We provide a solvability criteria for a depressed cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains Z * p , Zp, Qp are provided. Since Fp ⊂ Qp, we generalize J.-P. Serre's [27] and Z.H.Sun's [28,30] results concerning with depressed cubic equations over the finite field Fp. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the p−adic Cardano formula is provided for those cubic equations. Mathematics Subject Classification: 11Sxx

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study ξ (s) -QSO defined on 2D simplex. We first classify ξ (s) -QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.Mathematics Subject Classification 2010: 37E99; 37N25; 39B82, 47H60, 92D25. Key words: Quadratic stochastic operator; ℓ−Volterra quadratic stochastic operator; ξ (s) −quadratic stochastic operator; permuted ℓ−Volterra quadratic stochastic operator; dynamics10 = {V 11 , V 12 }, K 11 = {V 19 , V 31 }, K 12 = {V 20 , V 33 }, K 13 = {V 21 , V 32 }, K 14 = {V 22 , V 34 }, K 15 = {V 23 , V 36 }, K 16 = {V 24 , V 35 }, K 17 = {V 25 }, K 18 = {V 26 , V 27 }, K 19 = {V 28 }, K 20 = {V 29 , V 30 }.

In the present paper we study forward Quantum Markov Chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction we investigate QMC associated with XY -model on a Caylay tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY -model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators {K x,y }.

The most frequently asked question in the p−adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains Z