Volume 25, Issue 5

Let G be a finite group and f be an automorphism of the group G. The automorphism f specifies a recurrent sequence {a i } 0 ∞ on the group G by the rule a i+1 = f(a i ). If a 0 is the initial element of the sequence, then the period of the sequence does not exceed the number of elements having the same order as a 0 . Thus, it is interesting to find out whether there exist groups having automorphisms generating sequences with the largest possible period for any initial element. This work continues the study of groups possessing automorphisms of the maximal period. Earlier, the case of groups of odd orders was examined. It was established that such groups are necessarily Abelian, and their structure was completely described. This paper considers groups of even orders and completes the description of finite groups possessing automorphisms of the maximal period.

We study the process of variation of a player rating in an infinite series of games with the same adversary in the Elo rating model. This process is shown to have a stationary distribution, an upper estimate of the rate of convergence to which is established. In a previous paper by the author, the existence of a limit distribution was proved under more stringent assumptions on the parameters of a rating model.

The paper is concerned with an extremal problem of combinatorial analysis on finding the minimal possible number of edges in an n-regular hypergraph with chromatic number greater than r and girth greater than s. A new lower estimate of this extremal value is obtained and a number of related results is proved.

Let X 0 ,X 1 , . . . be a simple ergodic finite Markov chain.We prove limit theorems for the distribution of the number ξ̄(s, n) of events {X i−1 ≠ X j−1 , X i+k = X j+k , k = 0, . . . , s − 1}, 1 ≤ i < j ≤ n, when s, n → ∞. Limit theorems for distributions of some random variables connected with ξ̄(s, n) are derived as corollaries.

We study modules, in which all automorphisms of submodules may be extended to endomorphisms (automorphisms) of the whole module.

Transformations of independent random variables over a finite field by read-once formulas are considered. Subsets of probability distributions that are preserved by read-once transformations are constructed. Also we construct a family of distributions that may be arbitrarily closely approximated by a read-once combination of independent identically distributed random variables, whose distributions have no zero components.