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.

When solving practically the neutral type equations the derivatives are replaced by finite differences while the members of integral type are replaced by quadrature formulae. The paper deals with the convergence of a natural class of methods applied to the Cauchy problem for functional-differential neutral type equations. It is not obligatory for the approximated operators to be compact.

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.

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.

Let X be a set with infinite regular cardinality m. Within the full transformation semigroup ℑ(X) a subsemigroup Sm is described which is bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is also congruence-free. The semigroup contains an isomorphic copy of every semigroup having order less than m.

Sufficient conditions for oscillation of solutions to third order hyperbolic characteristic initial value problems are established. The results generalize known oscillation criteria for second order hyperbolic problems.

Upper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surfaces.

In this paper, appropriate definitions are given for the spectra of multiparameter problems in Hilbert space when the operators involved are not necessarily self-adjoint. Notions of adjoint multiparameter problems are introduced, and some properties of the spectrum of a single linear operator are generalised to multiparameter settings.

The results which are achieved are compared with those in the literature which have beendeveloped for multiparameter problems involving self-adjoint operators.

We consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

Boundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

We consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points, specified by a finite incidence matrix E. Similarly the allowable discontinuities of the piecewise polynomials are specified by a finite incidence matrix F. We first find necessary conditions on (E, F) for the problem to be poised, that is to have a unique solution for any given data. The main result gives sufficient conditions on (E, F) for the problem to be poised, generalising a well-known result of Atkinson and Sharma. To this end we prove some results involving estimates of the numbers of zeros of the relevant piecewise polynomials.

Asymptotic estimates for the eigenvalues and eigenfunctions of a directed set of Sturm–Liouville operators are obtained. Particular attention is paid to the influence of the diffusion coefficient, as it becomes arbitrarily large.

This note pursues two aims: the first is historical and the second is factual.

1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.

2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.

For the non-linear problem

where f is a discontinuous function at 1, we show that the number of non-trivial positive solutions, for a given real number λ≧0, is related to the graph of a continuous function g. Then, by studying the function g it is possible in some special cases to give, for any λ≧0, the minimal or exact number of non-trivial positive solutions.

Absence of non-trivial square integrable solutions of the Dirac equation

for values of λ in half-rays is proved under local conditions on q, curl b, and the radial derivative of q. The Coulomb potential is admitted. Another result does not contain any growth condition on the radial derivative of q. It states that there are no solutions of integrable square other than the trivial solution if λ ε ℝ\ [−1, 1] and

A perturbation theorem for symmetric differential expressions is given and applied to even-order expressions. We assume the coefficients p0(t), pn(t) to be eventually positive and to have real powers of t as dominating terms. Then we are able to admit for the absolute values of the other coefficients a rate of growth depending on the growth of both of the coefficients p0, pn in order to obtain minimal deficiency indices.