In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.

It is known that strong uniqueness can be used to prove second order convergence of the generalised Gauss-Newton algorithm. Formally this algorithm includes sequential linear programming as a special case. Here we show that the second order convergence result extends when the sequential linear programming algorithm is formulated appropriately. Also this discussion provides an example which shows that the assumption of Lipschitz continuity is necessary for the second order convergence result based on strong uniqueness.

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

Vacuum field equations in a scalar-tensor theory of gravitation, proposed by Ross, are obtained with the aid of a static plane-symmetric metric. A closed form exact solution to the field equations in this theory is presented which can be considered as an analogue of Taub's empty space-time in Einstein's theory.

Formulating a dust-filled spherically symmetric metric utilizing the 3 + 1 formalism for general relativity, we show that the metric coefficients are completely determined by the matter distribution throughout the spacetime. Furthermore, the metric describes both inhomogeneous dust regions and also vacuum regions in a single coordinate patch, thus alleviating the need for complicated matching schemes at the interfaces. In this way, the system is established as an initial boundary value problem, which has many benefits for its numerical evolution. We show the dust part of the metric is equivalent to the class of Lemaitre-Tolman-Bondi (LTB) metrics under a coordinate transformation. In this coordinate system, shell crossing singularities (SCS) are exhibited as fluid shock waves, and we therefore discuss possibilities for the dynamical extension of shell crossings through the initial point of formation by borrowing methods from classical fluid dynamics. This paper fills a void in the present literature associated with these collapse models by fully developing the formalism in great detail. Furthermore, the applications provide examples of the benefits of the present model.

Tsuno and Nodera proposed a new variant of the GMRES(m) algorithm. Their algorithm is referred to as the GMRES(≤ mmax) algorithm and performs the restart process adaptively, considering the distribution of the zeros of the residual polynomial. However, unless the zeros of the residual polynomial are distributed uniformly, mass is always chosen and their algorithm becomes almost the same as the GMRES(m) algorithm with m = mmax. In this paper, we include a convergence test for the residual norm in the GMRES(≤ mmax) algorithm and propose a new restarting technique based on two criteria. Even if the distribution of zeros does not become uniform, the restart can be performed by using the convergence test of the residual norm. Numerical examples simulated on a Compaq Beowulf computer demonstrate that the proposed technique accelerates the convergence of the GMRES(≤ mmax) algorithm.

We study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

This paper describes a SQP-type algorithm for solving a constrained maximum likelihood estimation problem that incorporates a number of novel features. We call it MLESOL. MLESOL maintains the use of an estimate of the Fisher information matrix to the Hessian of the negative log-likelihood but also encompasses a secant approximation S to the second-order part of the augmented Lagrangian function along with tests for when to use this information. The local quadratic model used has a form something like that of Tapia's SQP augmented scale BFGS secant method but explores the additional structure of the objective function. The step choice algorithm is based on minimising a local quadratic model subject to the linearised constraints and an elliptical trust region centred at the current approximate minimiser. This is accomplished using the Byrd and Omojokun trust region approach, together with a special module for assessing the quality of the step thus computed. The numerical performance of MLESOL is studied by means of an example involving the estimation of a mixture density.

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.

DeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

The boundary integral equation method is obtained by expressing a solution to a particular partial differential equation in terms of an integral taken round the boundary of the region under consideration. Various methods exist for the numerical solution of this integral equation and the purpose of this note is to outline an improvement to one of these procedures.

A quasi-steady-state apprcncimation to the Navier-Stokes equation is the corresponding equation with nonhomogeneous forcing term f(x, t), but with the term Vt deleted. For solutions that are zero on the boundary, the difference z between the solution of the Navier-Stokes equation and the solution of this quasi-steady-state approximation is estimated in the L2 norm ║z║ with respect to the spatial variables. For sufficiently large viscosity or sufficiently small body force f, the inequality

holds for 0 < t ≤ T and certain real numbres C, β > 0.

Some corrections of error bounds obtained by Chatelin and Lemordant for the first three terms of the asymptotic case of a strong approximation are given. The error bounds for the approximations of order 2 in the Galerkin method are compared with the Rayleigh quotients constructed with the eigenvectors in the Sloan method. A numerical experiment is also carried out.

This paper is concerned with the oscillation of first-order delay differential equations

where p(t) and τ(t) are piecewise continuous and nonnegative functions and τ(t) is non-decreasing. A new oscillation criterion is obtained.

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic equations. The unknown coefficient of the elliptic equations depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic equations are uniquely solvable for the given class of coefficients. Proof of the existence of a quasisolution of the inverse problems is obtained.