UK Nonlinear News, February 2000
Time delays and non-local nonlinearities are an important aspect of many biological and ecological systems. Not only are they highly relevant from an accurate modelling point of view, but their consequences can be extensive, including the introduction of new dynamics. In this thesis we consider simple models to derive generic results, concerned with novel aspects of delays and spatial effects. In addition we consider some specific models, in an attempt to make any existing non-local effects more relevant to the original problem. Since in many cases these are initial enquiries considering the possible dynamics of a system, we predominantly concern ourselves with the stability of, and solutions bifurcating from, (uniform) equilibria.
Source: David Schley
The problem is motivated by the need to provide artificial ventilation for neonatal infants. In the first year of the project, time was spent on a neonatal intensive care unit, and experiments were performed on the SLE2000 ventilator. The experiments showed that the waves travelled at a higher speed than generally thought leading to the conclusion that existing mathematical models were inappropriate.
The second and third years of the project were devoted to mathematical modelling and analysis with three parts. First, an acoustic analysis of the oscillating flow in a tube was performed and predicted shear stresses were compared favourably with experiment. Secondly, a transport model was developed based on shear dispersion in oscillatory flow, extending the Taylor model to compressible flow. Third, the Lagrangian transport of particle paths was studied. This theory extends chaotic advection into the compressible regime. The particles demonstrated a definite drift, with strong dependence on location in the pipe, and the dependence of the Womersley and Peclet number were determined. Some results on the role of molecular diffusion were also determined, by adding a stochastic input to the the Lagrangian transport equations. Overall the thesis conclusively established the importance of compressibility in models of jet ventilators.
The thesis research was supported by an EPSRC-CASE award in collaboration with SLE Ltd and the Royal Victoria Infirmary.
Source: Tom Bridges
I created a unique and general method for solving non-linear queries of any level of complexity. This method can be used in the calculation of optimization problems and solution of nonlinear equations for such areas as Modeling and Simulation, Operations Management, Machine Vision, Robotics & Automation.
Using it I am capable of solving quite complicated non-linear optimization problems of the classic form:
Wi(X) <= 0, i = 1, ... , m1,
Gi(X) = 0, i = 1, ... , m2,
where X is an n-vector and the functions F, Wi and Gi should not have gaps and can be linear, nonlinear, multi-extreme, convex or non-convex. My universal method of finding optimal solution for the above-enumerated programming queries consist of two well-known and simple algorithms: the method of proportionally deformed polyhedron and the method of gradient descending. It is well known that these two methods usually produce only local, separate solutions. But this is not true for my technique where these simple methods always acquire global solutions. Obviously there is no magic in it. The results obtained by me (proved theoretically) are the effects of extending the theory of convex functions.
My algorithm was checked during two years of investigations and illustrated by solving of several thousands of test and real nonlinear equations. Results obtained were always satisfactory which means that algorithm was able to find the global optimum for multi-extreme function (I was only calculating that type of function).
To my knowledge there is no other algorithm with such broad universality as mine.
Source: Kamyshnikov Vladimir
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Last Updated: 6th February 2000.