Discrete Geometrical Invariants in 3D Space: How Three Random Sequences Can Be Compared in Terms of “Universal” Statistical Parameters

Raoul R Nigmatullin

doi:10.3389/fphy.2020.00076

Raoul R Nigmatullin

Open Access

https://doi.org/10.3389/fphy.2020.00076

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Abstract

Abstract The statement that any random sequence has a set of deterministic components sounds as absurd and unacceptable. However, based on ideas of Prof. Yu. Babenko, who generalized the Pythagor theorem, it became possible to find a positive answer and state that any random sequence can be expressed quantitatively in terms of the discrete geometrical invariants (DGI). Earlier these DGIs were found for a couple of random sequences on 2D-plane [4,5]. In this paper, the author made a next logical step and obtained the DGI for three random sequences (representing a trajectory of an imaginary particle) in 3D-space. It becomes possible to receive the closed analytical form for the desired DGI of the fourth order in 3D-space and compare a triple of random sequences ({r1k, r2k, r3k}, k=1,2,…N) with each other. This unified and platform identifies (in total) 6 surfaces and 13 reduced and compact parameters having combination from 28 basic moments and their intercorrelations (up to the fourth order, inclusively). This platform reminds the universal form of the partition function proposed by the Gibbs in the statistical physics, when all microscopic parameters describing the trajectories of different micro-particles are transformed to the finite and compact set of thermodynamic variables. Similar idea have been realized earlier with the help of the DGI for a pair of random sequences belonging to 2D-space [4,5]. Based on available data, the author found the 3D images for two famous transcendental numbers as  and E (Euler constant) and their 13 quantitative parameters that differentiate them from each other. Besides, the author applied the DGI approach to analysis of the EQs data. Being a naive user in geophysics, nevertheless it becomes possible to classify available six Earthquakes (EQs) signals and discover their common statistical nature.

Highlights

AND FORMULATION OF THE PROBLEMThe statement that any random sequence has a set of deterministic components sounds absurd and unacceptable
[8, 9], it has been shown that the ideas expressed in the generalization of the Pythagorean theorem proposed by Babenko [10, 11], which was proved for 2D and 3D figures with different symmetry, can be developed to take into consideration different random sequences
The authors propose an extension of the discrete geometrical invariant (DGI) approach realized previously for 2D space

Summary

Introduction

AND FORMULATION OF THE PROBLEMThe statement that any random sequence has a set of deterministic components sounds absurd and unacceptable. Previous work has determined the discrete geometrical invariant (DGI) in 2D space, which allows the analysis of a pair of random sequences ([r1k, r2k], k = 1,2,...N) containing an equal number of data points to be reduced to eight “universal” parameters that present different inter-correlations between the two compared sequences.

Results

Conclusion