Fundamental Mathematical Methods and Concepts | Chapter 1 | The Paradigm of Complex Probability and Markov Chains, Edition 1

 

In the current manuscript, we will apply to the well-known and established theory of Markov Chains my groundbreaking and novel Complex Probability Paradigm (or CPP) which will lead to a completely and perfectly deterministic expression of these stochastic processes in the universe of probabilities C = R + M. Consequently, it is important before “probing the depths” of Markov Chains, that we define in this chapter some essential mathematical concepts and fundamental tools and theorems that will be extensively used in the whole manuscript algorithms and C++ programs.

 

Author(s) Details

Abdo Abou Jaoudé
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaizé, Lebanon.

 

Please see the book here:- https://doi.org/10.9734/bpi/mono/978-93-48006-18-9/CH1

The Paradigm of Complex Probability, Numerical Analysis, and Chaos Theory | Book Publisher International

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.

Author(s) Details:

Abdo Abou Jaoudé,
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon.

Please see the link here: https://stm.bookpi.org/TPCPNACT/article/view/14275