We consider an optimal control problem with, possibly time-dependent, constraints on state and control variables, jointly. Using only elementary methods, we derive a sufficient condition for optimality. Although phrased in terms reminiscent of the necessary condition of Pontryagin, the sufficient condition is logically independent, as can be shown by a simple example.

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.

The problem of estimating the limit f∞ of a sequence fn converging as fn − f∞ = O(n−λ) as n → ∞, where λ > 0, is discussed. Using the generalization of the ε-algorithm proposed recently by Vanden Broeck and Schwartz, an acceleration scheme is developed. The method is illustrated on several test sequences and compared to other acceleration procedures.

Duffing's differential equation in its simplest form can be approximated by a variety of difference equations. It is shown how to choose a ‘best’ difference equation in the sense that the solutions of this difference equation are successive discrete exact values of the solution of the differential equation.

For the completely stiff real homogeneous system

where e is a small positive parameter, a method is given for the construction of a basis for the solution space.

If A has n linearly independent eigenvector functions, then there exists a choice of these, {si}, with corresponding eigenvalue functions {λi}, such that there is a local basis for solution, that takes the form

where vi is a vector that tends to zero with e. In general, a basis of this form exists only on an interval in which the distinct eigenvalues have their real parts ordered. A construction is provided for continuing any solution across the boundaries of any such interval. These results are proved for a finite or infinite interval for which there are only a finite number of points at which the ordering of the real parts of eigenvalues changes.

In the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

When optical fibres are made by first constructing optical-fibre preforms, the fibre which is pulled from the heated preform is simply a scaled down version of the original preform structure. The expansion coefficient profile α(r) of the preform, which relates directly to the fabrication variables, can be determined from non-destructive optical retardation measurements δ(r) performed on the preform. In addition, the residual elastic stress distributions in a fabricated preform, which can be used to compare different fabrication procedures, have simple definitions as linear functionals of the expansion coefficients α(r). Thus, through the use of optical retardation data, an examination of different manufacturing procedures for preform fabrication is reduced to a problem in non-destructive manufacturing procedures for preform fabrication is reduced to a problem in non-destructive testing and analysis. The underlying numerical problem of evaluating the stres distributions reduces to solving and Abel-type integral equation for α(r), which involves an indeterminacy, followed by the evaluation of linear functionals defined on α(r). It is shown how the known inversion formulae for the Abel-type integral equation can be used formally to reduce the numerical problem of evaluating the radial stress to the evaluation of a linear functional defined on the data δ(r) which bypasses the indeterminacy. When only the radial stress is required, the problem of actually solving the Abel-type integral equation is avoided. Methods for evaluating the non-radial stresses, which avoid the indeterminacy, are also derived.

For finite dimensional linear systems it is known that in certain circumstances the input can be retrieved from a knowledge of the output only. The main aim of this paper is to produce explicit formulae for input retrieval in systems which do not possess direct linkage from input to output. Although two different procedures are suggested the fundamental idea in both cases is to find an expression for the inverse transfer function of the system. In the first case this is achieved using a general method of power series inversion and in the second case by a sequence of elementary operations on a Rosenbrock type system matrix.

A new method of field quantization, in which the number operator has the form a*a, is proposed. Representations of this method are considered, particularly in reference to what conditions unique-vacuum state representations are required to satisfy.

An expression for the impluse due to a vapour (cavitation) bubble is obtained in terms of an integral over a nearby boundary. Examples for a point source near a free surface, rigid boundary, inertial boundary and a fluid of different density are considered. It appears that the sign of the impluse determines the direction a cavitation bubble will migrate and the direction of the high speed liquid jet during the collapse phase. The theory may explain recent observations on buoyant bubbles near an interface between two fluids of different densities.

Solutions of the stiff system of linear differential equations

are obtained in a form yielding tight estimates of their properties, and conditions are obtained under which the operator norm of the map from r to the solution x does not become exponentially large for small values of ε. When these conditions are satisfied, the solutions are shown to be close to those of Ax + r = 0, save at any singular points of A, and in boundary layers. The behaviour of solutions near admissible singular points is also obtained.

