Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem

Given a submersion $\pi:Q \to M$ with an Ehresmann connection $\mathcal{H}$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\mathcal{H}$ along with a new form of parallel transport of covectors vanishing on $\mathcal{H}$. Using the Pontryagin maximum principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.Comment: 31 page

Asymptotics of Invariant Metrics in the normal direction and a new characterisation of the unit disk

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in $\mathbb C$ in terms of the asymptotic behaviour of quotients of invariant metrics

A Fatou-Bieberbach domain in C^2 which is not Runge

We give an example of a Fatou-Bieberbach domain which is not Runge in C^2

Funeral insurance

Funeral insurance has existed at least since antiquity, and it remains popular in many parts of Africa today. Yet the study of funeral insurance as a distinct form of insurance has hitherto been neglected. This paper presents a model in which funeral insurance combines regular life insurance with a restriction on how the payout is spent. The model predicts that there is an intermediate range of income and wealth where funeral insurance is demanded. The prediction is tested on a nationally representative sample of black South African households, a setting where both life and funeral insurance are widely available. The model also gives conditions under which funeral insurance is not demanded at any level of income and wealth. This may explain why funeral insurance is less popular in developed countries, even among the relatively poor.

Acoustic multipole sources from the Boltzmann equation

By adding a particle source term in the Boltzmann equation of kinetic theory, it is possible to represent particles appearing and disappearing throughout the fluid with a specified distribution of particle velocities. By deriving the wave equation from this modified Boltzmann equation via the conservation equations of fluid mechanics, multipole source terms in the wave equation are found. These multipole source terms are given by the particle source term in the Boltzmann equation. To the Euler level in the momentum equation, a monopole and a dipole source term appear in the wave equation. To the Navier-Stokes level, a quadrupole term with negligible magnitude also appears.Comment: 5 pages, to be published in Proceedings of the 36th Scandinavian Symposium on Physical Acoustic