In provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.

The problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

The exterior Dirichlet problem for the Helmholtz equation in two dimensions is reduced to a boundary integral equation which is soluble by iteration. A standard application of Green's theorem leads to boundary integral equations which are not uniquely soluble because the operator has an eigenvalue. The present approach modifies the operator in such a way that the former eigenvalue is in the resolvent spectrum for low frequencies. The results are applied to the inverse scattering problem wherein the far field is known for a limited frequency range and one seeks the curve on which a plane wave is incident and a Dirichlet boundary condition is assumed. The first iterate in the solution of the boundary integral equation is used to obtain a sequence of moment problems relating the Fourier coefficients of the far field to the coefficients of the Laurent expansion of the conformai transformation which maps the exterior of a circle onto the exterior of the unknown curve. These moment problems are soluble in terms of the mapping radius which in turn may be determined from scattered far field data for an incident plane wave from a second direction.

Asymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.

The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.

In this paper the reflection of singularities of solutions to the wave equation is studied using progressing wave expansions. The results are used to calculate the leading singularity of the Lax–Phillips scattering kernel, thus generalizing earlier results of Majda.

The isomorphism semigroup S(G) of a group G is the semigroup of isomorphic mappings between subgroups of G, with composition its operation. This paper will show that a periodic abelian group is uniquely determined by its isomorphism semigroup.

In the following paper, the horizontal line method (the Rothe method) is applied to Maxwell's initial boundary value problem. By means of results from abstract perturbation theory, convergence results and error estimates are established.

If |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).

Conditions are given on the coefficients p, q and r of the fourth-order, symmetric differentia expression

where y ∈ L2(0, ∞) and L[y] ∈ L2(0, ∞), such that some or all of the individual terms in L[y] are also in L2(0, ∞).

It is shown that eigenvalues of infinite multiplicity can exist for the Schrödinger equation holding in the whole N-dimensional space RN(N ≧ 2). In the example which is constructed, the potential is separable and bounded in RN, and the method is an application of inverse spectral theory.

We provide estimates of the form

for the length of gap centre μ in the essential spectrum of a self-adjoint operator generated by a matrix differential expression.

The paper deals with genetic algebras for populations in which given fractions of the population practise specified types of inbreeding. It is shown that the technique of linearising the quadratic transformation of genetic type frequencies, which has been studied for triangular algebras, can be extended to a much wider class, including those for inbreeding systems.

In this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

The non-negative weight functions U(x), V(x) for which the generalized Stieltjes transformation Sλ is a bounded operator from LP(V) into La(U) are characterized. This extends certain classical inequalities and some recent results of Erdélyi. In particular, our results provide weighted extensions of a classical inequality known as Hubert's double series theorem.

We study the nonlinear equation

in ℝ3, where Δ denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number λ. Because of our assumptions on R and K we shall look for solutions which are spherically symmetric, decrease with r = |x| and vanish at infinity.

The differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special case

but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series

Examples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problem

for given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

Linear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.

If z1, z2 … zn are complex numbers satisfying |zi−zj|≧1 for all i, j then the number of the 2n sums where ει = ±1, which lie in any circle of radius r cannot exceed αr2n/n3/2 where αr depends only on r.

In this paper the Sturm-Liouville regular boundary value problem and expansion theorem is extended to functions with values in some B*-algebras.

We show that for some closed graph theorems each countable codimensional subspace of a domain space may also serve as a domain space. This provides a general principle from which we are able to extract some of the known results on the inheritance of topological vector space properties by subspaces of countable codimension. We make use of a result of Savgulidze and Smoljanov on B-completeness for which we provide a new and simpler proof.