The method of asymptotic matching introduced by Buchwald [I] is adapted to the case of the diffraction of plane longitudinal and shear waves by cylindrical cavities with elliptic cross-sections. It is assumed that the dimensions of the cross section are small compared with the wavelength of the incident waves. Asymptotic formulae for the scattered wave potentials are obtained.

The method is valid when the cavity reduces to a two-dimensional stress free crack whose length is small compared with the wavelength. Formulae for the scattered waves, and for the stress-concentrations at the crack tips are obtained.

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 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.

During the late 1940s, T. M. Cherry published a series of research papers on uniform asymptotic formulae for transition points in ordinary differential equations. This work, together with his research into transonic gas flows, for which it was a necessary precursor, is probably his best known and most widely quoted piece of research. An analysis is made of the impact of Cherry's work on subsequent developments in this field, both in comparison to the work of others and with respect to Cherry's work on other topics.

The partially stiff system of ordinary differential equations

is studied by the methods developed in the earlier papers in this series. Here e is a small positive parameter, x and y are n- and m-vectors respectively, and A is nonsingular. A useful basis for the solution space of the homogeneous system is constructed and the method of variation of parameters is used to obtain useful representations of all solutions. Sufficient conditions are derived under which the formal approximation

is close to the actual solution. it is found that purely imaginary eigenvalues for A require more stringent requirements for the formal technique to be valid. A brief discussion of the case when A is singular shows that there are a great number of possibilities requiring consideration for a general theory. it is suggested that local computation of such cases is likely to be the most effective weapon for any specific system.

The steady, axisymmetric flow induced by a point sink (or source) submerged in an inviscid fluid of infinite depth is computed and the resulting deformation of the free surface is obtained. The effect of surface tension on the free surface is determined and is the new component of this work. The maximum Froude numbers at which steady solutions exist are computed. It is found that the determining factor in reaching the critical flow changes as more surface tension is included. If there is zero or a very small amount of surface tension, the limiting factor appears to be the formation of small wavelets on the free surface; but, as the surface tension increases, this is replaced by a tendency for the lowest point on the free surface to descend sharply as the Froude number is increased.

Sufficient conditions for an equilibrium point to be an attractor or a global attractor are derived for a class of first-order difference equations which need not be continuous at the equilibrium point. These conditions involve Lyapunovlike functions which need not be continuous and are applied to the logistic equation with a piecewise continuous control.

The fractal kinetics curve derived by Savageau is analysed to show that its parameters are not uniquely determined given four appropriately situated data points. Comparison is made with an alternate fractal Michaelis-Menten equation derived by Lopez-Quintela and Casado.

The laser Lorenz equations are studied by reducing them to a form suitable for application of an extension of a method developed by Kuzmak. The method generates a flow in a Poincaré section from which it is inferred that a certain Hopf bifurcation is always subcritical.

This paper studies the decentralized control and stabilization of two-input, two-output finite dimensional linear systems. A representation result for the system and a characterization of all stabilizing controllers are given in terms of certain fixed polynomial matrices and a stability constraint.

Numerical evidence is presented for the existence of unsteady periodic gravity waves of large height in deep water whose shape changes cyclically as they propagate. It is found that, for a given wavelength and maximum wave height, cyclic waves with a range of cyclic periods exist, with a steady wave of permanent shape being an extreme member of the range. The method of solution, using Fourier transforms of the nonlinear surface boundary conditions, determines the irrotational velocity field in the water and the water surface displacement as functions of space and time, from which properties of the waves are demonstrated. In particular, it is shown that cyclic waves are closer to the point of wave breaking than are steady permanent waves of the same wave height and wavelength.

In this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computed by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.

In the present paper the problem of reflection of water waves by a nearly vertical porous wall in the presence of surface tension has been investigated. A perturbational approach for the first-order correction has been employed as compared with the corresponding vertical wall problem. A mixed Fourier transform together with the regularity property of the transformed function along the positive real axis has been used to obtain the potential functions along with the reflection coefficients up to first order. Whilst the problem of water of infinite depth is the subject matter of the present paper, a similar approach is applicable to problems associated with water of finite depth.

