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.

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.

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.

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.

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.

A simple model for a problem in combustion theory has multiple steady state solutions when a parameter is in a certain range. This note deals with the initial value problem when the initial temperature takes the form of a hot spot. Estimates on the extent and temperature of the spot for the steady state solution to be super-critical are obtained.

The effects of iron on the uniformity of the field produced by a current-carrying axisymmetric conductor are considered. Using a perturbation analysis a simple analytic expression is obtained which describes the field close to the axis of symmetry. A Fourier series approach is also used to provide an analytical solution to the problem and the accuracy of the perturbation method is estimated by comparing results.