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.

Here eigenvalue problems A(λ)u = 0 and their approximations Ai(λ)νi = 0 are studied where the densely denned closed semi-Fredholm operators A and Ai depend holomorphically on the parameter λ. Two different kinds of approximations are established. One is based on a generalisation of the spectral projection and the other on a suitable linearisation of the problem. To this end generalised eigenvectors and suitable product spaces are introduced which provide a representation formula for the principal part of A -1 and A -1, respectively, in the neighbourhood of poles. The convergence of the methods is shown in the framework of discrete convergence theory. The results generalise the corresponding results for linearly dependent A, Ai in two directions: both methods are available for arbitrary holomorphic dependence on the parameter λ, and the first method provides convergent approximations to the whole generalised eigenspace which works also in cases of linear parameter dependence when the usual method fails.

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.

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.

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.

In the standard treatment of the harmonic solutions of Duffing's equation with small non-linearities and small forcing, all small parameters are assumed to be common multiples of some small parameter. As a consequence, the parameters do not vary in a full neighbourhood of zero and the bifurcation surfaces are not obtained. It is the purpose of this paper to give a complete description of the number of harmonic solutions for the parameters varying in a full neighbourhood of the origin in the parameter space.

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.

It is shown that exactly six non-isomorphic distributively generated near rings can be defined on the infinite dihedral group. One has the null multiplication, two have trivial multiplications in which the group of left annihilators has index two, and three are distributive and have their product sets equal to a subgroup of order 2.

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).

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.

Let S and T be formally symmetric ordinary differential operators defined on a real interval I. It is assumed that the order of S is constant and everywhere strictly higher than the possibly varying order of T. The main result of this paper (Theorem 2.3) gives necessary and sufficient conditions for maximality of the deficiency indices of the differential relation Su = Tv considered in a Hilbert space with a scalar product which is a Dirichlet integral (see section 2) belonging to S. The conditions generalise those given in [5] for less general choices of operators S and T. For certain choices of Dirichlet integral they are explicit integrability conditions on the coefficients of the Dirichlet integral and the operator T.

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 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.

Let (1) x′ = f(t, x) be any differential equation and S0 the set of solutions of (1) with open domain. It is known that for every g ∊ S0 a non-continuable (= saturated) ∊ S0 exists which is an extension of g. Usually is represented in the form is a sequence in S0 defined by some sort of a variant of what is called ‘recursive definition’ in set theory. It will be shown that a function

exists (P(S0) is the power set of S0) such that the above-mentioned variant can be given the form: There exists a sequence in S0 such that

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.

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.