The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

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We consider a nearest-neighbor inhomogeneous p-adic Potts (with q ≥ 2 spin values) model on the Cayley tree of order k ≥ 1. The inhomogeneity means that the interaction Jxy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p− adic) Gibbs measures of the model. We show that (i) if q / ∈ pN then there is unique Gibbs measure for any k ≥ 1 and ∀Jxy with |Jxy| < p −1/(p−1) . (ii) For q ∈ pN, p ≥ 3 one can choose Jxy and k ≥ 1 such that there exist at least two Gibbs measures which are translation-invariant.

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

We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over Z, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.

In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

In the present paper we investigate L 0 -valued states and Markov operators on C * -algebras over L 0 . In particular, we give representations for L 0valued state and Markov operators on C * algebras over L 0 , respectively, as measurable bundles of states and Markov operators. Moreover, we apply the obtained representations to study certain ergodic properties of C * -dynamical systems over L 0 .

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.