What is it about?

In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers and this by adding a supplementary three original axioms. Therefore, any experiment can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the "real" laboratory in R and hence to evaluate all the probabilities. Consequently, the probability in the entire set C = R + M is permanently equal to one no matter what the stochastic distribution of the input random variable in R is, therefore the outcome of the probabilistic experiment in C can be determined perfectly. This is due to the fact that the probability in C is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. Consequently, the purpose in this chapter is to join my complex probability paradigm to the analytic prognostic of buried petrochemical pipelines in the case of linear damage accumulation. Accordingly, after the calculation of the novel prognostic model parameters, we will be able to evaluate the degree of knowledge, the magnitude of the chaotic factor, the complex probability, the probabilities of the system failure and survival, and the probability of the remaining useful lifetime, after that a pressure time t has been applied to the pipeline, and which are all functions of the system degradation subject to random and stochastic influences.

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Why is it important?

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem considered here, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in C = R +M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my previous papers.

Perspectives

Although I have taught courses on probability and statistics at the university level for many years, I consider myself a beginner in this branch of knowledge; in fact an absolute beginner, always thirsty to learn and discover more. I think that the mathematician who proves to be successful in tackling and mastering the theory of probability and statistics has made it halfway to understanding the mystery of existence revealed in a universe governed sometimes in our modern theories by randomness and uncertainties. The probabilistic aspect is evident in the theories of the quantum world, of thermodynamics, or of statistical mechanics, for example. Hence, the universe’s secret code, I think, is written in a mathematical language, just as Galileo Galilei expressed it in these words: “Philosophy is written in this very great book which is the universe that always lies open before our eyes. One cannot understand this book unless one first learns to understand the language and recognize the characters in which it is written. It is written in a mathematical language and the characters are triangles, circles and other geometrical figures. Without these means it is humanly impossible to understand a word of it. Without these there is only clueless scrabbling around in a dark labyrinth.”

Dr. Abdo Abou Jaoude
Notre Dame University Louaize

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This page is a summary of: Analytic Prognostic in the Linear Damage Case Applied to Buried Petrochemical Pipelines and the Complex Probability Paradigm, December 2019, IntechOpen,
DOI: 10.5772/intechopen.90157.
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