A boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

Optimality conditions via subdifferentiability and generalised Charnes-Cooper transformation are obtained for a continuous-time nonlinear fractional programming problem. Perturbation functions play a key role in the development. A dual problem is presented and certain duality results are obtained.

This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative scheme is analyzed for solving such problems.

We present a general closed 4-point quadrature rule based on Euler-type identities. We use this rule to prove a generalization of Hadamard's inequalities for (2r)-convex functions (r ≥ 1).

Simple chemical reactions can be described by the Michaelis-Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the ∫ shaped curve cannot be too shallow and the upper part can't be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.

It is shown that an integrable class of helicoidal surfaces in Euclidean space E3 is governed by the Painlevé V equation with four arbitrary parameters. A connection with sphere congruences is exploited to construct in a purely geometric manner an associated Bäcklund transformation.

Using an estimate on the group velocity we give an independent proof of the existence of time translations for a large class of short range interactions. We demonstrate that these systems satisfy a strong form of causal propagation and that space-time algebras in suitable space-like directions are disjoint. Finally we derive criteria for dispersion of the interaction in terms of the algebraic density of the orbit of local subalgebras under the evolution or under the associated group of shifts. In this sense the Heisenberg and X-Y models are dispersive but the Ising model is not.

A homogeneous isotropic infinite elastic plate contains a circular cavity and a circular arc crack symmetrically situated about the x-axis. The cavity and crack are concentric but are of different radii. A circular inhomogeneity of radius slightly larger than that of the cavity is inserted into the cavity; thus generating a system of stresses in the outer material as well as in the inhomogeneity. The elastic field in the inhomogeneity and in the outer material outside the inhomogeneity is evaluated in this paper.

The method of Coullet and Spiegel [3], which derives ordinary differential equations describing the time evolution of a system of partial differential equations when the system is near critical, is applied to some simple problems. These problems serve to illustrate simply many features of the method.

In the paper King [8], a new class of source solutions was derived for the nonlinear diffusion equation for diffusivities of the form D(c) = D0cm/(l - vc)m+2. Here we extend this method for the nonlinear diffusion and convection equation

to obtain mass-conserving source solutions for a nonlinear conductivity function K(c) = K0cm+2/(l - vc)m+1. In particular we consider the cases m = -1,0, and 1, where fully analytical solutions are available. Furthermore we provide source solutions for the exponential forms of the diffusivity and conductivity as given by D(c) = D0c−2e−n/c and K(c) = K0ce−n/c.

In two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is The Poisson bracket of the two components of this vector leads to a third first-integral, cubic in the momenta. The Lie algebra of the three integrals under the operation of the Poisson bracket closes, and is shown to be so(3) for negative energy and so(2, 1) for positive energy. In the case of zero energy, the algebra is W(3, 1). The result does not have a three-dimensional analogue, apart from the usual Kepler problem.

In 1962 Lakshmikantham ([1], [2]) extended the concept of extreme stability (e.g. [4]) of a system described by an ordinary differential equation, not necessarily with uniqueness, to relative stability of two such systems. Here we show the restrictiveness of his definition of relative stability in that it implies not only are the solutions of two systems unique for each initial condition, they are in fact identical. We then introduce and give an example of a weaker version of relative stability which is of some interest for control systems. For greater simplicity and generality we use Roxin's attainability set defined General Control Systems [3] to describe the dynamics of our systems, as they subsume both ordinary differential equations without uniqueness and ordinary differential control equations.

This paper shows how to compute the trace of G(T) – G(T0), where G is an infinitely differentiable function with compact support, and where T and T0 are one-dimensional Schrödinger operators on (−∞, ∞) with potentials q and q0. It is assumed that q0 is a simple step potential and that q decays exponentially to q0. The trace is expressed in terms of the reflection and transmission coefficients for the scattering of plane waves by the potential q.

The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

New polynomial solutions of the Navier-Stokes equations for steady uni-directional flow of viscous incompressible fluid, with a free surface, down inclined channels of specialized cross-section are considered. An inverse method is uded to obtain the geometrical shape of the channel by equating the polynomials solution to zero (i.e. the no-slip condition) and thence determining the boundary shape.

In this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

In this paper, we discuss MDP-the moment optimal problem for the first-passage model. A policy improvement iteration algorithm is given for finding the k-moment optimal stationary policy.