Complex Bifurcation from Real Paths

A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions. A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given

Stabilization of Unstable Procedures: The Recursive Projection Method

Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a “black-box” time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

The quantum metrology triangle and the re-definition of the SI ampere and kilogram; Analysis of a reduced set of observational equations

We have developed a set of seven observational equations that include all of the physics necessary to relate the most important of the fundamental constants to the definitions of the SI kilogram and ampere. We have used these to determine the influence of alternative definitions being considered for the SI kilogram and ampere on the uncertainty of three of the fundamental constants (h, e and mu). We have also reviewed the experimental evidence for the exactness of the quantum metrology triangle resulting from experiments combining the quantum Hall effect, the Josephson effects and single-electron tunnelling.Comment: 16 pages, 3 figures & 5 table

Exponential ergodicity of the jump-diffusion CIR process

In this paper we study the jump-diffusion CIR process (shorted as JCIR), which is an extension of the classical CIR model. The jumps of the JCIR are introduced with the help of a pure-jump L\'evy process $(J_t, t \ge 0)$. Under some suitable conditions on the L\'evy measure of $(J_t, t \ge 0)$, we derive a lower bound for the transition densities of the JCIR process. We also find some sufficient condition guaranteeing the existence of a Forster-Lyapunov function for the JCIR process, which allows us to prove its exponential ergodicity.Comment: 14 page

SkyMapper and the Southern Sky Survey - a resource for the southern sky

SkyMapper is amongst the first of a new generation of dedicated, wide-field survey telescopes. The 1.3m SkyMapper telescope features a 5.7 square degree field-of-view Cassegrain imager and will see first light in late 2007. The primary goal of the facility is to conduct the Southern Sky Survey a six colour, six epoch survey of the southern sky. The survey will provide photometry for objects between 8th and 23rd magnitude with global photometric accuracy of 0.03 magnitudes and astrometry to 50 mas. This will represent a valuable scientific resource for the southern sky and in addition provide a basis for photometric and astrometric calibration of imaging data.Comment: 6 pages, 4 figures, proceedings of ESO Calibration Workshop 200