Vol. 7 No. 2 (2015): Arshag Hajian: By the 85th Anniversary.

We interpret the Pascal-adic transformation as a generalized induced automorphism (over odometer) and formulate the $\sigma$-finite analog of odometer which is also known as "Hajian-Kakutani transformation" (former "Ohio state example"). We shortly suggest a sketch of the theory of random walks on the groups on the base of $\sigma$-finite ergodic theory.

This article is concerned with ergodic theory for transformations which preserve an infinite measure. In the first part we present an overview of the invertible case with a focus on weakly wandering sequences and their applications to number theory as it has developed over the last fifty years. The second part presents a very preliminary investigation into extending weakly wandering sequences to the non-invertible case. This consists primarily of a few examples which illustrate the complexities which arise in the non-invertible case.

An increasing sequence of integers, $\mathbb{B}$, is given for which there exists a family of ergodic, infinite measure preserving transformations $T_\alpha$, $0 \leq \alpha \leq 1$ so that (1) $T_\alpha$ is of $\alpha$-type and (2) $\mathbb{B}$ is an exhaustive weakly wandering sequence for each $T_\alpha$.

In the paper we discuss the problem of description of random fields by means of systems of finite dimensional probability distributions. We present from a unified point of view a survey of the various such systems associated with random fields. Then general system of finite dimensional distributions with a suitable consistency condition is introduced. The properties, relations with other systems, as well as the problems of existence and uniqueness of the corresponding random fields are discussed.

We propose an algorithm to solve the two-phase obstacle problem by finite difference method. We prove the existence and uniqueness of the solution of the discrete nonlinear system and obtain an error estimate for the corresponding regularization. Also we prove the convergence of the proposed numerical algorithm. At the end of the paper we present some numerical simulations.

In 1949 M.~Sholander proved that every medial cancellation groupoid can be embedded into a medial quasigroup. In this paper we prove that it is also true for $n$-ary cancellation groupoids, namely, every medial cancellation $n$-ary groupoid can be embedded into a medial $n$-ary quasigroup.