We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

Understanding how seasonal patterns change from year to year is important for the management of infectious disease epidemics. Here, we present a mathematical formalization of the application of complex demodulation, which has previously only been applied in an exploratory manner in the context of infectious diseases. This method extracts the changing amplitude and phase from seasonal data, allowing comparisons between the size and timing of yearly epidemics. We first validate the method using synthetic data that displays the key features of epidemic data. In particular, we analyse both annual and biennial synthetic data, and explore the effect of delayed epidemics on the extracted amplitude and phase. We then demonstrate the usefulness of complex demodulation using national notification data for influenza in Australia. This method clearly highlights the higher number of notifications and the early peak of the influenza pandemic in 2009. We also identify that epidemics that peaked later than usual generally followed larger epidemics and involved fewer overall notifications. Our analysis establishes a role for complex demodulation in the study of seasonal epidemiological events.

In oscillatory shear experiments, the values of the storage and loss moduli, $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$, respectively, are only measured and recorded for a number of values of the frequency $\unicode[STIX]{x1D714}$ in some well-defined finite range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$. In many practical situations, when the range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$. For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum $H(\unicode[STIX]{x1D70F})$ has been determined through the inversion of the measured storage and loss moduli $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$. For the recovery of an approximation to $H(\unicode[STIX]{x1D70F})$, in order to cope with the limited range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that $H(\unicode[STIX]{x1D70F})$ has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous $H(\unicode[STIX]{x1D70F})$. In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous $H(\unicode[STIX]{x1D70F})$ from limited measurements of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$.

We present a Rayleigh–Ritz method for the approximation of fluid flow in a curved duct, including the secondary cross-flow, which is well known to develop for nonzero Dean numbers. Having a straightforward method to estimate the cross-flow for ducts with a variety of cross-sectional shapes is important for many applications. One particular example is in microfluidics where curved ducts with low aspect ratio are common, and there is an increasing interest in nonrectangular duct shapes for the purpose of size-based cell separation. We describe functionals which are minimized by the axial flow velocity and cross-flow stream function which solve an expansion of the Navier–Stokes model of the flow. A Rayleigh–Ritz method is then obtained by computing the coefficients of an appropriate polynomial basis, taking into account the duct shape, such that the corresponding functionals are stationary. Whilst the method itself is quite general, we describe an implementation for a particular family of duct shapes in which the top and bottom walls are described by a polynomial with respect to the lateral coordinate. Solutions for a rectangular duct and two nonstandard duct shapes are examined in detail. A comparison with solutions obtained using a finite-element method demonstrates the rate of convergence with respect to the size of the basis. An implementation for circular cross-sections is also described, and results are found to be consistent with previous studies.

Steady state solutions for spontaneous thermal ignition in a unit sphere are considered. The multiplicity of unstable, intermediate, steady state, temperature profiles is calculated and shown for selected parameter values. The crossing of the temperature profiles corresponding to the unstable, intermediate, steady states is exhibited in a particular case and is proven in general using elementary ideas from analysis. Estimates of the location of crossing points are given.

The flow between two eccentric rotating cylinders with a slotted sleeve placed around the inner cylinder is determined numerically using an exponentially fitted finite-volume method. The flow field is determined for various Reynolds numbers, eccentricities and rotational speeds for the cases when the cylinders rotate in the same sense and rotate in opposite senses. The flow field developed when both cylinders rotate in the same sense is characterised, for sufficiently large eccentricity and rotational rate, by two counter-rotating eddies. Only one eddy is observed when the cylinders rotate in opposite senses. The presence of these eddies restricts the flow through the slotted sleeve in the former case but encourages through flow in the latter. For both cases, the eccentricity affects the location of the eddies, while changing the relative rotational rate only affects the eddy location for the case when the cylinders rotate in opposite directions. The change in Reynolds number has little effect on the flow field for the problems considered here. The vorticity generated by the slotted sleeve is convected into the main body of the flow field. No inviscid core within the main body of the flow field is observed for the range of Reynolds number considered.

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.