An analysis is made of the Daley-Kendall and Maki-Thompson rumour models starting from general initial proportions of ignorants, spreaders and stiflers in the population. We investigate as a function of the initial conditions the composition of the final population when the rumour has run its course.

Our purpose in this paper is to display the stability analysis of Runge–Kutta methods applied to a Volterra integral equation of a simple form. As prerequisite we define, and then develop the structure of, the class of Runge–Kutta methods considered. The test equation is taken as the “;basic” equation ; the simple form of this equation permits ready insight into features which are more obscure when considering (as elsewhere [1], [2], [6]) equations of a more complicated form. Due to the structure of the methods and the nature of the test equation, the stability analysis reduces to the study of recurrence relations of the form Фk + 1 = MФ k + γk (k = 0, 1, 2, …) which are common in stability discussions in numerical analysis.

The linear long-wave equations with (and without) small ground motion are considered. The governing equations are represented in a matrix from and transformations are sought which reduce the system to (for example) a form associated with the conventional wave equation. Integration of the system is then immediate. It is shown that such a reduction may be acheived provided the variation in water depth is specified by certain multi-parameter forms.

In this paper various two-dimensional motions are determined for waves in a stratified region of infinite total depth with a free surface containing two superposed liquids, allowing for the effects of surface and interfacial tension. The fundamental set of wave-source potentials for the two layers is used to construct the set of slope potentials that produce discontinuous free-surface and interface slopes. The latter potentials are then utilized to obtain the potentials for waves due to both heaving vertical plates and incident progressive waves against a vertical wall. The underlying assumption of small time-harmonic motion pertains, described by a pair of velocity potentials for the two layers satisfying coupled linearized boundary-value problems, and all solutions are obtained in terms of their matching basic solutions. The technique for applying Green's theorem in the two layers is developed for use with the wave-source potentials, which themselves are found to obey a generalised reciprocity principle. Familiar results for a single liquid of infinite depth are hereby extended, but the new feature emerges of there being two types of progressive waves in all solutions. For ease of presentation the solutions are obtained for a particular relationship between surface and interfacial tension.

A proof is given for the existence and uniqueness of a stationary vacuum solution (M, g, ξ) of the boundary value problem consisting of Einstein's equations in an exterior domain M diffeomorphic to R × Σ (where Σ = R3\B(0, R)) and boundary data depending on the Killing field ξ on ∂Σ. The boundary data must be sufficiently close to that of a stationary, spatially conformally flat vacuum solution .

We consider a mesh grading quadrature method for real constant-coefficient Cauchy singular integral equations of index 0. The quadrature method is based on the trapezoidal rule. A complete stability and convergence analysis is given by the use of the noncompact perturbation analysis as in Jeon [10] and Elschner and Stephan [7]. The order of convergence can be arbitrarily high if the order of mesh grading is high enough. We also provide an efficient way of evaluating asymptotics of the solution at the end points. Experimentally, we observe that the method also works well for Cauchy singular integral equations with variable coefficients.

We consider the stability of high Reynolds number flow past a heated, curved wall. The influence of both buoyancy and curvature, with the appropriate sense, can render a flow unstable to longitudinal vortices. However, conversely each mechanism can make a flow more stable; as with a stable stratification or a convex curvature. This is partially due to their influence on the basic flow and also due to additional terms in the stability equations. In fact the presence of buoyancy in combination with an appropriate local wall gradient can actually increase the wall shear and these effects can lead to supervelocities and the promotion of a wall jet. This leads to the interesting discovery that the flow can be unstable for both concave and convex curvatures. Furthermore, it is possible to observe sustained vortex growth in stably stratified boundary layers over convexly curved walls. The evolution of the modes is considered in both the linear and nonlinear régimes.

In this work the modified Green's function technique for an exterior Dirichlet and Neumann problem in linear elasticity is investigated. We introduce a modification of the fundamental solution in order to remove the lack of uniqueness for the solution of the boundary integral equations describing the problems, and to simultaneously minimise their condition number. In view of this procedure the cases of the sphere and perturbations of the sphere are examined. Numerical results that demonstrate the effect of increasing the number of coefficients in the modification on the optimal condition number are also presented.

