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.

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.

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.

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.

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

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.

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.

Let f, g be two functions of two Besov spaces (or Sobolev spaces), we look for the Besov spaces to which the product f × g belongs so that the multiplication is a continuous mapping.

Many known results about the stability of selfadjointness are extended to results about the stability of the deficiency index of closed symmetric operators on Hilbert space under perturbation.

Let D be an open connected subset of the open unit ball in Ṟd, d ≧ 2. We give an estimate of the harmonic measure of ∂D∩{|x| = 1} with respect to D. This estimate depends in a simple way on the geometry of D. An essential tool is a rearrangement theorem for differential inequalities. When d = 2, examples are given which illustrate the precision of the results.

The paper deals with boundary value problems for second-order vector differential equations x″ = f (t, x, x′). Given a region Ω in (t, x)-space we ask whether there exists a solution x(t) of the problem satisfying (t, x(t)) ∊Ω. We arrive at a rather general type of conditions which are sufficient in order that Ω has the desired property. One of these conditions is geometric in nature and depends upon the boundary data only. The second condition can be expressed in terms of inequalities and depends upon the values of f on ∂Ω. These inequalities turn out to be the common background of a variety of conditions which can be found in the literature on boundary value problems and which in the case of a scalar equation reduce to the well-known properties of upper and lower solutions.

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.