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.