In the second, Helen Billinge suggests that some reformulation of Gleason's theorem in three-space may have a constructive proof. It turns out that Gleason's formulation admits a constructive proof as it stands. In this paper we discuss this seemingly anomalous situation. Gleason's theorem itself is somewhat peripheral to the discussion. What is interesting is the relationship of classical mathematics to constructive mathematics that is highlighted by this misunderstanding.
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The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest. Pointwise differentiability , LaTeX file , dvi file , pdf file , ps file .