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Erling Folner proved that the amenability or nonamenability of a countable group depends on the complexity of its finite subsets. Complexity has three measures: maximum Folner ratio, optimal cooling function, and minimum cooling norm. Our first aim is to show that, for a fixed finite subset, these three measures are tightly bound to one another. We then explore their algorithmic calculation. Our intent is to provide a theoretical background for algorithmically exploring the amenability and nonamenability of discrete groups.

This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite subdivision rules with one tile type. An example is given of a Lattes map which is not the subdivision map of a finite subdivision rule with either i) two tile types and 1-skeleton of the subdivision complex a circle or ii) one tile type., Comment: 28 pages, 17 figures

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f^n is the subdivision map of a finite subdivision rule., Comment: 10 pages

By use of a simple function to be known as a Natural Classification Function, multispectral signatures of 13910 LANDSAT pixels were classified into categories each of which when plotted on a character map, was found to correspond with a definable ground feature. Classification of the multispectral signatures was done in two stages. In the first stage, 65 categories were generated on the basis of the Natural Classification Function in which the independent variables are the reflectance values of multispectral signatures of pixels. In the second stage the 65 categories were merged into seven realistic categories on the basis of the variance of pixels within each category. The ground features represented on the character map are six plant associations and open bodies of water. At a cost of less than $50 for analysis of data and generating character maps for about 20 square miles, the method is considered to be a good tool for rapid exploration using LANDSAT Multispectral Data.