We give a definition of “quasispectral maximalč subspaces for a quasinilpotent, but not nilpotent, bounded Banach space operator. The definition applies to a class of operators, close to the Volterra operator.

A Titchmarsh-Weyl matrix function W(λ) is defined for the differential equation of order 2n

with po>0, pk≧0, k = 1, 2, …, n on 005B;0, b), λєℂ and an indefinite weight function r. It is shown that this function W(λ) belongs to some class and that some operators associated with the above equation are definitizable in the Krein space . In the particular case n = 1, these results are contained in an earlier paper by the present author and H. Langer.

This paper is a sequel to [1-4]. We consider the problem of G-closure, i.e. the description of the set GU of effective tensors of conductivity for all possible mixtures assembled from a number of initially given components belonging to some fixed set U. Effective tensors are determined here in a sense of G-convergence relative to the operator ∇· D · ∇, of the elements DeU ∈ [5, 6].

The G-closure problem for an arbitrary initial set U in the two-dimensional case has already been solved [3, 4]. It remained, however, unclear how to construct, in the most economic way, a composite with some prescribed effective conductivity, or, equivalently, how to describe the set GmU of composites which may be assembled from given components taken in some prescribed proportion. This problem is solved in what follows for a set U consisting of two isotropic materials possessing conductivities D+ = u+E and D− = u−E where 0<u−<u+<∞ and E ( = ii+jj) is a unit tensor.

We give two generalizations of a theorem of L. Amerio and G. Prouse concerning uniqueness of the almost-periodic solution of the wave equation with a local multivalued damping term and almost-periodic forcing.

Counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped. They also show that the two generalizations are relatively independent.

Let T, V1,…, Vk denote compact symmetric linear operators on a separable Hilbert space H, and write W(λ) = T + λ1V1 + … + λkVk, λ = (λ1, …, λk) ϵ ℝk. We study conditions on the cone

related to solubility of the multiparameter eigenvalue problem

with W(λ)−I nonpositive definite. The main result is as follows.

Theorem. If 0 ∉ V, then (*) is soluble for any T. If 0 ∈ V, then there exists T such that (*) is insoluble.

We also deduce analogous results for problems involving self-adjoint operators with compact resolvent.

The result that any monoid is isomorphic to some endomorphism monoid is generalized to super-associative systems.

The implication of Dirac's large numbers hypothesis (LNH) that there are two cosmological spacetime metrics, gravitational (E) and atomic (A), is used to formulate the gravitational laws for a general mass system in atomic scale units within such a cosmology. The metric is constrained to be asymptotic to the cosmological A metric at large distance. The gravitational laws are illustrated in application to the case of a single spherical mass immersed in the smoothed out expanding universe. The condition is determined for such a metric to apply approximately just outside a typical member of a cosmic distribution of such masses. Conversely, the condition is given when the influence of the universe as a whole can be neglected outside such a mass. In the latter situation, which applies in particular to stars, a Schwarzschild-type metric is derived which incorporates variable G in accordance with the LNH. The dynamics of freely moving particles and photons in such a metric are examined according to the theory and observational tests are formulated.

We obtain asymptotic solutions of odd-order formally self-adjoint differential equations with power coefficients and discuss possible values for the deficiency indices of the associated operators.

We answer some questions about the semicontinuity in L∞ and in Lp of functionals defined on W1,n (Ωℝn) as integrals of non-convex functions of the gradient matrix, and we give a conjecture for the general case.

Landau's inequality ∥y′∥2≦4∥y∥∥y″∥ is extended to ∥y′∥2≦K(a)∥y∥1−a ∥y″ ∣y∣a∥, K(a) = 4/(l−a), 0≦ a<1. The proof is elementary and new even in the case a = 0 considered by Landau.

Let F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.

Let s and t be normal elements of the Calkin algebra, and let (s) denote the unitary orbit of s. A formula ρ(s,t) is defined to measure the distance between unitary orbits, and satisfies

Let

be, for each λ∈ℝk, a compact symmetric operator on a complex Hilbert space. Let the“fundamental” eigenset Z be denned by the relation λ∈Z if and only if W(λ) has maximal eigenvalue one. Conditions are given for Z to be the boundary of an open convex set P. A detailed investigation is given of the structure of P, including its recession cone and its representations as intersections of half-spaces.

A coefficient difference bound for k-fold symmetric and close-to-convex functions in the unit disc is established in this paper.

We discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

For the differential equation Lf = λMf on an open interval of ℝ, a theory in terms of relations in a Hilbert space associated with M was developed in a paper by Coddington and de Snoo, and eigenfunction expansions were derived in a paper by Dijksma and de Snoo. In the case of a regular problem on a compact interval, pointwise convergence of the expansions was shown in another paper by Coddington and de Snoo. Here, we show pointwise convergence in the general singular case.

An inequality whose origins date to the work of G. H. Hardy is presented. This Hardy-type inequality applies to derivatives of arbitrary order of functions whose domain is a subset of ℝn. The Friedrichs inequality is a corollary. The result is then used to establish lower bounds on the essential spectra of even-order elliptic partial differential operators on unbounded domains.