9626575 Burton ABSTRACT Original research is pursued by the investigator and his colleagues on: (i) topics of neural networks, such as feed-forward networks and unsupervised networks, (ii) algorithms to estimate the expected extreme value of a time series given some record of the past, (iii) dynamical and metric properties of various continued fraction type expansions, and (iv) stationary processes and random fields that arise as limits of random substitutions with a view toward random tiling systems. The methods employed are taken from ergodic theory, probability, and statistical physics. This work consists of four topics: (i) mathematical properties of neural networks, (ii) extreme value estimation, (iii) continued fraction algorithms, and (iv) random substitution processes. (i) A neural network is a type of self-programming computer that learns by example, essentially adjusting itself to adapt to its environment. Neural networks--though not yet well understood, mathematically--are already in commercial use in such areas as handwriting readers, DNA classifiers, and financial forecasters. With greater understanding will come more enlightened use. (ii) The investigator and his colleagues are developing extreme value estimation algorithms to help solve an estimation problem needed in the design of structures such as off-shore oil platforms which require a high probability of withstanding storms. Typically, there is limited historical data from which to make these estimates. The algorithms are used to estimate the most powerful storm that is likely to occur in the vicinity of the structure in the next 20 years. (iii) Continued fractions have the property of being the most economical way to approximate a real number by a fraction. They consist of fractions containing fractions containing fractions, etc., etc. The investigator and his colleagues look at these as dynamical systems to study their properties. This work ties together ideas from probability theory, group theory, and flows on surfaces. (iv) The investigator is studying random substitution schemes as they are natural ways of creating strings of symbols with random properties, using a simple recipe. This idea may also be used to generate tilings of the plane or space that have properties of randomness and determinism. The construction begins with a set of tiles (or simple shapes) and decompositions of these tiles so that each piece of each decomposition is a scale replica of one of the original tiles. This procedure is continued, decomposing to finer and finer scales, zooming the scale. This may give models of materials in nature, as has been the case for non-random versions of this procedure.
