On new classes of chains of evolution algebras

Year 2021, , 146 - 158, 04.02.2021

Manuel Ladra , Sherzod N. Murodov

https://doi.org/10.15672/hujms.597790

Cited By: 2

Abstract

The paper is devoted to studying new classes of chains of evolution algebras and their time-depending dynamics and property transition.

Keywords

Evolution algebra, time, Chapman-Kolmogorov equation

Supporting Institution

Agencia Estatal de Investigación (Spain) and Xunta de Galicia (European FEDER support included, UE).

Project Number

MTM2016-79661-P and ED431C 2019/10

Thanks

We sincerely acknowledge Professor U.A. Rozikov for helpful discussions.

References

• [1] V.M. Abraham, Linearizing quadratic transformations in genetic algebras, Proc. Lon- don Math. Soc. (3), 40 (2), 346–363, 1980.
• [2] Y. Cabrera Casado, Evolution algebras, Ph.D. thesis, Universidad de Málaga, 2016, http://hdl.handle.net/10630/14175.
• [3] M.I. Cardoso Gonçalves, D. Gonçalves, D. Martín Barquero, C. Martín González and M. Siles Molina, Squares and Associative Representations of two Dimen- sional Evolution Algebras, J. Algebra Appl., 2020, doi: https://doi.org/10.1142/ S0219498821500900.
• [4] J.M. Casas, M. Ladra and U.A. Rozikov, A chain of evolution algebras, Linear Algebra Appl. 435 (4), 852–870, 2011.
• [5] J.M. Casas, M. Ladra, B.A. Omirov and U.A. Rozikov, On evolution algebras, Algebra Colloq. 21 (2), 331–342, 2014.
• [6] I.M.H. Etherington, Genetic algebras, Proc. Roy. Soc. Edinburgh, 59, 242–258, 1939.
• [7] I.M.H. Etherington, Duplication of linear algebras, Proc. Edinburgh Math. Soc. (2), 6, 222–230, 1941.
• [8] I.M.H. Etherington, Non-associative algebra and the symbolism of genetics, Proc. Roy. Soc. Edinburgh. Sect. B. 61, 24–42, 1941.
• [9] O.J. Falcón, R.M. Falcón and J. Núñez, Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field, J. Algebra, 472, 573–593, 2017.
• [10] V. Glivenkov, Algèbre Mendelienne, C. R. (Doklady) Acad. Sci. URSS, 4, 385–386, 1936.
• [11] H. Gonshor, Contributions to genetic algebras. II, Proc. Edinburgh Math. Soc. (2), 18, 273–279, 1973.
• [12] I. Heuch, Sequences in genetic algebras for overlapping generations, Proc. Edinburgh Math. Soc. (2), 18, 19–29, 1972.
• [13] P. Holgate, Sequences of powers in genetic algebras, J. London Math. Soc. 42, 489– 496, 1967.
• [14] P. Holgate, Selfing in genetic algebras, J. Math. Biology, 6, 197–206, 1978.
• [15] V.A. Kostitzin, Sur les coefficients mendéliens d’hérédité, C. R. Acad. Sci. Paris, 206, 883–885, 1938.
• [16] Y.I. Lyubich, Mathematical structures in population genetics, Springer-Verlag, Berlin, 1992.
• [17] G. Mendel, Experiments in plant-hybridization, 1865. The Electronic Scholarly Pub- lishing Project http://www.esp.org/foundations/genetics/classical/gm-65.pdf.
• [18] Sh.N. Murodov, Classification dynamics of two-dimensional chains of evolution alge- bras, Internat. J. Math. 25 (2), 1450012, 23 pp., 2014.
• [19] Sh.N. Murodov, Classification of two-dimensional real evolution algebras and dynamics of some two-dimensional chains of evolution algebras, Uzbek. Mat. Zh. 2014 (2), 102–111, 2014.
• [20] M.L. Reed, Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc. (N.S.), 34 (2), 107–130, 1997.
• [21] O. Reiersöl, Genetic algebras studied recursively and by means of differential opera- tors, Math. Scand. 10, 25–44, 1962.
• [22] U.A. Rozikov and Sh.N. Murodov, Dynamics of two-dimensional evolution algebras, Lobachevskii J. Math. 34 (4), 344–358, 2013.
• [23] U.A. Rozikov and J.P. Tian, Evolution algebras generated by Gibbs measures, Lobachevskii J. Math. 32 (4), 270–277, 2011.
• [24] R.D. Schafer, An introduction to nonassociative algebras, Academic Press, New York, 1966.
• [25] A. Serebrowsky, On the properties of the Mendelian equations, C. R. (Doklady) Acad. Sci. URSS 2, 33–39, 1934 (in Russian).
• [26] J.P. Tian, Evolution algebras and their applications, Lecture Notes in Mathematics 1921, Springer-Verlag, Berlin, 2008.
• [27] J.P. Tian and P. Vojtechovsky, Mathematical concepts of evolution algebras in non- Mendelian genetics, Quasigroups Related Systems 14, 111–122, 2006.
• [28] A. Wörz-Busekros, Algebras in genetics, Lecture Notes in Biomathematics 36, Springer-Verlag, Berlin-New York, 1980.