Browsing by Browse by FOR 2008 "010203 Calculus of Variations, Systems Theory and Control Theory"

We study properties of covariant derivatives of vector fields along curves on Riemannian manifolds mapped by rolling the manifolds without slip or twist. A natural definition of Riemannian polynomials arises from these properties.

In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval. We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.

We study rolling of one submanifold upon another submanifold, both isometrically embedded in a Riemannian manifold.We generalise the definition of rolling in Sharpe (1997). In this new definition, the Euclidean group of motions is replaced by the Lie group of orientation preserving isometries. We show that rolling in this general situation is unique. We prove a theorem that enables us to learn how to roll non-Euclidean manifolds that result from deformations of Euclidean submanifolds from the knowledge of the kinematic equations of rolling these Euclidean submanifolds. Taking into account that the ellipsoid is a deformed sphere, we apply the above mentioned theorem and the kinematic equations for the rolling sphere to derive the kinematic equations for rolling the ellipsoid. This example serves as a motivation to roll other manifolds.

Different methods for analyzing the sensitivity of the direct signal transduction pathway of receptor-induced apoptosis to parameter changes are presented. Apoptosis is a form of programmed cell death, removing unwanted cells within multicellular organisms to maintain a proper balance between cell reproduction and death. The results indicate the importance of controlling activated caspases by direct inhibition to prevent apoptosis. A misregulation of IAP molecules, one of the main inhibitors, appears to be especially critical. The results indicate how an increased production of this molecule promotes survival and might promote cancer progression, while a reduced degradation might not, thereby providing insight of potential pharmaceutical relevance and also stimulating experimental verification. The different engineering methods applied, nicely complement each other to provide valuable insight into this important process. Because IAPs, among others, are also an important connection to other signaling pathways, the results will enable a more efficient extension of the current model. This is outlined at the example of Tumor Necrosis Factor induced signaling pathways.

Linkages relating to the teaching of Rural Science and of Agriculture at the University of New England (UNE) are diverse; together with the wide range of degrees and units, these linkages are rapidly evolving in response to both internal and external pressures. Some of these linkages within and outside UNE-include cross-School and cross-Faculty co-operation, as well as many industry linkages. UNE's involvement in seven Cooperative Research Centres has ensured for it great relevance to the agricultural industries of the broader region. Many agencies, including CSIRO and the NSW Department of Primary Industries, provide the university with substantial assistance in teaching, both at the undergraduate and postgraduate level. The University of New England also encourages close links between its students and the fanning community, including with the university's rural properties. UNE's linkages with other universities, both within Australia and internationally, are usually dependent on specific funded projects. Several initiatives from UNE have seen uptake by other universities, such as the use of multimedia and decision-support tools in teaching. Postgraduate training benefits from linkages with many Australian and overseas universities, as well as from industry connections. There are many opportunities for enhancement of our teaching through linkages, but most depend on securing funding. Successful partnerships will be best achieved through the development and implementation of carefully designed strategies that build upon the substantial goodwill that exists between staff and students across the university and from various industry sectors.

We are interested in knowing whether or not the motion of two smooth surfaces rolling on each other, without slip or twist, can be controlled. We present a few cases of surfaces rolling on a tangent plane where we show that controllability fails and why. The control system associated to a rolling motion defines a distribution in the configuration space. If this rolling distribution is bracket generating, local controllability is guaranteed. After deriving the kinematic equations for rolling Euclidean submanifolds of co-dimension one, we derive a condition for local controllability.