This paper examines the role of import tariffs and consumption taxes when a product is supplied to a domestic market by a foreign monopoly via a subsidiary. It is assumed that there is no competition in the domestic market from internal suppliers. The home country is able to levy a profits tax on the subsidiary. The objective of our analysis is to determine the mix of tariff and consumption tax which simultaneously maximizes national welfare. We show that national welfare does not have an internal maximum, but attains its maximum on a boundary of the consumption tax–tariff parameter space. Furthermore, the optimal value of national welfare increases as the tariff decreases and the consumption tax increases. The results obtained generalize the results of an earlier paper in which national welfare was maximized with respect to either a tariff or consumption tax, but not both.

An interior layer problem posed by an elliptic partial differential equation of the type ε∇2φ - x∂φ/∂y = f(x, y, ε), 0 < ε ≪ 1, is investigated. This equation arises, for example, in the theory of rotating fluids and the important feature of the problem is an interior layer of width O(ε1/3) in which the solution has a relatively large magnitude.

The paper considers the simplest case which involves an interior layer, that is, where the domain is rectangular and f(x, y, ε) = εA for A constant. A leading approximation is derived and it is shown to be asymptotic to the exact solution in nearly all of the domain as ε → 0. The error estimates are derived using an a priori estimate for the solution of elliptic equations and a technique which optimizes the estimates is introduced. The applicability and limitations of the estimation technique are discussed briefly.

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.

In this paper, we consider denumerable state continuous time Markov decision processes with (possibly unbounded) transition and cost rates under average criterion. We present a set of conditions and prove the existence of both average cost optimal stationary policies and a solution of the average optimality equation under the conditions. The results in this paper are applied to an admission control queue model and controlled birth and death processes.

This paper derives key equations for the determination of optimal control strategies in an important engineering application. A train travels from one station to the next along a track with continuously varying gradient. The journey must be completed within a given time and it is desirable to minimise fuel consumption. We assume that only certain discrete throttle settings are possible and that each setting determines a constant rate of fuel supply. This assumption is based on the control mechanism for a typical diesel-electric locomotive. For each setting the power developed by the locomotive is directly proportional to the rate of fuel supply. We assume a single level of braking acceleration. For each fixed finite sequence of control settings we show that fuel consumption is minimised only if the settings are changed when certain key equations are satisfied. The strategy determined by these equations is called a strategy of optimal type. We show that the equations can be derived using an intuitive limit procedure applied to corresponding equations obtained by Howlett [9, 10] in the case of a piecewise constant gradient. The equations will also be derived directly by extending the methods used by Howlett. We discuss a basic solution procedure for the key equations and apply the procedure to find a strategy of optimal type in appropriate specific examples.

Numerical solutions for travelling combustion waves of a solid material are sought. The algorithm of computation is based on a two-sided shooting method. It is found that there is a lower bound of the wave speed c, say c*, such that for c < c* no numerical solution can be constructed. This c* is a function of the activation energy of the medium.

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:

where the funtion f and g satisfy

for some η: X0 × X0 → ℝn

It is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

Wavelet systems with a maximum number of balanced vanishing moments are known to be extremely useful in a variety of applications such as image and video compression. Tian and Wells recently created a family of such wavelet systems, called the biorthogonal Coifman wavelets, which have proved valuable in both mathematics and applications. The purpose of this work is to establish along with direct proofs a very neat extension of Tian and Wells' family of biorthogonal Coifman wavelets by recovering other “missing” members of the biorthogonal Coifman wavelet systems.

The flexibacters are a form of gliding bacteria which are often found on the surfaces of solid bodies in fresh and salt water. An individual organism lacks motility in the bulk aqueous phase but glides over a solid surface with its rod-like body aligned with and nearly touching the surface. It has been suggested that this gliding motion in Flexibacter strain BH3 may be caused by waves moving down the outer surface of the rod-shaped cell [2]. This paper is concerned with the fluid mechanical aspects of this form of propulsion.

Formulae for the velocity of the organism and for the power dissipation are obtained by using a lubrication theory analysis in the small gap between the bacterium and the wall. It is found that for any progressive waveform there is an optimum distance from the wall at which the flexibacter may maximize its speed for a given power output. Assuming that the flexibacter sits at this optimum distance and taking the waveform to be sinusoidal we calculate the power required for the flexibacter to move at the maximum observed speed. It is found that this power requirement represents only a small fraction of the power available to the cell.

In this paper we present a Kaleckian-type model of a business cycle based on a nonlinear delay differential equation. A numerical algorithm based on a decomposition scheme is implemented for the approximate solution of the model. The numerical results of the underlying equation show that the business cycle is stable.

The main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

Generalizations of the Green-Lanford-Dollard theorem on scattering into cones and Ruelle-Amerin-Georgescu theorem characterizing bound states and scattering states are derived. The first is shown to be an easy consequence of the Kato-Trotter theorem on semi-group convergence whilst the latter is corollary of Wiener's version of the mean ergodic theorem.

In this paper we consider a natural extension of the minimum time problem in optimal control theory which we refer to as the minimum trapping time problem. The minimum trapping time problem requires a fixed time interval [0, T], where T is finite. The aim is to determine a control for which the system trajectory not only reaches a specified target in minimum time but also remains trapped within the target until time T. Our aim is to devise a computational procedure for solving the minimum trapping time problem. The computational procedure we adopt uses control parametrisation in which the class of controls is approximated by a class of piecewise constant functions. The problem we are solving is therefore an approximation to the original minimum trapping time problem. Some properties for the approximate problem are then established. These lead to an extremely efficient iterative procedure for calculating the minimum trapping time.

Boundary value problems where resonance phenomena are studied are most often transformable to parameter dependent Sturm-Liouville (SL) eigenproblems with interior singularities. The parameter dependent Sturm-Liouville eigenproblem with interior poles is examined. Asymptotic approximations to the solutions are obtained using an extended Langer's method to take care of the resulting complex eigenvalues and eigenfunctions.

Based on the theory of difference equations, we derive necessary and sufficient conditions for the existence of eigenvalues and inverses of Toeplitz matrices with five different diagonals. In the course of derivations, we are also able to derive computational formulas for the eigenvalues, eigenvectors and inverses of these matrices. A number of explicit formulas are computed for illustration and verification.

A simple lumped hydraulic model of knee drainage following arthroplasty is developed incorporating a pressure-volume equation of state for the knee capsule and a wound healing rate dynamically retarded by the blood flow-induced shear stress. The resulting second-order nonlinear ordinary differential system is examined numerically and qualitatively to map the parameter space. In the model, moderate suction or a slight back-pressure promotes gradual drainage and healing whereas excessive suction can lead to a bifurcation in which healing is retarded or even prevented. Guided, then, by the model, the literature, and experience, continuous drainage with a small constant back-pressure appeared beneficial so we prospectively evaluated a series of ten patients. The results are consistent with the model and promising.