BP International: complex probability set C

Showing posts with label complex probability set C. Show all posts

Monday, 11 August 2025

The Paradigm of Complex Probability and Regular Markov Chains | Chapter 3 | The Paradigm of Complex Probability and Markov Chains, Edition 1

The crucial job of the theory of classical probability is to compute and assess probabilities. A deterministic expression of probability theory will be achieved by the addition of new dimensions to the stochastic experiments. This is the original and novel idea at the foundation of my paradigm. As a matter of fact, since the events' outcomes are due to randomness and chance, then the theory of probability is a nondeterministic system in its essence. A deterministic experiment and hence a stochastic event will have a certain result in the complex probability set C after encompassing novel imaginary dimensions to the chaotic experiment occurring in the real set R. Thus, we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes if the random event becomes completely predictable. Hence, extending the real probabilities set R to the deterministic complex probabilities set C = R + M by including the contributions of the set M which is the imaginary set of probabilities, is the work that has been accomplished here. Therefore, a novel paradigm of stochastic sciences and prognostic was laid down in which all stochastic phenomena in R were expressed deterministically in C since this extension was found to be successful. I coined this original model with the term: “The Complex Probability Paradigm” or CPP for short. Knowing that it was illustrated and initiated in my previous research publications. Henceforth, this original probability paradigm will be applied in this work to Regular Markov Chains and Processes.

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/CH3

The Paradigm of Complex Probability and Markov Chains Transition Matrices | Chapter 2 | The Paradigm of Complex Probability and Markov Chains, Edition 1

In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms and an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that luck and chance in R are substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.

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/CH2

Wednesday, 13 March 2024

The Paradigm of Complex Probability and Metarelativity | Book Publisher International

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, 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 imaginary dimensions to the event in the real set of probabilities R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe G = C. It is of great importance that the stochastic system, like in the real-world problems, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature like 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 G = C = R + M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and in science in general is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This paradigm was initiated and developed in my previous 25 publications and research works. Moreover, this model will be related to my Theory of Metarelativity which takes into account faster-than-light matter and energy. This is what I called ‘The Metarelativistic Complex Probability Paradigm (MCPP)’ which will be elaborated in the present book.

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/TPCPM/article/view/13537

Keywords: Chaotic factor, Degree of our knowledge, Complex random vector, Probability norm, Complex probability set C, Metarelativistic transformations, Imaginary number, Imaginary dimensions, Superluminal velocities, Metaparticles, Dark matter, Metamatter, Dark energy, Metaenergy, Metaentropy, Universe G1, Metauniverse G2, luminal universe G3, the total universe G

Tuesday, 12 March 2024

The Paradigm of Complex Probability and the Quantum Entropic Uncertainty Principle | Chapter 7 | The Paradigm of Complex Probability and Quantum Mechanics

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon 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. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R + M independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be determined perfectly in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. My innovative ‘Complex Probability Paradigm’ (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum entropic uncertainty principle in order to verify it and to extend it to the universes M and C.

The Paradigm of Complex Probability and Heisenberg’s Quantum Uncertainty Principle | Chapter 6 | The Paradigm of Complex Probability and Quantum Mechanics

The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum uncertainty principle in order to verify it and to extend it to the universes M and C.

The Paradigm of Complex Probability and Quantum Mechanics: The Quantum Harmonic Oscillator with Gaussian Initial Condition – The Momentum Wavefunction and The Wavefunction Entropies | Chapter 5 | The Paradigm of Complex Probability and Quantum Mechanics

The system of probability axioms of Andrey Nikolaevich Kolmogorov put forward in 1933 can be developed to encompass the set of imaginary numbers after adding to his established five axioms a supplementary three axioms. Therefore, any probabilistic phenomenon can thus be performed in what is now the set of complex probabilities C which is the sum of the real set of probabilities R and the complementary and associated and corresponding imaginary set of probabilities M. The aim here is to compute the complex probabilities by taking into consideration additional novel imaginary dimensions to the phenomenon that occurs in the ‘real’ laboratory. Hence, the corresponding probability in the entire probability set C = R + M is, whatever the random distribution of the input random variable considered in R, permanently and constantly equal to 1. Thus, the result of the stochastic experiment in C can be foretold perfectly and completely. Subsequently, the consequence shows that luck and chance in R is substituted now by absolute determinism in C. Accordingly, this is the consequence of the fact that the probability in C is got by subtracting from the degree of our knowledge of the random system the chaotic factor. Henceforth, I will apply to the established and well-known theory of quantum mechanics my innovative and original Complex Probability Paradigm (CPP) which will yield a completely deterministic expression of quantum theory in the universe of probabilities C = R + M.

The Paradigm of Complex Probability and Quantum Mechanics: The Quantum Harmonic Oscillator with Gaussian Initial Condition – The Position Wavefunction | Chapter 4 | The Paradigm of Complex Probability and Quantum Mechanics

In the current work, we extend and incorporate in the five-axioms probability system of Andrey Nikolaevich Kolmogorov set up in 1933 the imaginary set of numbers and this by adding three supplementary axioms. Consequently, any stochastic experiment can thus be achieved in the extended complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. The purpose here is to evaluate the complex probabilities by considering additional novel imaginary dimensions to the experiment occurring in the “real” laboratory. Therefore, the random phenomenon outcome and result in C = R + M can be predicted absolutely and perfectly no matter what the random distribution of the input variable in R is since the associated probability in the entire set C is constantly and permanently equal to one. Thus, the following consequence indicates that chance and randomness in R is replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge the chaotic factor in the probabilistic experiment. Moreover, I will apply to the established theory of quantum mechanics my original Complex Probability Paradigm (CPP) in order to express the quantum mechanics problem considered here completely deterministically in the universe of probabilities C = R + M.

The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem – The Momentum Wavefunction and The Wavefunction Entropies | Chapter 3 | The Paradigm of Complex Probability and Quantum Mechanics

The mathematical probability concept was set forth by Andrey Nikolaevich Kolmogorov in 1933 by laying down a five-axioms system. This scheme can be improved to embody the set of imaginary numbers after adding three new axioms. Accordingly, any stochastic phenomenon 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. Our objective now is to encompass complementary imaginary dimensions to the stochastic phenomenon taking place in the “real” laboratory in R and as a consequence to calculate in the sets R, M, and C all the corresponding probabilities. Hence, the probability is permanently equal to one in the entire set C = R + M independently of all the probabilities of the input stochastic variable distribution in R, and subsequently the output of the random phenomenon in R can be determined perfectly in C. This is due to the fact that the probability in C is calculated after the elimination and subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic phenomenon. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M.

The Paradigm of Complex Probability and Quantum Mechanics: The Infinite Potential Well Problem – The Position Wavefunction | Chapter 2 | The Paradigm of Complex Probability and Quantum Mechanics

The Paradigm of Complex Probability and Quantum Mechanics | Chapter 1 | The Paradigm of Complex Probability and Quantum Mechanics

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence 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. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M.