number of subgroups of given index in the modular group. The formula is used to prove a recent conjecture on the parity of the number of subgroups.

The group of a harmonic elliptic quartic has exceptional action on the edges of the self-polar tetrahedron, singling out one pair of opposite edges. Geometrical explanations are given. One concerns the possession by each of these two edges of a certain harmonic tetrad constructible in three ways. The same constructions yield, for a general quartic, three distinct tetrads on each edge of its tetrahedron, with a related fourth. The cases of exceptional overlap of these tetrads are examined: they occur for quartics of moduli 0, ∞ and — 32/49.

The Stieltjes transformation is extended to generalised functions both by the direct approach and the method of adjoints, and the resulting extensions are correlated. Inversion formulae are developed, as is the application of fractional integration to these transforms. An integral transformation with a hypergeometric kernel is also briefly considered.

A 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

This paper is concerned with the asymptotic behaviour of solutions of the differential-difference equation w'(s) = g(s)[w(s— 1)—w(s)], where g(s) is a continuous real-valued function. g(s) is assumed to have one of the following asymptotic behaviours: algebraic, exponential algebraic, periodic or zero.

Second-order expressions –d2/dx2+q whose squares are of limit-3 type are constructed. The construction is simpler than others in the literature. The function q can be chosen to be infinitely differentiable and decreasing and to satisfy q(x)≧ –x2(log x)2+ε (x→∞) for a given ε>0.

In the publication [2] we obtained some structure theorems for certain Dubreil-Jacotin regular semigroups. A crucial observation in the course of investigating these types of ordered regular semigroups was that the (ordered) band of idempotents was normal. This is characteristic of a class of semigroups studied by Yamada [5] and called generalised inverse semigroups. Here we specialise a construction of Yamada to obtain a structure theorem that complements those in [2], The important feature of the present approach is the part played by the greatest elements that exist in each of the components in the semilattice decompositions involved.

We determine a set of conditions that are necessary and sufficient for an ordered regular semigroup to be a perfect Dubreil-Jacotin semigroup; and a necessary and sufficient condition for a perfect Dubreil-Jacotin semigroup to be orthodox.

The aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.

The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

Abstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.

The existence of zeroes of an operator in Banach space near a. known zero is examined, when the non-degeneracy condition of Magnus [1] is violated, and replaced by a condition which is called simple-degeneracy. Criteria are given which help determine the number and the structure of curves of zeroes of the operator, passing through the known zero.

These conditions and results are interpreted in terms of bifurcation theory. By considering several different cases, it is shown that the failure of various conditions frequently employed in the Lyapunov–Schmidt method may be overcome using the idea of simple-degeneracy.