It is well known that the trapezoidal rule of quadrature is exact for linear functions on [0, 1], and easy to see that it is exact for functions of the form f = l+g where l is linear and g is odd about ½. Not so well known is an example of a continuous function for which the trapezoidal rule is exact but which does not have this form. In this paper we show that if the trapezoidal rule is exact for f then f has the form above provided it has either absolutely convergent Fourier series or continuous second derivative. We consider one-sided versions in which the approximate integrals are non-negative, and also characterize those sequences arising as the approximate integrals of a function with absolutely convergent Fourier series.

This paper organizes in a systematic manner the major features of a general theory of m-tone rows. A special case of this development is the twelve-tone row system of musical composition as introduced by Arnold Schoenberg and his Viennese school. The theory as outlined here applies to tone rows of arbitrary length, and can be applied to microtonal composition for electronic media.

A class of partially structured nonlinear programming problems, containing the capacitated nonlinear minimum cost multicommodity flow problem, is considered. Such problems, although large, can often be solved efficiently and with minimal computational storage by gradient projection methods.

When the first two elements of a sequence satisfying a second order difference equation are prescribed, the remaining elements are evaluated from a continued fraction and a simple product.

A study is made of a non-linear diffusion equation which admits bifurcating solutions in the case where the bifurcation is asymmetric. An analysis of the initial-value problem is made using the method of multiple scales, and the bifurcation and stability characteristics are determined. It is shown that a suitable interpretation of the results can lead to determination of the choice of the bifurcating solution adopted by the system.

In a previous paper the authors have shown that the classical barrier function has an O(r) rate of convergence unless the problem is degenerate when it reduces O(r½). In this paper a modified barrier function algorithm is suggested which does not suffer from this problem. It turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.

Recent hypercircie estimates for non-linear equations are extended to include a new class of boundary value problems of monotone type. The results are illustrated by the boundary value problem for the equilibrium-free surface of a liquid with prescribed contact angle.

The main aim of this paper is to derive a condition whose satisfaction ensures that the Euler–Lagrange equations of a non-linear scale covariant Lagrangian are satisfied by the metric of a space reciprocal to an arbitrary static Einstein space.

This paper presents an algorithm to solve the least squares problem when the parameters are restricted to be non-negative. The algorithm is based on the branch and bound method which has been suggested for this problem, and shares with it the property that an unrestricted least squares subproblem is solved at each step. However, improvements have been made to the branching rules by making use of the convexity of the problem, and the Kuhn–Tucker conditions are used to test for optimality. The resulting algorithm becomes essentially iterative in nature, and linearity of the number of subproblems solved can be shown under assumptions which have always been observed in practice.

It is shown in this note that the supremum and infimum of all causal topologies on Minkowski space are not causal. It is further shown that maximal and minimal elements exist in the set of all principal causal spaces.

The perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.

The note added in proof on p. 495 should read:

The author has discovered that R. P. Kanwal (J. Math. Phys. 44 (1965), 273–283) proposed a technique similar to the one in this paper. Unfortunately, Kanwal's scheme implies that the elastostatic problem can be reduced generally to the solution of Laplace's equation, a result which is well-known to be incorrect.