The allometric hypothesis which relates the shape (y) of biological organs to the size of the plant or animal (x), as a function of the relative growth rates, is ubiquitous in biology. This concept has been especially useful in studies of carcass composition of farm animals, and is the basis for the definition of maintenance requirements in animal nutrition.

When the size variable is random the differential equation describing the relative growth rates of organs becomes a stochastic differential equation, with a solution different from that of the deterministic equation normally used to describe allometry. This is important in studies of carcass composition where animals are slaughtered in different sizes and ages, introducing variance between animals into the size variable.

This paper derives an equation that relates values of the shape variable to the expected values of the size variable at any point. This is the most easily interpreted relationship in many applications of the allometric hypothesis such as the study of the development of carcass composition in domestic animals by serial slaughter. The change in the estimates of the coefficients of the allometric equation found through the usual deterministc equation is demonstrated under additive and multiplicative errors. The inclusion of a factor based on the reciprocal of the size variable to the usual log - log regression equation is shown to produce unbiased estimates of the parameters when the errors can be assumed to be multiplicative.

The consequences of stochastic size variables in the study of carcass composition are discussed.

Ginzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.

A study is made of the branching of time periodic solutions of a system of differential equations in R2 in the case of a double zero eigenvalue. It is shown that the solution need not be unique and the period of the solution is large. The stability of these solutions is analysed. Examples are given and generalizations to larger systems are discussed.

Page 242, bottom line.

Page 243, right side of equation (6).

Page 243, right side of equations (8) and (9).

Page 250, line 4 of Section 7.

A perturbation model is used to predict the distance jumped by a long-jumper for a range of tailwinds and headwinds. The zeroth-order approximation is based on gravity being the only force present, the effects of drag and lift only being included in the first-order corrections. The difference in predicted distances produced by the zeroth and first-order approximations is less than 2% for headwinds or tailwinds upto 4 ms−1. Most increases or decreases due to wind are caused by changes in the run-up speed, and consequently the take-off angle and speed.

Recently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.

In this paper we consider a simple, nonlinear optimal control problem with sufficient convexity to enable us to formulate its dual problem. Both primal and dual problems will include constraints on both the states and controls. The constraints in one problem may cause the “optimal” dual states to be discontinuous. However, we will look at conditions under which the presence of constraints does not force discontinuities and the optimal states and costates are absolutely continuous.

Various problems related to the propagation of small amplitude long waves on the surface of superfluid helium (helium II), usually called third sound, are studied on the basis of the appropriate governing equations. The two-fluid continuum model due to Landau is considered, with the effects of healing and relaxation incorporated, and viscosity, heat conduction and compressibility terms retained. The helium vapour is treated as a classical (Newtonian) compressible gas and the exact jump conditions across the liquid/vapour interface are employed. These liquid, vapour and jump equations constitute the exact problem although, in an effort to reduce the complexity of the equations, a simplified set of ‘model’ surface boundaiy conditions is also introduced. This full set of equations is non-dimensionalised taking care that all physical parameters are defined using only the undisturbed depth of the layer as the appropriate length scale. The ratio depth/wave-lenght (δ) is then a separate parameter as is the wave amplitude/depth ratio (ɛ). The limit which corresponds to the wave under discussion is then ɛ, δ → 0 with all the other parameters fixed.

A number of analyses are presented, four of which describe various aspects of the linearised theory and two examine the nature of the far-field nonlinear problem. Using the simplified surface boundary conditions we discuss in turn: the wave motion in the absence of healing: the rôle of a second wave speed leading to a wave hierarchy; and the effects of healing. The final linearised problem makes use of the full vapour model, but again the healing terms are ignored. This latter analysis suggests that if the upper boundary of the vapour is sufficiently close to the liquid surface then third sound is suppressed.

The complexity of the equations, particularly when the nonlinear terms are to be examined, is such that the incompressible limit is now taken in the absence of both healing and relaxation. Imposing the physically realistic limiting process (ɛ, δ → 0) we show that the only equation valid in the far-field is the Burgers equation. However, we also demonstrate that allowing the other parameters to be functions of e (which is not physically realisable in practice) it is easy to derive, for example, the Korteweg-de Vries equation.