The results are used to characterize those boundary-value problems for the above system in which the solution defines maps from the data that are of “moderate” operator norm. This leads to a constructive existence theory for a limited class of boundary-value problems for the nonlinear system

It is suggested that the treatment of more general classes of boundary-value problems may be simplified using these results. By the use of simple examples, the problems involving large operator norms are shown to be related to the stability properties of the possible branches of the outer solutions close to those of

In Part I of this series, surface tension was included in the classical two-dimensional planing-surface problem, and the usual smooth-detachment trailing-edge condition enforced. However, the results exhibited a paradox, in that the classical results were not approached in the limit as the surface tension approached zero. This paradox is resolved here by abandoning the smooth-detachment condition, that is, by allowing a jump discontinuity in slope between the planing surface and the free surface at the trailing edge. A unique solution is obtainable for any input planing surface at fixed wetted length if one allows such jumps at both leading and trailing edges. If, as is the case in practice, the wetted length is allowed to vary, uniqueness may be restored by requiring either, but not both, of these slope discontinuities to vanish. The results of Part I correspond to the seemingly more-natural choice of making the trailing-edge detachment continuous, but it appears that the correct choice is to require the leading-edge attachment to be continuous.

Reacting systems with Arrhenius chemistry can exhibit multiple steady states for γ > 4(1 + β)/β where γ is activation energy and β is the heat of reaction. We give a detailed analysis of the extremely rapid jump which takes place as catalyst activity slowly decays through criticality. Previous analyses are asymptotic in the limit γ → ∞; here we relax the large γ assumption and only require that passage through criticality is slow.

Using geometric quantization, and accepting the quantum Hamiltonian of previous authors, we propose some candidate formulae for the quantum operator of an observable which is a quadratic form in the momenta.

The method of the Lie theory of extended groups has recently been formulated for Hamiltonian mechanics in a manner which is consistent with the results obtained using the Newtonian equation of motion. Here the method is applied to the three-dimensional time-independent harmonic oscillator and to the classical Kepler problem. The expected constants of motion are obtained. Previously unobserved relations between generators and invariants are also noticed.

Duffing's equation, in its simplest form, can be approximated by various non-linear difference equations. It is shown that a particular choice can be solved in closed form giving periodic solutions.

For the class of continuous games where σi and fi {σi, φ(σ1, …, σN)} are the strategy of and payoff to player i for i = 1, …, N, it is proved that the set of weak type I optima defined in Paper I conicide with the set of solution of a matrix condition. The latter condition restricts the equilibrium solutions of an adjustment process. Numerical results for N = 2 and N = 3 indicate that the set of all equilibrium solutions coincides with the above sets. The optima of types I to IV from Paper I are described fairly completely for the given class of games.

The paper is concerned with periodic solutions of the difference equation un + 1 = 2aun, where a and b are constants, with and b > 0. A new method is developed for dealing with this problem and, for period lengths up to 6, polynomial equations are given which allow the periodic solutions to be determined in a precise and practical manner. These equations apply whether the periodic solutions are stable or unstable and the elements of the cycle can be determined with an accuracy which is not affected by instability of the cycle.

A simple transformation puts the equation into the form , where A = a2 − a, and the detailed discussion is based on this simpler form. The discussion includes details such as the number of cyclic solutions for a given value of A, the pattern of the cycles and their stability. For practical purposes, it is enough to consider a restricted range of values of A, namely , although the equations obtained are valid for A > 2.

This paper demonstrates feasibility of aerodynamically-supported motion of a thin sheet near to a plane wall. Steady equilibrium is possible for uniform sheets only if they are deformable, and a set of possible equilibrium shapes is determined.

The Rapp formula of teletraffic dimensioning is generalized to admit an arbitrary renewal stream of offered traffic. The derivation proceeds from a heavy traffic approximation and provides also an estimate of the order of error involved in the Rapp formula. In principle, the method could be used to seek convenient higher order approximations.

Our equations give an incidental theoretical substantiation of an empirical result relating to marginal occupancy found recently by Potter.