„Toute information est la valeur moyenne, étendue à l’ensemble des états, de la résultante de l’application d’un opérateur linéaire S sur le logarithme de la probabilité a priori de chaque état.“ (p. 43) #Schützenberger #information #probabilité
„Théorème: L’information est une fonction additive pour la composition des variables aléatoires indépendantes.“ (p. 46) #Schützenberger #information
„C’est la première propriété commune aux informations de SHANNON et de FISHER que d’être relatives à des processus dans lequels joue un élément de nature aléatoire.“ (p. 39) #Schützenberger #Shannon #Fisher #information
„To permit failure as an independent logical entity means that one does not state the axioms in a rigorous manner. The axioms are not of the form: if A and B happen, C will follow. The axioms are always of this variety: if A and B happen, Cwill follow with a certain specified probability, D will follow with another specified probability, and so on. In other words, in every situation several alternatives are permitted with various probabilities. Mathematically it is simplest to say that anything can follow upon anything in accordance with a probability matrix. You may put your question in this manner: If A and B have happened, what is the probability that C will follow? This probability pattern gives you a probabilistic system of logics. Both artificial and natural automata should be discussed in this system as soon as there is any degree of involvement.“ (p. 58) #Neumann #failure #probability #automata
„Outside of statistical mechanics proper, the thermodynamic formalism and its mathematical methods have now been used extensively in constructive quantum field theory and in the study of certain differentiable dynamical systems (notably Anosov diffeomorphisms and flows).“ (p. 1) #Ruelle #thermodynamic #formalism