On ξ-quadratic stochastic operators and related algebras

Abstract

In this thesis, we start to study a class of quadratic stochastic operators called (s) ξ - QSO. We first classify them into 20 non-conjugate classes. Moreover, we investigate the dynamics of four classes of (s) ξ -QSO. Furthermore, we study another class of quadratic stochastic operator called (a) ξ -QSO. We also classify (a) ξ -QSO into two non-conjugate classes. Further, we investigate the dynamics of these classes. After that, we move to study the existence of associativity and derivations of genetic algebras generated by the four classes of (s) ξ . Moreover, we figure out the connection between genetic and evolution algebras. Thereafter, we reduce the study of arbitrary evolution algebra of permutations into two special evolution algebras. Furthermore, we establish some properties of three-dimensional evolution algebras whose each basis element has infinite period. At end, we classify three dimension nilpotent and solvable evolution algebras.