So this means that it becomes possible to make a wide range of new models for biological systems—and potentially to see how to emulate the essence of their operation, say for medical purposes.

And I have no doubt that in time this will yield a remarkably broad range of applications.

Measurements suggest that these values are uniformly distributed in the range 0 to 1, but despite a fair amount of mathematical work since the 1940s, there has been no substantial progress towards proving this.

If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis.

Multivalued logic As noted by Jan Łukasiewicz and Emil Post in the early 1920s, it is possible to generalize ordinary logic to allow k values Range[0, 1, 1/(k - 1)] , say with 0 being False , and 1 being True .

Randomness in markets After the somewhat tricky process of correcting for overall trends, empirical price data from a wide range of markets seem to a first approximation to follow random walks and thus to exhibit Gaussian fluctuations, as noted by Louis Bachelier in 1900.

there are general principles that govern the behavior of a wide range of systems, independent of the precise details of each system.

But it turns out that if one picks a number at random subject only to the constraint that its size be in a certain range, then it is overwhelmingly likely that the number one gets will have a digit sequence that is essentially random.

Partial Differential Equations By introducing continuous cellular automata with a continuous range of gray levels, we have successfully removed some of the discreteness that exists in ordinary cellular automata.

Considering the amount of mathematical work that has been done on partial differential equations, one might have thought that a vast range of different equations would by now have been studied.