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.

We consider a hybrid switch which provides integrated packet (asynchronous) and circuit (isochronous) switching. Queue size and delay distribution of the packet switched traffic in the steady state are derived by modelling the packet queue as a queue in a Markovian environment. The arrival process of the packets as well as of the circuit allocation requests are both modelled by a Poisson process. The analysis is performed for several circuit allocation policies, namely repacking, first-fit (involving static or dynamic renumbering) and best-fit. Both exact results and approximations are discussed. Numerical results are presented to demonstrate the effect of increase in packet and circuit loading on the packet delay for each of the policies.

Experimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.

We consider a single server queueing system where each customer visits the queue a fixed number of times before departure. A customer on his j th visit to the queue is defined to be a class-j -customer. We obtain the joint probability generating function for the number of class-j-customers and also obtain the Laplace-Stieltjes transform for the total response time of a customer.

The paper investigates the effect of a static magnetic field on the helical flow of an incompressible cholestenc liquid crystal with director of unit magnitude between two coaxial circular cylinders rotating with different angular velocities about their common axis and moving with different axial velocities. At low shear rates with a weak magnetic field in the axial direction, the axial velocity, the angular velocity and the orientation of molecules between the two cylinders have been obtained. It is found that the magnetic field has influenced the orientation of molecules while the axial velocity and the angular velocity remain uneffected by the magnetic field.

In this paper we establish some generalizations of a weighted trapezoidal inequality for monotonic functions and give several applications for the r-moments, the expectation of a continuous random variable and the Beta and Gamma functions.

A new numerical method is applied to the problem of inviscid irrotational flow past a semi-infinite stern-like body of general shape. Both smooth-detachment and stagnant-detachment flows are considered, in the context of varying the geometry of the stern to generate very small waves, with the eventual aim of eliminating waves altogether. The results of this work confirm previously published results for the smooth-detachment case, but cast doubt on the existence of waveless solutions for stagnant detachment.