From mcbsszsf@fs1.ma.umist.ac.uk Wed Sep 22 17:19:20 1999 Date: Wed, 22 Sep 1999 14:19:11 +0100 From: Sergei Fedotov Reply-To: Sergei.Fedotov@umist.ac.uk To: Theodore Voronov Subject: Re: Talk Fedja! Posylau ABSTRACT **************************************************************************************************** The problem of wavefront propagation for the n-dimensional reaction-diffusion system involving Kolmogorov-Petrovskii-Piskunov kinetics and the diffusion transport with a finite velocity has been considered. By using a scaling procedure we have given an asymptotic derivation of the equation governing the evolution of a reaction front in the long-time large-distance limit. It has been found that this equation is identical in form to the relativistic Hamilton-Jacobi equation. The effects of an anisotropic diffusion have been examined in the framework of Hamilton-Jacobi theory. We have found that in the large scale asymptotic limit Hamiltonian dynamical system associated with reaction-diffusion system has a structure identical to that of general relativity. We have shown that the function determining the position of reaction front and its speed can be interpreted as the action functional for a particle in both gravitational and electromagnetic fields. The metric tensor of the 4-dimensional Riemannian space of general relativity has been determined through the diffusivity tensor, while the speed of light corresponds to the finite speed of ''diffusion'' waves. The analogy with the general relativity theory has allowed us to find the explicit formula for the reaction front position. ***************************************************************************************************** Best -- Sergei --------------------------------------- Dr. Sergei Fedotov Mathematics Department UMIST - University of Manchester Institute of Science and Technology Manchester M60 1QD UK Tel.: ++44 0161 200 3659 E-mail: sergei.fedotov@umist.ac.uk http://www.ma.umist.ac.uk/sf/index.html ---------------------------------------- From Bill.Lionheart@umist.ac.uk Wed Sep 22 17:19:44 1999 Date: Tue, 21 Sep 1999 14:10:09 +0100 From: Bill Lionheart To: Theodore Voronov Subject: Re: Talk Title: Geometric Methods in Inverse Problems A typical inverse problem for a system of partial differential equations is to recover the coefficients of the system from over specified boundary data. For example to determine information about a Riemannian metric on a manifold with boundary from the Dirichlet to Neumann mapping for Laplace's equation. This data does not completely determine the metric, an ambiguity arises due to the action of diffomorphisms which fix points on the boundary. In practical situations, such as electrical imaging where the metric is determined by the conductivity, one might know additional geometric information, such as a conformal structure (CO(n)-structure). One then investigates the group of diffeomorphisms preserving this structure. As an example we prove that a conformal diffeomorphism which is the identity on the boundary is the identity. We go on to discuss geometric aspects of other inverse problems for sytems of PDEs in electromagnetics and elasticity.. . Theodore Voronov wrote: > > Dear Bill, > > Could you provide some short and informal abstract of your talk, so that I > could post it on the web? It shouldn't be binding you in no way -- just to > give the audience some preliminary idea. > > Thank you a lot! > > Best, > Ted > > > -------------------------------------------------------------------- > Dr Theodore Voronov > Department of Mathematics > UMIST > PO Box 88 > Manchester M60 1QD, UK > > Phone: (0161) 200-3682 from the UK, (44 161) 200-3682 from overseas, > FAX: (0161) 200-3669 from the UK, (44 161) 200-3669 from overseas, > Email: theodore.voronov@umist.ac.uk > Home page: http://www.ma.umist.ac.uk/tv/PUBLIC -- ------------------------------------------------------------- Dr W.R.B. Lionheart, Department of Mathematics UMIST PO Box 88 Manchester M60 1QD UK Tel +44- 161-200-8978 Fax +44-161-200 3669 Moved to UMIST July 19th new email works now so please update your address book: Bill.Lionheart@umist.ac.uk -------------------------------------------------------------- British Workshops on Inverse Problems: EIDORS and Electrical Impedance Tomography web sites will soon be moving to http://www.ma.umist.ac.uk/bl/