It has been known for some time that the Boltzmann weights of the chiral Potts model can be parametrised in terms of hyperelliptic functions. but as yet no such parametrisation has been applied to the partition and correlation functions. Here we show that for N = 3 the function S(tp) that occurs in the recent calculation of the order parameters can he expressed quite simply in terms of such a parametrisation.

We consider the following model that describes the spread of n types of epidemics which are interdependent on each other:

Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions (u1, u2, …, un), that is, for each 1 ≤ i ≤ n, ui, is periodic and θiui ≥ 0 where θi, ε (1, −1) is fixed. Examples are also included to illustrate the results obtained.

A small gas bubble in a liquid, when driven by intense ultrasound, collapses and emits light in a process called Single-Bubble Sonoluminescence (SBSL). While the dynamics of driven bubbles are well studied, less is known of the physical conditions in the gas or whether it is necessary to include ionisation in simpler studies of bubble dynamics. In this study, a model was derived from Rayleigh-Plesset dynamics, a van der Waals equation of state and the first law of thermodynamics (including interfacial heat transfer and ionisation). Stronger model ionisation reduced the maximum collapse temperature, and altered other collapse characteristics. Chaotic parameter regions are proximal to, but not coincident with, known stable SL regions. Resonant behaviour was only markedly affected by ionisation close to these chaotic regions.

This paper gives explicit, applicable bounds for solutions of a wide class of third-order difference equations with nonconstant coefficients. The techniques used are readily adaptable for higher-order equations. The results extend recent work of the authors for second-order equations.

The results obtained by A. J. Roberts and N. Ujević in a recent paper are generalised. A number of inequalities for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or R-integrable functions are derived. Also, some error estimates for the derived formulae are obtained.

Recursive parametric series solutions are developed for polynomial ODE systems, based on expanding the system components in series of a form studied by Weiss. Individual terms involve first-order driven linear ODE systems with variable coefficients. We consider Lotka-Volterra systems as an example.

We examine the transmission problem in a two-dimensional domain, which consists of two different homogeneous media. We use boundary integral equation methods on the Maxwell equations governing the two media and we study the behaviour of the solution as the two different wave numbers tend to zero. We prove that as the boundary data of the general transmission problem converge uniformly to the boundary data of the corresponding electrostatic transmission problem, the general solution converges uniformly to the electrostatic one, provided we consider compact subsets of the domains.

A nutrient-consumer model involving a distributed delay in material recycling and a discrete delay in growth response has been analysed. Various easily verifiable sets of sufficient conditions for global asymptotic stability of the positive equilibrium solution of the model equations have been obtained and the length of the delay in each case has been estimated.