卷 233, 编号 1 (2018)

This article discusses the structure of cyclic semirings with nonidempotent noncommutative addition. Also given are some general results on cyclic semirings with noncommutative addition.

Let H be an \( \mathfrak{N} \)-prefrattini subgroup of a soluble finite group G and Δ(G) be its Gaschütz subgroup. In this paper, it is proved that there exist elements x, y ∈ G such that the equality H∩H^x∩H^y = Δ(G) holds.

The paper is devoted to circuits implementing parity functions. A review of results establishing exact values of the complexity of parity functions is given. The structure of optimal circuits implementing parity functions is described for some bases. For one infinite basis, an upper bound for the complexity of parity functions is given.

Various generalizations of the classical problem of the fastest raising to a power (or the so-called problem on addition chains) are studied in the asymptotic sense. Under weak restrictions, we demonstrate asymptotically tight solutions of the two best known generalizations, namely, Bellman’s problem on the computational complexity (on the minimal number of multiplication operations) of a normed monomial of several variables and Knuth’s problem on the computational complexity of a power system of one variable. We also briefly review some results on the computational complexity for three problems, namely, the computation of p-element systems of normed monomials in q variables, additive computations for systems of p integer linear forms over q variables, and the computation of p-element systems of the free Abelian group with q generators.

In this paper, we study generalized soluble groups with restriction on normal closures of cyclic subgroups. A group G is said to have finite Hirsch–Zaitsev rank if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors is finite. It is not hard to see that the number of infinite cyclic factors in each of such series is an invariant of a group G. This invariant is called the Hirsch–Zaitsev rank of G and will be denoted by r_hz(G). We study the groups in which the normal closure of every cyclic subgroup has the Hirsch–Zaitsev rank at most b (b is some positive integer). For some natural restrictions we find a function k₁(b) such that r_hz([G/Tor(G),G/Tor(G)]) ≤ k₁(b).

We implement Leibniz’s idea about the differential as the length of an infinitesimally small elementary interval (a monad) in a form satisfying modern standards of rigor. The concept of sequential differential introduced in this paper is shown to be in good alignment with the standard convention of the integral calculus. As an application of this concept we simplify and generalize the construction of the Perron–Stieltjes integral.