These polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.

In this paper we obtain some new results on a nonlinear parabolic system related to the equations of the nematic liquid crystals and introduced in earlier papers by J. P. Dias.

These results mainly concern the existence and uniqueness of generalized solutions for discontinuous data and also their asymptotic behaviour in various cases.

Given the linear hyperbolic evolution equation (P0) on a reflexive Banach space, we present a new method for an existence proof of unbounded solutions admitting an exponential growth rate as time tends to infinity. Utilizing abstract Wiener—Hopf techniques, an operational calculus is developed for the construction of the resolving operator associated with the problem under consideration. The results are based upon the fundamental hypothesis that the spectral set of the time-independent mapping A is contained in the interior of a parabola. The distance of the focus from the vertex of this parabola turns out to be a measure for the growth rate. Applicability of the results is shown in the case where A is a non-symmetric perturbation of a self-adjoint partial differential operator.

The present note is concerned to develop the principle of limiting absorption for the Laplacian Δ on a two-point homogeneous noncompact space M = G/H subject to a real-valued potential perturbation V. Such a property depends on the detailed structure of the Laplacian in a suitable coordinate system while V is assumed to satisfy a short-range condition.

We consider the question: When do two ordinary linear differential expressions commute? It turns out that the set of all expressions which commute with a given one form a commutative ring. Here we study the algebraic structure of these rings. As an application a complete characterization of normal differential expressions is obtained.

Dual similarity solutions in the context of mixed convection are presented. In contrast to the Falkner–Skan solutions the bifurcation point is found to be distinct from the point of vanishing skin friction. The eigenvalue problem arising out of a stability analysis of these solutions is examined numerically. The numerical evidence would seem to indicate that the margin of stability is associated with the onset of reverse flow as opposed to the bifurcation point, as conjectured by Banks and Drazin in 1973.