We consider ordinary linear differential systems of first order with distributional coefficients and distributional nonhomogeneous terms. Firstly the coefficients are assumed to be functions, secondly to be first order distributions (i.e. first derivatives of functions which are integrable or of bounded variation), and thirdly to be distributions of higher order.

The proof of Lemma 6 (and thus of Theorem 9) has a gap in it. While m(B) → 0 as r → ∞ for each fixed h in Ni, it is not clear (and probably false) that this holds uniformly for h in . However Lemma 6 (and thus Theorem 9) holds with only trivial modifications of the given proof if one of the following holds: (i) ygi(y) → ∞ as |y| → ∞; (ii) m{x∈Ω: h(x) = 0} = 0 for every h in (iii) Niis one dimensional; (iv) there is a subset A of Ω such that h(x) = 0 if x ∈ A and h ∈ and m{x∈Ω\A: h(x) = 0} = 0 for every h ∈ . Assumption (ii) holds under very weak conditions. For example, the methods in [1] and regularity theorems imply that (ii) holds if there is a closed subset T of Ω of measure zero such that either (a) Ω\T is connected and aij (i, j = 1, …, n) are locally Lipschitz continuous on Ω\T or (b) for each component A of Ω\T, the aij have Lipschitz extensions to Ā and T is a “nice” set. (For example, it suffices to assume that T is a smooth submanifold of Ω though much weaker conditions would suffice.) Remember that we are assuming Ω is connected.

We obtain an explicit formula for the essential norm of a Hankel operator with its symbol in the space PC, which is the closure in L∞ of the space of piecewise continuous functions on the unit circle . It follows from this formula that functions in PC can be approximated as closely by functions in C, the continuous functions on the circle, as by functions in the much larger space H∞ + C. This is an example of the way in which properties of the Hardy spaces can be derived from properties of Hankel operators.

Let H be a Hilbert space in which a symmetric operator S with a dense domain Ds is given and let S have a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov type

and a method for calculating the best possible constants Cn,m(S).

Moreover, let φ be a symmetric bilinear functional with a dense domain Dφ such that Ds ⊂ Dφ and φ(f, g) = (Sf, g) for all f ∈ Ds, g ∈ Dφ. A necessary and sufficient condition for validity of the inequality

as well as a method for calculating the best possible constant K are obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form

The paper is concluded by establishing the best possible constants in the inequalities

where T is an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

The main result asserts that the base of an infinite dimensional Dedekind complete space with unit contains an infinite set of disjoint elements. From this result it can be shown that the dimension of Dedekind σ -complete spaces with unit is not countably infinite.

In an earlier paper we considered periodic Dirac operators and obtained criteria for them to be self-adjoint and for their spectra to be devoid of eigenvalues of finite multiplicity. The question of the existence of eigenvalues of infinite multiplicity was left open. In this article we obtain further criteria for self-adjointness and show that under these conditions periodic Dirac operators do not possess eigenvalues of infinite multiplicity. We also obtain a spectral gap result.

This note treats some questions about analytic continuation in several variables. The first theorem in effect determines the envelops of holomorphy of certain domains in ℂn. The second main result is a continuity theorem: If a bounded holomorphic function f on a convex domain ∆ in ℂn has boundary values that are continuous on the complement (in b∆) of a set of the form int∏ (b∆∩∏) where ∏ is a real hyperplane in ℂn that misses ∆, then f is continuous on . In addition, we obtain what may be regarded as a local version of the theorem in our earlier paper concerning the one-dimensional extension property. Our methods depend on Hartogs' theorem (n ≧ 3) and directly on the BochnerMartinelli formula (n = 2).

Given a Banach space X, we investigate the behaviour of the metric projection PF onto a subset F with a bounded complement.

We highlight the special role of points at which d(x, F) attains a maximum. In particular, we consider the case of X as a Hilbert space: this case is related to the famous problem of the convexity of Chebyshev sets.

The purpose of this paper is to examine the content of Chapter 4 of Ramanujan's second notebook. The first half of this chapter is on iterates of the exponential function. The second half focuses upon an interesting formal procedure which Ramanujan, in particular, used in the theory of integral transforms.

Let y(n) + a1y(n−1 +…+ an−1y(1) + an = 0 (*) be a linear ordinary differential equation of order n. A (relative) differential invariant of (*) is a differential polynomial function π(xi) defined on the solution space of (*) satisfying: there is an integer g such that for all invertible linear transformations α of V into itself, π(αxi) = (det α)βπ(xi). We prove in a purely algebraic manner the following two theorems: A. The differential invariants of (*) are generated algebraically by the Wronskian W and the coefficients ala2, …, an of (*). B. Every generic differential relation (i.e. differential relation which holds for every linear ordinary differential equation of order n) among W, a1 …, an can be deduced algebraically from Abel's identity, W′ = −a1W. The second theorem may be considered as an algebraic version of the existence theorem for linear ordinary differential equations.

In this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

A Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

We study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Here

is bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.