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.

We clear the decks for an attack on the problem of the breadth and class of a finite p-group by exhibiting some examples with properties relevant to breadth. These were constructed by a mixture of hand computation, machine search and serendipity.

This paper gives simple proofs of two integral inequalities recently proved by Brodlie and Everitt[1].

Let T be the formally self-adjoint second-order elliptic differential expression

in ℝn, where the coefficients bj, ajk, q are real-valued and ajk ≈ akj. In this paper sufficient conditions for all positive integer powers Tm of T to be essentially self-adjoint on are obtained.

A scheme devised by Chandrasekhar for investigating the transformations between various differential equations of the second order governing perturbations of the Schwarzschild black hole demands further investigation. The transformation between two differential equations in normal form is considered, and a wide survey of the properties of the transformation is given. It is shown how Chandrasekhar's equations fit into the scheme, after which some examples with particular properties are considered. A detailed investigation of Bessel's equation is undertaken using various devices, in particular by employing asymptotic methods for products of Bessel functions, and employing matrix methods for dealing with large numbers of matrix equations which necessitates an interesting method of solution, the results being reinterpretations of the standard recurrence relations for Bessel functions.

In 1932, Hardy and Littlewood [1] proved the inequality

The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where

y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)

and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

In this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

This paper studies the boundary-value problem arising from the behaviour of a fluid occupying the region 0 ≦ x ≧ 1 between two rotating discs, rotating about a common axis perpendicular to their planes, when the discs, are rotating in the same sense with speeds 0 ≦ Ω0<Ω1. The equations which describe the axially symmetric similarity solutions of this problem are

with the boundary conditions

where ε = v/2Ω1 and v is the kinematic viscosity.

The major result is: There is an ε0 such that for 0<ε≦ε0 there does not exist a solution 〈H(x,ε), G(x,ε)〉 with G′(x,ε)≧0.

We consider the generalisation from central series to marginal series in groups and set up firstly various basic results. The main section of the paper is concerned with the study of which group theoretical properties may be transferred from a marginal factor in a group to the corresponding verbal subgroup and which properties may be transferred from one factor of a lower marginal series to successive factors of the series.

We study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

Spectral properties of the singular Sturm-Liouville equation –(p−1y′)′ + qy = λry with an indefinite weight function r are studied in . The main tool is the theory of definitisable operators in spaces with an indefinite scalar product.

The Weyl limit-point, limit-circle nature of the equation y″(x)–q(x)y(x) = 0(0≦ x< ∞) is analysed When q(x) has the form q(x) = xαp(xβ), where α and β are positive constants and p(t) is a continuous periodic function of t.

Some continuous dependence theorems are presented for classical solutions of initial-boundary value problems in viscoelastic materials which occupy the exterior of a bounded domain in Euclidean three space. Two broad classes of anisotropic viscoelastic materials are considered. The theorems are proved by a combination of the Protter and the Graffi methods.

Using Hilbert space methods, existence and uniqueness are proved for the solution of some strongly non-linear partial differential equations of elliptic and parabolic type.

They are associated with quasi-linear operators of the form: -div(β(x, grad u)) + β0(x, u) where β (resp β0) is a maximal monotone subdifferential on ℝN(resp ℝ) depending smoothly on x in a bounded domain Ω of ℝN

These operators are shown to be the subdifierentials over Lp(Ω) of convex functional of the following type:

where j is a normal convex integrand over Ω×ℝN+1 satisfying a coerciveness condition.

This method avoids the theory of Sobolev-Orlicz spaces. An application is given also forthe gas-diffusion equation over ℝ+.

The existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.