This is an application of the characteristic identity satisfied by matrices whose elements are also elements of a semi-simple Lie algebra. Generalized eigenvectors are determined for matrices consisting of generators of GL(n), O(n) and Sp(n), and it is shown how to resolve the identity into idempotents constructed from such eigenvectors. By this means rather general functions of the matrices may be defined. It is also shown how to determine traces of such functions, in terms of the invariants of the Lie algebra.

Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

We show that a sequence of generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichiet-Dirichlet boundary conditions forms a Riesz basis for the state Hilbert space. This develops a parallel result for the same system with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions.

Some comparison theorems and oscillation criteria are established for the neutral difference equation

as well as for certain neutral difference equations with coefficients of arbitrary sign. Neutral difference equations with mixed arguments are also considered.

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.

Using a mixed-type Fourier transform of a general form in the case of water of infinite depth and the method of eigenfunction expansion in the case of water of finite depth, several boundary-value problems involving the propagation and scattering of time harmonic surface water waves by vertical porous walls have been fully investigated, taking into account the effect of surface tension also. Known results are recovered either directly or as particular cases of the general problems under consideration.

Given the data (pi, ti, fi), i = 1,…,m, we consider the existence problem for the best least squares approximation of parameters for the 3-parametric exponential regression model. This problem does not always have a solution. In this paper it is shown that this problem has a solution provided that the data are strongly increasing at the ends.

The most fundamental characteristic of a physical system can often be deduced from its behaviour under discrete symmetry transformations, such as time reversal, parity and chirality. Here, we review some of the basic symmetry properties of the relativistic quantum theories for free electrons in ($2+1$)- and ($1+1$)-dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in ($2+1$) dimensions, and are generally present in physical realizations of such systems, for example in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour ($2+1$)-dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.

We investigate two routing problems that arise when order pickers traverse an aisle in a warehouse. The routing problems can be viewed as Euclidean travelling salesman problems with points on two parallel lines. We show that if the order picker traverses only a section of the aisle and then returns, then an optimal solution can be found in linear time, and if the order picker traverses the entire aisle, then an optimal solution can be found in quadratic time. Moreover, we show how to approximate the routing cost in linear time by computing a minimum spanning tree for the points on the parallel lines.

This note describes a simple numerical method for solution of the lifting surface integral equation of aerodynamics, and provides benchmark computations of up to 7 figure accuracy for flat rectangular wings of arbitrary aspect ratio. The nature of the large aspect ratio limit is also investigated numerically and asymptotically. This enables determination of the limiting behaviour near the wing tips, which is compared to the predictions of lifting line theory. Generalisations to non-rectangular wings are discussed.

In management science and system engineering, problems with two incommensurate objectives are often detected. Bicriterion optimization finds an optimal solution for the problems. In this paper it is shown that bicriterion discrete optimal control problems can be solved by using a parametric optimization technique with relaxed convexity assumptions. Some necessary optimality conditions for discrete optimal control problems subject to a linear state difference equation are derived. It is shown that in this case no adjoint equation is required.

A substance carried convectively through the liver by the blood undergoes two successive enzymatic transformations. The resulting concentrations of the three forms of the substance are determined, as functions of position along the blood flow in the steady state, by coupled ordinary differential equations of the first order on a finite interval. The densities along the blood flow of the activities of the two (immobile) transforming enzymes are described by two non-negative and normalised functions of position.

The problem, suggested by recent experimental results, is to choose these two functions so as to minimise the concentration of the once-transformed (intermediate) form of the substance at one boundary (the liver outlet). That minimisation is particularly significant biologically when the intermediate form is toxic and the second transformation renders it harmless. In this problem of optimal control (exerted perhaps by natural selection), the classical approach through Euler's equations is inapplicable because of the constraints on the two density functions. Moreover, when the enzyme kinetics and hence the differential equations are non-linear, the functional to be minimised is not obtainable explicitly. Instead it appears, after some manipulation of the coupled equations, as the terminal boundary value of the solution of a non-linear Volterra integral equation, which involves the two density functions (one explicitly and one implicitly) as control variables.

Appropriate existence, uniqueness and boundedness results are obtained for the solution of this integral equation, and the problem is then solved rigorously for one class of non-linearities (including saturation kinetics). Some unanswered questions are posed for another class (including substrate-inhibition kinetics).

In this paper, Antczak's η-approximation approach is used to prove the equivalence between optima of multiobjective programming problems and the η-saddle points of the associated η-approximated vector optimisation problems. We introduce an η-Lagrange function for a constructed η-approximated vector optimisation problem and present some modified η-saddle point results. Furthermore, we construct an η-approximated Mond-Weir dual problem associated with the original dual problem of the considered multiobjective programming problem. Using duality theorems between η-approximation vector optimisation problems and their duals (that is, an η-approximated dual problem), various duality theorems are established for the original multiobjective programming problem and its original Mond-Weir dual problem.

This paper discusses robust stochastic stability and stabilization of time-delay discrete Markovian jump singular systems with parameter uncertainties. Based on the restricted system equivalent (RES) transformation, a delay-dependent linear matrix inequalities condition for time-delay discrete-time Markovian jump singular systems to be regular, causal and stochastically stable is established. With this condition, problems of robust stochastic stability and stabilization are solved, and delay-dependent linear matrix inequalities are obtained. A numerical example is also given to illustrate the effectiveness of this method.2000Mathematics subject classification: primary 39A12; secondary 93C55.

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.

A linear programming model for optimally assigning diameters to a gas pipeline network is discussed. Computational results for a real life situation are presented, and certain properties that have to be satisfied by an optimal assignment are derived.

Line distributions of Stokes flow singularities are used to model the flow around a slender body which is straddling a flat interface between two viscous fluids. Motion of the slender body parallel to the interface and normal to the interface is considered where the axis of symmetry of the slender body is always perpendicular to the undisturbed interface. Asymptotic approximations to the force distributions on the slender body are evaluated and the relative contributions of that part of the slender body in one fluid to the force distribution in the other fluid and of the interface interaction to the force distribution are examined. It is observed that a shielding region exists about the interface which is due to the interaction with that part of the slender body in the other fluid. Finally, for parallel motion, the first order interface deformation is calculated.

Consider the forced differential equation with variable delay

where

We establish a sufficient condition for every solution to tend to zero. We also obtain a sharper condition for every solution to tend to zero when is asymptotically constant.

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

An analytical expression for the voltage response to current stimulation at relatively short and long times is used to obtain estimates of the passive electrical constants of a neuron that is electrotonically coupled at the soma and dendritic terminals to other neurons in a neural network.