Symmetric swallows

A simple problem relating to birds chasing each other gives rise to a homogeneous differential equation. The solution draws on student skills in differential equations and basic co-ordinate geometry.     

Constitutive relations describing creep deformation for multi-axial time-dependent stress states

A theory of primary and secondary creep deformation in metals is presented, which is based upon the concept of tensor internal state variables and the principles of continuum mechanics and thermodynamics. The theory is able to account for both multi-axial and time-dependent stress and strain states. The wellknown concepts of elastic, anelastic and plastic strains follow naturally from the theory. Homogeneous stress states are considered in detail and a simplified theory is derived by linearizing with respect to the internal state variables. It is demonstrated that the model can be developed in such a way that multi-axial constant-stress creep data can be presented as a single relationship between an equivalent stress and an equivalent strain. It is shown how the theory may be used to describe the multi-axial deformation of metals which are subjected to constant stress states. The multi-axial strain response to a general cyclic stress state is calculated. For uni-axial stress states, square-wave loading and a thermal fatigue stress cycle are analysed.     

The routes of unity

A model for car following on a closed loop is defined. The stability of the solutions of the model is investigated by considering the evolution of the roots of the corresponding characteristic equation in the complex plane. The solution provides a motivation for investigating the behaviour of the roots of a simple class of algebraic equation.     

Lamé hypersurfaces and Lagrange multipliers

Starting with a standard calculus problem of finding the rectangle of maximum area, which can be inscribed in an ellipse, we generalizethe result to Lamé curves and hypersurfaces, before investigating the volume enclosed by a generalized Lamé hypersurface. Suggestions of classroom use and a selection of possible classroom exercises are also given.     

Molecular distribution sensing in a fluorescence resonance energy transfer based affinity assay for glucose

A newly developed method for determining molecular distribution functions is applied to a widely researched glucose affinity sensor. The reduction in fluorescence resonance energy transfer (FRET) to a malachite green (MG)-dextran complex from allophycocyanin (APC) bound to concanavalin A (ConA) due to displacement of the complex by glucose from ConA provides the basis of the assay. The higher sensitivity and specificity of a new approach to fluorescence decay analysis, over the methods based on conventional Forster-type models, is demonstrated and critical parameters in competitive binding FRET sensing derived.