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Host: Institute for Mathematics and its Applications
Homepage: http://www.ima.umn.edu/geoscience/fall/g2.html
Email: staff@ima.umn.edu
Organizers: William I. Newman, V.I. Keilis-Borok, Jean Carlson
Description:
The solid earth, oceans and atmospheres are profoundly nonlinear. While the oceans and
atmosphere are well-described by first principles equations, many nonlinear processes in the
solid earth lack such a description. Many geophysical problems possess an underlying discrete
character, in contrast with a continuous one, or alternatively do not offer a well-posed PDE
description but appear to be easy to characterize in a discrete fashion. Fracture of Earth
materials provides a good example. Grains in a rock, approximately 1 mm in size, constitute
the basic unit in this otherwise heterogeneous medium. These scenarios lend themselves in a
natural way to a cellular automaton or a lattice gas formulation depending on whether the time
dependence is intrinsically discrete or continuous, respectively. In an important subcategory of
cellular automaton problems, the accessible states in the problems are discrete, and especially
subject to delayed influences. Equations governing this class of problems are often called
Boolean Delay Equations. Illustrative examples include percolation problems, with the
attendant possibility of critical point behavior; earthquake and avalanche problems, including
the possibility of self organized criticality and scaling; and the modeling of complex transport
processes, which blend fluids with granular materials, and provide important insights in to
complicated problems in the establishing of landforms and fluvial drainage patterns.
Keywords: earthquakes and avalanches, percolation, cellular automata, Boolean difference equations, sandpiles
Date received: January 31, 2001
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