A Girsanov approach to slow parameterizing manifolds in the presence of noise

We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure

Fuzzy Fluid Mechanics in Three Dimensions

We introduce a rotation invariant short distance cut-off in the theory of an ideal fluid in three space dimensions, by requiring momenta to take values in a sphere. This leads to an algebra of functions in position space is non-commutative. Nevertheless it is possible to find appropriate analogues of the Euler equations of an ideal fluid. The system still has a hamiltonian structure. It is hoped that this will be useful in the study of possible singularities in the evolution of Euler (or Navier-Stokes) equations in three dimensions.Comment: Additional reference

Solospun : the long staple weavable singles yarn

SolospunTM is a spinning technology which provides the means to produce a singles yarn that can be successfully woven as either warp or weft. The technology is a versatile and cost effective alternative to two-folding, Sirospun or sizing. It offers significant benefits in terms of efficiency and productivity. SolospunTM is a simple, inexpensive, clip-on attachment for the spinning of long staple, weavable singles yarns. The technology is the result of a joint development between CSIRO Textile and Fibre Technology, The Woolmark Company and WRONZ and was commercially released in 1998. It is now successfully operating in worsted mills worldwide. The SolospunTM hardware consists of a pair of rollers held in a bracket, which is clipped onto the front of the pendulum-drafting arm. Each roller is positioned immediately below, and parallel to, each top front draft roller where it interacts with the emerging drafted fibre strand before twist insertion. The roller-fibre interaction subtly changes the structure of the yarn, which dramatically increases its abrasion resistance. This allows Solospun yarns to be woven without twofolding or sizing, providing significant cost savings. Depending on fabric structure, weaving performance is equivalent to that of conventional two-fold yarns. Fabrics woven from SolospunTM yarns are of good quality and appearance. Some aspects, including preferred hand and FAST results, are equal or superior to two-fold yarn fabrics. As with compact and condensed spinning, SolospunTM offers significant reductions in yarnhairiness. However, Solospun can be simply implemented on existing spinning frames and goes much further by making the singles yarns weavable.<br /

Could Fire and Rescue Services identify older people at risk of falls?

Protecting or improving the efficiency and effectiveness of services while reducing costs in response to public sector funding reductions is a significant challenge for all public service organisations. Preventing falls in older people is a major public health objective. We propose here an innovative model of community partnership with Fire and Rescue Services assisting falls prevention services to enhance the safety and well-being of older people in local communities through early identification of those who are at risk of injury from a fall or accidental domestic fire