Abstract
The most important problem in the theory of phenomenologically symmetric geometries of two sets is that of classification of these geometries. In this paper, complexifying the metric functions of some known phenomenologically symmetric geometries of two sets (PSGTS) with the use of associative hypercomplex numbers, we find metric functions of new geometries in question. For these geometries, we find equations of the groups of motions and establish phenomenological symmetry, i.e., find functional relations between metric functions for certain finite number of arbitrary points. In particular, for one-component metric functions of PSGTS’s of ranks (2, 2), (3, 2), (3, 3), we find (n + 1)-component metric functions of the same ranks. For these metric functions, we find finite equations of the groups of motions and equations that express their phenomenological symmetry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mikhailichenko, G. G. and Muradov, R. M. Physical Structures as Geometries of Two Sets (GASU, Gorno-Altaisk, 2008) [in Russian].
Mikhailichenko, G. G. “On a Problem in the Theory of Physical Structures”, Sib. Mat. Z. 18, No. 6, 1342–1355 (1977) [in Russian].
Mikhailichenko, G. G. “The Solution of Functional Equations in the Theory of Physical Structures”, Sov. Math., Dokl. 13, 1377–1380 (1972).
Kantor, I. L. and A.S.Solodovnikov, A. S. Hypercomplex Numbers (Nauka, Moscow, 1973) [in Russian].
Mikhailichenko, G. G. and Muradov, R.M. “Hypercomplex Numbers in the Theory of Physical Structures”, Russian Mathematics 52, No. 10, 20–24 (2008).
Kostrikin, A. I. Introduction to Algebra (Nauka, Moscow, 1977) [in Russian].
Mikhailichenko, G. G. “Phenomenological and Group Symmetry in the Geometry of Two Sets (Theory of Physical Structures)”, Sov. Math., Dokl. 32, 371–374 (1985).
Mikhailichenko, G. G. “Group Properties of Physical Structures”, Preprint No. 1589–B89 (VINITI, 1989).
Kulakov, Yu. I., Vladimirov, Yu. S. and Karnaukhov, A.V. Introduction to the Theory of Physical Structures (Arkhimed, Moscow, 1992) [in Russian].
Vladimirov, Yu. S. Relational Theory of Space-Time. Vol. 2. Theory of Physical Interactions (Moscow University, Moscow, 1999) [in Russian].
Kyrov, V. A. “Affine Geometry as a Physical Structure”, Zhurn. Sib. Fed. Univ. Matem. i Fizika 1, No. 4, 460–464 (2008) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © G.G. Mikhailichenko, V.A. Kyrov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 7, pp. 19–29.
About this article
Cite this article
Mikhailichenko, G.G., Kyrov, V.A. Hypercomplex numbers in some geometries of two sets. I. Russ Math. 61, 15–24 (2017). https://doi.org/10.3103/S1066369X17070039
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X17070039