### Journals Information

**
Mathematics and Statistics Vol. 8(2), pp. 173 - 180 DOI: 10.13189/ms.2020.080213 Reprint (PDF) (456Kb) **

## Analysis of the Element's Arrangement Structures in Discrete Numerical Sequences

**Anton Epifanov ^{*}**

Institute of precision mechanics and control, Russian Academy of Sciences, Russia

**ABSTRACT**

Paper contains the results of the analysis of the laws of functioning of discrete dynamical systems, as mathematical models of which, using the apparatus of geometric images of automatons, are used numerical sequences which interpreted as sequences of second coordinates of points of geometric images of automatons. The geometric images of the laws of the functioning of the automaton are reduced to numerical sequences and numerical graphs. The problem of constructing an estimate of the complexity of the structures of such sequences is considered. To analyze the structure of sequences, recurrence forms are used that give characteristics of the relative positions of elements in the sequence. The parameters of recurrent forms are considered, which characterize the lengths of the initial segments of sequences determined by recurrence forms of fixed orders, the number of changes of recurrent forms required to determine the entire sequence, the place of change of recurrence forms, etc. All these parameters are systematized into the special spectrum of dynamic parameters used for the recurrent determination of sequences, which is used as a means of constructing estimates of the complexity of sequences. In this paper, it also analyzes return sequences (for example, Fibonacci numbers), for the analysis of the properties of which characteristic sequences are used. The properties of sequences defining approximations of fundamental mathematical constants (number e, pi number, golden ratio, Euler constant, Catalan constant, values of Riemann zeta function, etc.) are studied. Complexity estimates are constructed for characteristic sequences that distinguish numbers with specific properties in a natural series, as well as for characteristic sequences that reflect combinations of the properties of numbers.

**KEYWORDS**

Discrete Dynamical System, Mathematical Model, Discrete Deterministic Automaton, Geometrical Image of Automata Mapping, Numerical Sequence, Complexity Estimation, Recurrent Form, Spectrum of Dynamical Parameters, Characteristic Sequence

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Anton Epifanov , "Analysis of the Element's Arrangement Structures in Discrete Numerical Sequences," Mathematics and Statistics, Vol. 8, No. 2, pp. 173 - 180, 2020. DOI: 10.13189/ms.2020.080213.

(b). APA Format:

Anton Epifanov (2020). Analysis of the Element's Arrangement Structures in Discrete Numerical Sequences. Mathematics and Statistics, 8(2), 173 - 180. DOI: 10.13189/ms.2020.080213.