A method for determining the upper and lower bounds for performance measures for certain types of Generalised Semi-Markov Processes has been described in Taylor and Coyle [8]. A brief description of this method and its use in finding an upper bound for the time congestion of a GI/M/n/n queueing system will be given. This bound turns out to have a simple form which is quickly calculated and easy to use in practice.

An interesting and challenging problem in robotics is the off-line determination of the minimum cost path along which an end effector should move from a given initial to a given final state. This paper presents a discrete minimum cost path/trajectory planner which provides a general solution and allows for a range of constraints such as bounds on joint coordinates, joint velocities, joint torques and joint jerks. To demonstrate the practicability and feasibility of the planner, simulation results are presented for the Stanford manipulator using three and then the full six of its degrees of freedom. Simulation runs with two-link planar arms are also presented to enable a comparison with previously published results.

Design of an interior point method for linear programming is discussed, and results of a simulation study reported. Emphasis is put on guessing the optimal vertex at as early a stage as possible.

A general condition is provided from which an error bound can be concluded for approximations of queueing networks which are based on modifications of the transition and state space structure. This condition relies upon Markov reward theory and can be verified inductively in concrete situations. The results are illustrated by estimating the accuracy of a simple throughput bound for a closed queueing network with alternate routing and a large finite source input. An explicit error bound for this example is derived which is of order M—1, where M is the number of sources.

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.

A finite transverse shock wave propagates through an unbounded medium consisting of two joined incompressible elastic half-spaces of different material properties, in the direction normal to the plane interface. A semi-inverse method is used to examine the reflection-transmission of this wave at the interface. It is found that, depending on the material properties, the reflected wave is either a simple wave or a shock; the transmitted wave is always a shock.

The existence, uniqueness and regularity of solutions are proved for the obstacle problem with semilinear elliptic partial differential equations of second order. Computationally effective algorithms are provided and application made to steady state problem for the logistic population model with diffusion and an obstacle to growth.

This paper describes a mathematical model for a broadband integrated services network offered traffic of many different types. Performance measures are introduced related to revenue generation and overall grade-of-service, providing criteria for the optimal management of resources. Simple asymptotic expressions are derived for quantities termed the “implied costs”, which measure the effect on performance of changes in parameters that are controllable by network management, or that are subject to variation. These implied costs may be used, both to implement optimal bandwidth allocation polices, and also to indicate which services may share a single facility without adversely affecting performance, and which might require a dedicated facility. Asymptotic results are also used to examine how to make efficient use of capacity that is shared between calls with fluctuating bit-rate requirements.

Stochastic Petri Nets are used extensively to find performance measures for communication protocols. This paper illustrates how equilibrium distributions for the markings of a wide class of nets can be found directly without the need to generate a large state space and then resort to equilibrium balance equations.

A centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.

This paper studies degenerate forms of Maxwell's equations which arise from approximations suggested by geophysical modelling problems. The approximations reduce Maxwell's equations to degenerate elliptic/parabolic ones. Here we consider the questions of existence, uniqueness and regularity of solutions for these equations and address the problem of showing that the solutions of the degenerate equations do approximate those of the genuine Maxwell equations.

The propagation of a flame front in a combusting gas is considered in the limit in which the width of the reaction-zone is small compared with some overall flow dimension. In this approximation, the front propagates along its normals at a speed dependent on the local curvature of the front and is governed by a nonlinear equivalent of the geometric optics equations. Some exact solutions of this equation are found and a numerical scheme is developed to solve the equation for more complicated geometries.

Systems of coupled nonlinear differential equations with an externally controlled slowly-varying bifurcation parameter are considered. Canonical equations governing the transition between bifurcated solutions are derived by making use of methods of “steady” bifurcation theory. It is found that, depending on the initial amplitudes, the solutions of the transition equations are either asymptotically equivalent to the bifurcated solutions or the solutions develop algebraic singularities at some positive finite time. These singularities correspond to a transition to a solution of a fully nonlinear problem.

A problem in combustion theory with tenperature-dependent conductivity is considered. It is shown that information regarding criticality dependence on data and parameters can be obtained from a transformed equation in which the conductivity is constant, while the nonlinear source term is modified. Some previous work can then be used in such a study.

We consider the nonlinear evolution of a Hamiltonian system as the system passes through a linear resonance (as the system parameters vary). Two cases are considered. In the first case the linearized problem (at resonance) possess a full complement of normal mode solutions. This case is presented in the context of the interaction between modes which may have oppositely signed energy. The second case considered has an additional degeneracy in that the linearized problem (at resonance) has a single normal mode solution.

Both cases are analysed using normal form theory and in both cases the systems governing the transition through resonance are shown to be completely integrable in the classical sense. Possible bifurcations as the resonance is traversed are discussed. Conditions for the existence of algebraic singularities at some finite positive time are also presented.

In the paper we study the conditions under which multiconnection networks are nonblocking. A multiconnection network deals with the connections of pairs {(T1, T2)} where T1 is a subset of the input terminals and T2 is a subset of the output terminals. We investigate networks composed of digital switching matrices. Such networks can be treated as a very general case encompassing many kinds of networks used in practice as well as studied theoretically.

We present four routing strategies and then develop conditions under which multiconnection networks are nonblocking when each of these strategies is used. We also show that the obtained conditions reduce to known results for some values of network parameters.

In this paper we review a simple class of fixed point models for loss networks. We illustrate how these models can readily deal with heterogeneous call types and with simple dynamic routing strategies, and we outline some of the recent mathematical advances in the study of such models. We describe how fixed point models lead to a natural and tractable definition of the implied cost of carrying a call, and how this concept is related to issues of routing and capacity expansion in loss networks.

We study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −ε|⋅ṡ| at the free boundary x = s(t). The problem is posed on the semi-infinite strip [0,∞) with unit Stefan number and bounded initial temperature ϕ(x) ≤ 0, such that ϕ → −1 − δ as x → ∞, where δ is constant. Special solutions and the asymptotic behaviour of the free boundary are considered for the cases ε ≥ 0 with δ negative, positive and zero, respectively. We show that, for ε > 0, the free boundary is asymptotic to , δt/ε if < δ > 0 respectively, and that when δ = 0 the large time behaviour of the free boundary depends more sensitively on the initial temperature. We also give a brief summary of the corresponding results for a radially symmetric spherical crystal with kinetic undercooling and Gibbs-Thomson conditions at the free boundary.

Minkowski space-time is specified with respect to a single coordinate frame by the set of timelike lines. Isotropy mappings are defined as automorphisms which leave the events of one timelike line invariant. We demonstrate the existence of two special types of isotropy mappings. The first type of isotropy mapping induce orthogonal transformations in position space. Mappings of the second type can be composed to generate Lorentz boosts. It is shown that isotropy mappings generate the orthochronous Poincaré group of motions. The set of isotropy mappings then maps the single assumed coordinate frame onto a set of coordinate frames related by transformations of the orthochronous Poincaré group.

In recent papers we have considered the numerical solution of the Hammerstein equation

by a method which first applies the standard collocation procedure to an equivalent equation for z(t):= g(t, y(t)), and then obtains an approximation to y by use of the equation

In this paper we approximate z by a polynomial zn of degree ≤ n − 1, with coefficients determined by collocation at the zeros of the nth degree Chebyshev polynomial of the first kind. We then define the approximation to y to be

and establish that, under suitable conditions, , uniformly in t.