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.

Associated with a Cauchy problem for an eikonal equation is a structure of caustics and wave fronts. The geometrical properties of these structures are of considerable interest and one would like to classify them. The first step in such a programme is to identify those solutions whose structures have certain (structural) stability properties or robustness. These are structures which survive small perturbations of the initial conditions. (It turns out that they also survive small perturbations of the equation.) This problem for caustics has been tackled with considerable success by a number of authors, in particular by Arnold. Here we develop the analogous, but different, theory for wave fronts and give an algebraic characterization of those local solutions of eikonal equations which have stable wave front structures. From this result it is easy to deduce the qualitative form of those stable wave front structures which occur when the propagation occurs in a space of dimension not greater than five. They correspond to Thorn's elementary catastrophes, as demonstrated by Zakalyukin.

In this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation

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.

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).

In a previous publication we investigated certain idempotent residuated mappings and showed how these could be used to provide a solution to the problem of finding a Baer semigroup coordinatisation of bounded modular lattices. Here we use essentially the same idempotents to provide a coordinatisation of bounded distributive lattices. Specifically, we prove that a bounded lattice L is distributive if and only if it can be coordinatised by a Baer semigroup S such that if eS, fS, gS ∈ R(S) with eS ∩ fS = eS ∩ gS then there are idempotents ē, , ∈ S such that ēS = eS, S = fS, S = gS and ē commutes with both and .

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.

Dual extremum problems associated with an infinite family of admissible loops on a Riemann surface are shown to be solvable by Jenkins-Strebel differentials. Then the inequalities associated with these problems are used to calculate the first derivatives of extremal length functionals on Teichmüller space and to estimate the difference quotients for the second derivatives of these functions.

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

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