We describe a computer proof of the 17-point version of a conjecture originally made by Klein-Szekeres in 1932 (now commonly known as the “Happy End Problem”) that a planar configuration of 17 points, no 3 points collinear, always contains a convex 6-subset. The proof makes use of a combinatorial model of planar configurations, expressed in terms of signature functions satisfying certain simple necessary conditions. The proof is more general than the original conjecture as the signature functions examined represent a larger set of configurations than those which are realisable. Three independent implementations of the computer proof have been developed, establishing that the result is readily reproducible.

The high Reynolds number flow past a circular cylinder with a trailing wake region is considered when the wake region is bounded and contains uniform vorticity. The formulation allows only for a single vortex pair trapped behind the cylinder, but calculates solutions over a range of values of vorticity. The separation point and length of the region are determined as outputs. It was found that using this numerical method there is an upper bound on the vorticity for which solutions can be calculated for a given arclength of the cavity. In some cases with shorter cavities, the limiting solutions coincide with the formation of a stagnation point in the outer flow at both separation from the cylinder and reattachment at the end of the cavity.

In this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. By defining appropriate sub-and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solutions.

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.

Conditions are given for a Ck map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton's method. For finitely differentiable maps and functions, these conditions are only necessary, but in the smooth case, that is, for k = ∞, they are also sufficient. The characterisation rests upon the structure of the fixed point set of T and the value of the derivative T′ there, and it is best possible as is demonstrated through examples.

Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

We study the existence of extremal solutions for an infinite system of first-order discontinuous functional differential equations in the Banach space of the bounded functions I∞(M).

We consider the stability of solutions for a family of Dirichlet problems with (p, q)-growth conditions. We apply the results obtained to show continuous dependence on a functional parameter and the existence of an optimal solution in a control problem with state constraints governed by the p(x)-Laplacian equation.

This paper studies a class of impulsive switched systems with persistent bounded disturbance using robust attractor analysis and multiple Lyapunov functions. Some sufficient conditions for internal stability of the systems are obtained in terms of linear matrix inequalities (LMI). Based on the results, a simple approach for the design of a feedback controller is presented to achieve a desired level of disturbance attenuation. Numerical examples are also worked out to illustrate the obtained results.

A k-out-of-N:G reparable system with an arbitrarily distributed repair time is studied in this paper. We translate the system into an Abstract Cauchy Problem (ACP). Analysing the spectrum of the system operator helps us to prove the well-posedness and the asymptotic stability of the system.

This paper is aimed at establishing sufficient computable criteria for the Euclidean null controllability of an infinite neutral differential system, when the controls are essentially bounded measurable functions on finite intervals, with values in a compact subset U of an m-dimensional Euclidean space with zero in its interior. Our results are obtained by exploiting the stability of the free system and the rank criterion for properness of the controlled system. An example is also given.

The following figure has been reprinted since the one on page 30 of the previous issue (ANZIAM J. vol. 48 part 1) is unsatisfactory.