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But it seems that over most of a typical weather map there is no such sensitivity—so that in the end the difficulties of weather prediction are probably much more a result of computational irreducibility and of the sophisticated kinds of computations that the Principle of Computational Equivalence implies should often occur even in simple fluids.
And in general the existence of a universal Diophantine equation implies that any set obtained by any finite computation must correspond to the positive values of some polynomial.
Note that the fact that a number is normal in one base does not imply anything about its normality in another base (unless the bases are related for example by both being powers of 2).
Boltzmann then showed that his equation implied the so-called H Theorem, which states that a quantity equal to entropy in equilibrium must always increase with time. … But then it was noticed that since molecular collisions were assumed reversible, his derivation could be run in reverse, and would then imply the opposite of the Second Law.
And this implies that to find a particular digit of m t in base k will take altogether about t Log[t] 2 steps.
At a single step, standard static game theory from the 1940s implies that a player should always defect, but in the 1960s a folk theorem emerged that if a whole sequence of steps is considered then a possible strategy for perfectly rational players is always to cooperate—in apparent agreement with some observations on human and animal behavior.
But I strongly suspect that computational irreducibility prevents outcomes in systems like rule 30 and rule 110 from being found by computations that are in NC—implying in effect that allowing arbitrary connections does not help much in computing the evolution of such systems.
An important step was taken in 1941 by Andrei Kolmogorov who argued that if the eddies in such a cascade were in a statistical equilibrium, then dimensional analysis would effectively imply that the spectrum of velocity fluctuations associated with the eddies must have a k -5/3 distribution, with k being wavenumber.
The basic properties of concatenation then also imply that (a ∘ b) ∘ c a ∘ (b ∘ c) .
The Riemann Hypothesis then states that all r[i] satisfy Re[r[i]] 1/2 , which implies a certain randomness in the distribution of prime numbers, and a bound of order √ n Log[n] on PrimePi[n] - LogIntegral[n] .