Development of V. L. Rvachev’s Non-Archimedean Calculus

DOI
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute of Power Machines and Systems
of National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 29, no. 3, 2026 (September)
Pages 8-16
Cited by J. of Mech. Eng., 2026, vol. 29, no. 3, pp. 8-16

 

Authors

Mykola S. Syniekop, State Biotechnological University (44, Alchevskykh str., Kharkiv, 61002, Ukraine), ORCID: 0000-0003-2046-7997

Lidiia K. Kravchenko, State Biotechnological University (44, Alchevskykh str., Kharkiv, 61002, Ukraine)

Tetiana I. Sheiko, Anatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine (2/10, Komunalnykiv str., Kharkiv, 61099, Ukraine), e-mail: sheyko@ipmach.kharkov.ua, ORCID: 0000-0003-3295-5998

 

Abstract

This paper was prepared by students of Academician V. L. Rvachev, the creator of non-Archimedean calculus. Structurally, the paper is divided into three sections. The first section examines the relativistic formula for n-ary addition. Transformations of this formula yield four results: two pairs of mutually inverse functions which, given their properties, define the construction of non-Archimedean calculus; the structure of a relativistic series; and two mappings between the sets ℜ=(-∞, ∞) and ӿ=(-c, c). The second section presents relativistic trigonometry, including formulas for addition, the conversion of a sum of trigonometric functions into a product, and the conversion of a product of trigonometric functions into a sum. In the third section, relativistic hyperbolic functions are derived with the use of the relativistic exponential function, and their properties are outlined.

 

Keywords: non-Archimedean arithmetic, arithmetic with a maximal number, non-Archimedean algebra, finite infinity, relativistic sequence of natural numbers.

 

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References

  1. Rashevskiy, P. K. (1973). O dogmate naturalnogo ryada [On the dogma of the natural series]. Uspekhi matematicheskikh nauk – Russian Mathematical Surveys, vol. 28, iss. 4 (172), pp. 243–246 (in Russian).
  2. Rvachev, V. L. (1990). Relyativistskiy vzglyad na razvitiye konstruktivnykh sredstv matematiki [Relativistic view on the development of constructive means of mathematics]. Kharkiv, 44 p. (in Russian).
  3. Rvachev, V. L. (1991). Nearkhimedova arifmetika i drugiye konstruktivnyye sredstva matematiki, osnovannyye na ideyakh spetsialnoy teorii otnositelnosti [Non-Archimedean arithmetic and other constructive tools of mathematics based on the ideas of the special theory of relativity]. Doklady Akademii nauk SSSR – Reports of the USSR Academy of Sciences, vol. 316, no. 4, pp. 884–889 (in Russian).
  4. Rvachev, V. L. (1992). Relyativistskoye i drugiye nearkhimedovy ischisleniya [Relativistic and other non-Archimedean calculi]. Kharkiv, 47 p. (in Russian).
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  6. Rvachev, V. L. & Avinash, K. (2001). Quadratic red shift law and the non-Archimedean Universe. Radio Physics and Radio Astronomy, vol. 6, no. 2, pp. S34–S36. https://doi.org/10.15407/rpra.

 

Received 01 June 2026
Accepted 01 August 2026
Published 30 September 2026