BP International: degree of our knowledge

Showing posts with label degree of our knowledge. Show all posts

Monday, 16 March 2026

The Paradigm of Octonionic Probability: A Model of Transcendent Order |Book Publisher International

Probability, once formalised as a scalar measure, has undergone successive enlargements: complex amplitudes introduced interference, quaternionic vectors encoded multi‑component epistemic states. The Octonionic Probability Paradigm (OPP) inaugurates the next generative turn, situating probability within the non‑associative eight‑dimensional algebra of the octonions.

In OPP, a probabilistic state is not merely a number or vector but an authored octonionic entity. Its components inscribe magnitude, orientation, resonance, generative memory, and the twist of non‑associativity. This construction reframes probability as a sovereign epistemic artefact – an authored compass that records the act of knowing alongside the algebraic resonance of its unfolding.

Classical stochastic structures – limit theorems, Markov chains, diffusion processes – are reinterpreted through octonionic dynamics. By embedding chaotic factors and degrees of knowledge into the octonionic field, convergence acquires new dimensions of determinism and relationality. Simulation itself is transformed: Monté Carlo procedures become octonionic algorithms whose outputs carry orientation, resonance, and non‑associative trace, turning randomness into generative inscription.

The paradigm’s operational depth is exemplified through the octonionic reformulation of Thomas Bayes’ problem. Here, stochastic unpredictability is lifted into the octonionic domain, revealing how non‑associativity encodes epistemic entanglement and relational depth. Probability ceases to be a passive measure; it becomes a participatory geometry of uncertainty.

In sum, OPP declares uncertainty as an eight‑fold phenomenon: directional, resonant, twisting, entangled, authored, and generative. This book establishes the canonical principles, develops operator machinery, and demonstrates applied exemplars, inviting scholars to engage with the octonionic turn as both mathematical innovation and epistemic revolution.

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 book here :- https://doi.org/10.9734/bpi/mono/978-81-69006-87-3

Saturday, 30 August 2025

The Framework of the Paradigm of Complex Probability and Monté Carlo Methods | Book Publisher International

Calculating probabilities is a crucial task of classical probability theory. Adding supplementary dimensions to nondeterministic experiments will yield a deterministic expression of the theory of probability. This is the novel and original idea at the foundation of my complex probability paradigm. As a matter of fact, probability theory is a stochastic system of axioms in its essence; that means that the phenomena outputs are due to randomness and chance. By adding novel imaginary dimensions to the non-deterministic phenomenon happening in the set R will lead to a deterministic phenomenon and thus a stochastic experiment will have a certain output in the complex probability set C. If the chaotic experiment becomes completely predictable then we will be fully capable of predicting the output of random events that occur in the real world in all stochastic processes. Accordingly, the task that has been achieved here was to extend the random real probabilities set R to the deterministic complex probabilities set C = R + M and this by incorporating the contributions of the set M which is the associated and complementary imaginary set of probabilities to the set R. Hence, the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. Consequently, since this extension was revealed to be successful, then an innovative paradigm of stochastic sciences and prognostic was put forward in which all nondeterministic phenomena in R were expressed deterministically in C. I coined this novel model with the term "The Complex Probability Paradigm (or CPP)" which was initiated and established in my earlier research works. Moreover, this pioneering paradigm will be applied in a creative manner to the stochastic procedures and algorithms of the famous and historical Buffon’s needle method to compute PI, to the renowned neutron shielding problem, and to numerous and various topics that arise in Monté Carlo Methods.

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/978-93-48006-59-2

Monday, 11 August 2025

The Paradigm of Complex Probability and Absorbing Markov Chains | Chapter 4 | The Paradigm of Complex Probability and Markov Chains, Edition 1

The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to Absorbing Markov Chains Theory in order to express it totally and absolutely deterministically in the complex universe C of probabilities.

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/CH4

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

Monday, 29 April 2024

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.

Saturday, 30 March 2024

The Paradigm of Complex Probability and Analytic Nonlinear Prognostic for Unburied Petrochemical Pipelines – A Relation to Dynamic Logic | Chapter 4 | The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

Andrey Nikolaevich Kolmogorov put forward in 1933 the five fundamental axioms of classical probability theory. The original idea in my complex probability paradigm is to add new imaginary dimensions to the experiment real dimensions which will make the work in the complex probability set totally predictable and with a probability permanently equal to one. Therefore, adding to the real set of probabilities R the contributions of the imaginary set of probabilities M will make the event in C = R + M absolutely deterministic. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Hence, my purpose is to link my complex probability paradigm to unburied petrochemical pipelines analytic prognostic in the nonlinear damage accumulation case. Consequently, by calculating the parameters of the novel prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of knowledge, the complex probability, the system failure and survival probabilities, and the remaining useful lifetime probability, after that a pressure time t has been applied to the pipeline and which are all functions of the system degradation subject to random effects. Furthermore, we will apply the new paradigm to my novel ‘Dynamic Logic’ model.

The Paradigm of Complex Probability and the Novel Dynamic Logic – The Simulations | Chapter 3 | The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

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 a newly defined logic that I called “Dynamic Logic”.

The Paradigm of Complex Probability and the Novel Dynamic Logic – The Model | Chapter 2 | The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set C totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the chaotic factor, the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Accordingly, my purpose here is to link my complex probability paradigm to logic. Hence, after adding the time dimension, we will apply this novel paradigm to a newly defined logic that I called ‘Dynamic Logic’.

The Paradigm of Complex Probability, Prognostic, and Dynamic Logic | Chapter 1 | The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

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 logic in order to express it completely deterministically in the probability universe C = R + M. Therefore, after adding the time dimension, we will relate and join this original paradigm to a newly defined logic that I called ‘Dynamic Logic’ and it will be also implemented to pipeline prognostic with the aim of illustrating CPP and this novel kind of logic.

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

The Paradigm of Complex Probability and The Central Limit Theorem | Chapter 3 | The Paradigm of Complex Probability, the Law of Large Numbers, and the Central Limit Theorem

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. We will apply this innovative paradigm to the well-known Central Limit Theorem and to prove as well its convergence in a novel way.

The Paradigm of Complex Probability and the Weak and Strong Law of Large Numbers | Chapter 2 | The Paradigm of Complex Probability, the Law of Large Numbers, and the Central Limit Theorem

The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to the law of large numbers in order to demonstrate it in an innovative way and to prove as well in an original way an important property at the foundation of statistical physics.

The Paradigm of Complex Probability, the Law of Large Numbers, and the Central Limit Theorem | Chapter 1 | The Paradigm of Complex Probability, the Law of Large Numbers, and the Central Limit Theorem

The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to the Law of Large Numbers in order to demonstrate it in an innovative manner and to prove as well in an original fashion an important property at the foundation of statistical physics, addtionally it will be applied to the well-known Central Limit Theorem and to demonstrate thus its convergence in a novel way.

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 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.