In this paper we prove the existence of classical solutions for all t ≧ 0 for parabolic equations u′ + A(t)u = –f(u, ∇y, …, ∇2m–2u) of arbitrary order. 2m is the order of the elliptic principal part. f must satisfy some monotonicity and growth conditions. Moreover, similar results are also valid for semilinear elliptic equations.

Groups having two generators and two relations are studied. The Reidemeister-Schreier algorithm is used to determine presentations for their derived groups. This enables the orders of the groups to be found. Necessary and sufficient conditions are given for the groups to be metabelian. Certain classes closely related to the class are also discussed.

In answer to a question raised by John Barrett in 1961, conditions are established under which the existence of a systems-conjugate point of a fourth-order equation also assures the existence of a 2–2 conjugate point. These results lead to new conjugacy criteria and apply to certain non-selfadjoint equations more general than those previously considered.

Let X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

In this paper we show that some of Bass' results on the normal structure of the stable general linear group can be extended to infinite dimensional linear groups over non-commutative Noetherian rings.

The inequality considered is

where p and q are given real-valued coefficients on the interval [a, b), with b ≦ ∝, of the real line; here D is a linear manifold of the Hilbert function space L2(a, b), and μ is a real number characterised in terms of the spectrum of a uniquely determined self-adjoint differential operator in L2(a, b).

In a previous publication [1] we considered the structure of a fragmented ring and of its multiplicative semigroup. In the present publication we consider, in a purely semigroup-theoretic context, some ideas related to those in [1]. As shown by an example, the semigroups we consider need not be the multiplicative semigroup of a ring.

A new proof of the Holley-Preston generalisation of the Fortuin-Kastelyn-Ginibre inequalities is given, and Batty's extension to the case of infinite products is discussed briefly. An application of the theorems in combinatorial probability theory is described.

The publication of the results in this paper has been delayed, for non-mathematical reasons. However, the author has given lectures on these results in Smolenice [2], Gainesville, Tulane. Riverside (1971), Milan (1972) and in Paris. The main consideration in this paper is the notion of a fragmented ring and its multiplicative semigroup. A fragmented ring is a ring with an identity having a finite set of idempotents, these commuting and therefore being central. In a subsequent paper we shall consider associated ideas in a purely semigroup-theoretic context and, in so doing, point out some differences between general semigroups and those semigroups that are the multiplicative semigroups of rings.

In this paper we study the propagation of weak discontinuities in quasi-linear hyperbolic systems of equations with discontinuous coefficients when one or more speeds of propagation o f the discontinuity wave is coincident with the speed of propagation of the strong discontinuity.

Existence and uniqueness theorems are established for dual trigonometric equations having right-hand sides that are given functions of bounded variation. The first equation in each pair has coefficients, say {Jn(n + h)} or (jn(n + h – ½)}, and the second equation coefficients {jn)}, where h is a nonnegative constant. A potential problem involving mixed boundary conditions of first and third kind is associated with each dual series. The potential problem is analysed using a stepwise perturbation procedure involving solutions in powers of h. The analysis demonstrates that the present dual series problem can be resolved if the dual series problem associated with the case h = 0 is solvable, the latter being a result obtained earlier.

We consider the set S(f) of λ-values for which there exists a function g ∊ L(t0, ∞) such that

has a solution not tending to 0; here f is a fixed function which is positive, non-decreasing and tends to ∞ with t. It is shown that if the jumps of logf(t) at its discontinuities are uniformly bounded, then S(f) is an additive group. This group is determined in some cases; some related groups are noted, which may coincide with S(f).