Dartboard arrangements with a concave penalty function

This paper investigates combinatorial arrangements of the dartboard to maximize a penalty function derived from the differences of adjacent sectors. The particular penalty function is constructed by summing the absolute differences of neighbouring sectors raised to a power between zero and one. The arrangement to give the maximum penalty is foun

On the nonlinear Neumann problem with critical and supercritical nonlinearities

We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent. In the first part of this work it is assumed that the coeffcients Q and h are at least continuous. Moreover Q is positive on overline Omega and lambda > 0 is a parameter. We examine the common effect of the mean curvature and the shape of the graphs of the coeffcients Q and h on the existence of low energy solutions. In the second part of this work we consider the same problem with Q replaced by - Q. In this case the problem can be supercritical and the existence results depend on integrability conditions on Q and h

Optimal strategies in political elections

In the Majoritarian Parliamentary System, the government has a constitutional right to call an early election. This right provides the government a control to achieve its objective to remain in power for as long as possible. We model the early election problem mathematically using opinion polls data as a stochastic process to proxy the government's probability of re-election. These data measure the difference in popularity between the government and the opposition. We fit a mean reverting Stochastic Differential Equation to describe the behaviour of the process and consider the possibility for the government to use other control tools, which are termed `boosts' to induce shocks to the opinion polls by making timely policy announcements or economic actions. These actions improve the government's popularity and have some impact upon the early-election exercise boundary

An innovative learning model for computation in first year mathematics

MATLAB is a sophisticated software tool for numerical analysis and visualisation. The University of Queensland has adopted Matlab as its official teaching package across large first year mathematics courses. In the past, the package has met severe resistance from students who have not appreciated their computational experience. Several main factors contribute: Firstly, the software is numerical rather than symbolic, providing a departure from the thinking patterns presented in lectures and tutorials. Secondly, many students cannot see a direct connection between the laboratory exercises and core course material from lectures. Thirdly, the students find hurdles to entry as commands often return annoying error messages and don't execute, and programs are difficult to write and debug. Overall, the details of the mathematics are lost in trying to negotiate the software. After considerable effort in tuning, it appears that a sequence of innovations has captured student support and added considerable value to both the computational and traditional learning process

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators

A continuous time model for election timing

We consider a continuous time model for election timing in a Majoritarian Parliamentary System where the government maintains a constitutional right to call an early election. Our model is based on the two-party-preferred data that measure the popularity of the government and the opposition over time. We describe the poll process by a Stochastic Differential Equation (SDE) and use a martingale approach to derive a Partial Differential Equation (PDE) for the government’s expected remaining life in office. A comparison is made between a three-year and a four-year maximum term and we also provide the exercise boundary for calling an election. Impacts on changes in parameters in the SDE, the probability of winning the election and maximum terms on the call exercise boundaries are discussed and analysed. An application of our model to the Australian Federal Election for House of Representatives is also given

An early political election problem

Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. Whereas the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining period in power. We find the optimal control for the government by locating an exercise boundary, which indicates whether or not a premature election should be called. This problem draws upon the methods of optimal stopping and stochastic control