The Paradigm of Complex Probability, Numerical Analysis, and Chaos Theory

What is it about?

The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. This novel complex probability paradigm will be applied to numerical analysis and to chaos theory to prove henceforth that chaos vanishes totally and completely in the probability universe C = R + M.

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Why is it important?

Calculating probabilities is the main task of the theory of classical probability. In fact, if we add new dimensions to a random phenomenon it will result to a deterministic expression of the theory of probability. This is the original and novel idea at the foundations of “The Complex Probability Paradigm (or CPP for short)”. As a matter of fact, probability theory is a nondeterministic theory in its core; that means that the outcomes of events are due to chance and randomness. If we add imaginary and new dimensions to a random experiment occurring in the set R it will result to a deterministic experiment and thus a nondeterministic phenomenon will have a certain outcome in the complex probability set C. If the random event becomes completely predictable then we will have perfect knowledge to predict the outcome of random experiments that arise in the real world in all random processes. Consequently, the work that has been accomplished in CPP was to extend the set R of real probabilities to the set C = R + M of deterministic complex probabilities by incorporating the contributions of the set M which is the imaginary probabilities set. Therefore, because this extension was found to be fruitful, then a novel paradigm of prognostic and nondeterministic sciences was established in which all random phenomena in R was defined deterministically. I called this original model "the Complex Probability Paradigm" that was initiated and illustrated in my 36 earlier research publications.

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