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.

In industrial applications of microwave heating, it has been observed that rather than the heating taking place uniformly, regions of high temperature, called hot-spots, tend to form. Depending on the industrial application, these can be either desirable or undesirable, and hence a theoretical understanding of the properties of the material that lead to hotspot formation is necessary. It has been shown in previous studies that hotspot formation is a product of the nonlinear dependence of microwave energy absorption by the material on temperature. It is shown in the present work that the conductivity of the material can have a significant effect on hotspot formation and can, if large enough, stop a hotspot from forming.

This paper considers a single-product industries, fixed capital model discussed briefly by Sraffa in Chaapter X if his book. The analysis is elementary, being based on the direct calculation of a certain martrix inverse. This generalises the approach adopted for circulating capital models. It also demonstrates that common economic and mathematical frameworks exist for both circulating and fixed capital models.

A formula is given for assigning sums to divergent series where the ratio of adjacent terms varies slowly along the series. This formula consistş of a weighted average of partial sums and is shown to be a general formula which can be easily calculated using a simple recurrence relation. It appears to be more powerful than a repeated Aitken or Shanks e1 process as long as the transformed series remains divergent and it is also compared with the Padé approximants. It is demonstrated on a factorial series, on a nearly geometric divergent series and for the extrapolation of a velocity formula for small amplitudes of motion.

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

Here we discuss the stability and convergence of a quadrature method for Symm's integral equation on an open smooth arc. The method is an adaptation of an approach considered by Sloan and Burn for closed curves. Before applying the quadrature scheme, we use a cosine substitution to remove the endpoint singularity of the solution. The family of methods includes schemes with any order O(hp) of convergence.

How to obtain a workable initial guess to start an optimal control (control parametrization) algorithm is an important question. In particular, for a system of multi-link vertical planar robot arms moving under the effect of gravity and applied torques (which can exhibit chaotic behaviour), a non-workable initial guess of torques may cause integration failure regardless of what numerical packages are used. In this paper, we address this problem by introducing a simple and intuitive “Blind Man” algorithm. Theoretical justification as well as a numerical example is provided.

In this paper we use the theory of generalized geometric programming to develop a dual for a discrete time convex optimal control problem. This has interesting interpretational implications. Further it is shown that the variables in the dual problem are intimately related to the costate vector in the usual Maximum Principle approach.