Representation of complex probabilities

2020

Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle's random motion in the complex plane. The probability distribution ρc(t,x,y) of the particle's position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution ρc(t,x,y) is verified by the solution of the complex Fokker-Planck equation. It...

Journal of Statistical Physics, 1997

The average density of zeros for monic generalized polynomials, P n (z) = φ(z) + n k=1 c k f k (z), with real holomorphic φ, f k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |Im z|. We present the low and high disorder asymptotic behaviors. Then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form P n (z) = z n + n k=1 c k z n−k with real independent, identically distributed Gaussian coefficients having zero mean and dispersion δ = 1 √ nλ. The average density tends to a simple, universal function of ξ = 2nlog |z| and λ in the domain ξ coth ξ 2 ≪ n| sin arg(z)| where nearly all the roots are located for large n. 1 Which follow by comparing the polynomial Pn(z) = n k=0 a k z n−k with its roots expansion Pn(z) = a0 n j=1 (z − zj). The Jacobi determinant for this transformation was computed by Girschick and Hammersley [7-9]. 2 We thank an anonymous referee for bringing this paper to our attention 3 The analogous result for complex coefficients was obtained by Shparo and Shur [11].

The intensive week spent at the Banff Research Center was focused on discussions of a series of precise quantitative conjectures and pointed questions referring to the Euclidean distance geometry of the critical points of complex polynomials versus the locations of their zeros. The subject goes back to some early studies in electrostatics by Gauss and Maxwell and has penetrated into modern mathematics via approximation theory, specifically the Ilieff-Sendov conjecture.

2008

where c q J is a real or complex random variable with associated measure dγ for each q. We show that for these m independent functions in m variables, the expected density of non-real zeros in the real coefficients case rapidly approaches the expected density in the complex coefficients case as the degree of the polynomials gets large. In fact, we show

Journal of Mathematical Analysis and Applications, 1998

2011

We show that the use of techniques from algebra and algebraic geometry can be highly beneficial for tackling machine learning problems, where the set of desired solutions can be described in terms of approximate polynomial equations. Namely, they allow one to directly solve learning problems using algebraic operations thus avoiding iterative optimization which may converge to local minima. In this

Systems Science & Control Engineering, OA, 2016

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. This novel complex probability paradigm will be applied to the established theorem of Pafnuty Chebyshev's inequality and to extend the concepts of expectation and variance to the complex probability set C.

IntechOpen, 2023

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.

IntechOopen, 2023

In the current work, we extend and incorporate 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 are replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge of 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.

Systems Science & Control Engineering, OA, 2015

Andrey N. Kolmogorov's system of axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now 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. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the 'real' laboratory. Whatever the probability distribution of the input random variable in R is, the corresponding probability in the whole set C is always one, so the outcome of the random experiment in C can be predicted totally. The result indicates that chance and luck in R is replaced now by total determinism in C. This is the consequence of the fact that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to the classical theory of Brownian motion and to prove as well the law of large numbers in an original way. Nomenclature R real set of events M imaginary set of events C complex set of events i the imaginary number, where i 2 = −1 EKA extended Kolmogorov's axioms P rob probability of any event P r probability in the real set R P m probability in the imaginary set M corresponding to the real probability in R Pc probability of an event in R with its associated event in M probability in the complex set C z complex number = sum of P r and P m = complex random vector DOK |z| 2 = the degree of our knowledge of the random experiment it is the square of the norm of z. Chf the chaotic factor of z MChf magnitude of the chaotic factor of z N number of random variables Z the resultant complex random vector DOK Z = |Z| 2 N 2 the degree of our knowledge of Z Chf Z = Chf N 2 the chaotic factor of Z MChf Z magnitude of the chaotic factor of Z * Email: abdoaj@idm.net.lb S R entropy in R S R entropy of the complementary real probability set to R S M entropy in M S C entropy in C