Let θ(t, λ) and (t,λ) be the solutions to the differential equation y“+(λ − q(t))y =0, − ∞ <t < ∞ such that θ(0, λ)) = l, θ'(0, λ)) = 0, ø(0, λ) = 0 and ø'(0, λ) = −1. It is known that for a function f(t)∈ L2(−∞,∞), we have

where for some measures dξ, dη; and dζ independent of f.

In the first part of this paper we devise a technique to locate the singular points of f(t) (t is complex) under the assumptions that E(λ), F(λ) are of order O(e-cvλ), as λ →∞and q(t) is a sufficiently nice function. In this case all the singularities of fare off the real axis. In the second part of the paper we relax the restriction on E(λ) and f(λ) so that they are of order O(λk) as λ →∞ for some constant k, and we show that in this case f(t) is a generalised function whose analytic representation is holomorphic in the upper and lower half planes. We then devise a technique to continue analytically across the real axis and locate its singularities thereon.

In this paper we study a problem in multilinear algebra which consists of finding small values of a certain quotientμ/α. Here μ is the minimal eigenvalue of a positive definite operator determinant Δ of the type introduced by F. V. Atkinson, and α is the minimum of the quadratic form corresponding to Δ with respect to all decomposable tensors of unit norm. Our results are connected with earlier results of P. Binding.

Liouville type theorems are obtained for the solutions to elliptic equations of the form Δ2u −q(x)Δu + p(x)f(u)=0 by means of two subharmonic functionals and Green type inequalities.

Limit-point and limit-circle criteria are given for the generalised Sturm-Liouville differential expression

where

(i) p, q, and w are real-valued on [a, b),

(ii) p−1, q, w are locally Lebesgue integrable on [a, b),

(iii) w > 0 almost everywhere on [a, b) and the principal coefficient p is allowed toassume both positive and negative values.

The equation {r(x)y'}' = q(x)y is considered with the coefficients q and rtaking complex values. Conditions are obtained for there to be solutions with the asymptotic form , . These involve the coefficients and their derivatives up to some order N.

The correspondence between radicals of associative rings and A-radicals is studied. It is shown that corresponding to each A-radical there is an interval of radicals and that each radical belongs to exactly one such interval. The question of the nature of the radical of a one-sided ideal is considered. It is shown that the radicals such that the radical of a one-sided ideal is always a one-sided ideal are those which contain their associated A-radicals. Radicals such that the radical of a one-sided ideal always equals the intersection of a left ideal and a right ideal are described, as are those A-radicals such that every associated radical has this property.

Assume f ∈ C(RM, SN−1) (M ≧ 1, N ≧ 1) is periodic with period one in each variable and satisfies:

Then, every probability measure on [0, 1)M such that

is a Dirac measure at some point of [0, 1)M.

This paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

In this paper we study the following boundary value problem

where Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .

By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

A ring epimorphism θ:A →B extends in a natural way to a homomorphism γn: GLn(A)→GLn(B) and, when A is commutative, to a homomorphism σn:SLn(A)→SLn(B), where n ≧ 1. In this paper we consider the question: when are γn and σn surjective (or non-surjective)?

In this paper we prove that every precompact subset in any (LF)-space has a metrizable completion. As a consequence every (LF)-space is angelic and in this way the answer to a question posed by K. Floret [3] is given. Some contributions to the general problem of regularity in inductive limits posed by K. Floret [3] are also given. Particularly, extensions of well-known results of H. Neuss and M. Valdivia are provided in the general setting of (LF)-spaces. It should also be noted that our results hold for inductive limits of an increasing sequence of metrizable spaces.

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

The shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.

We give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,

where T is the Hankel-or the K-transformation.

The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.

For many purposes, and in particular for the calculation of upper and lower bounds to bilinear forms 〈g0,f〉, where f is the solution to an operator equation and g0 is known, it isuseful to obtain an approximation to the inverse of the operator.

For a normal operator A acting in a complex Hilbert space with bounded inverse A−1, we use a direct approach through fundamental results in functional analysis and derive a recipe for the ‘best formula’ for A−1 of form B = βI with β constant and I the identity operator. Examples illustrate that this leads to improved results for certain classes of operator.

The investigation is extended to the representation of A−1 by a polynomial in A withcomplex coefficients. For polynomials of first and higher orders the application is restricted to bounded self-adjoint operators; at first order, explicit formulae are given.

Application of the method to Fredholm operators is considered in detail, and its use for the point-wise solution to Fredholm integral equations is illustrated.

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Page 58: Lemma 7.1 is incorrect. It is claimed that Ao =cocl for c =k−l when in fact the result is proved for c = (dim H)−1. Thus while Corollaries 7.4 and 7.6 still stand (as does Corollary 7.3 by alternative arguments) the remaining results in Section 7 must be withdrawn.