Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Maľcev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

Group polynomials over rings

We consider polynomials over rings such that the polynomials represent Latin squares and define a group operation over the ring. We introduce the notion of a group polynomial, describe a number of properties of these polynomials and the groups generated. For the case of residue rings$\mathbb{Z}_{r^n},$where r is a prime number, we give a description of groups specified by polynomials and identify a class of group polynomials that can be used to construct controlled cryptographic transformations.

Mathematical model for the analysis of group modes of induction motors when fed from a common source.

The article analyzes the operating modes and transients that occur when starting, stopping and changing the load in electric drives with induction motors. The relevance of the need for such studies for drives with powerful asynchronous motors that are connected to the same energy source and start or stop at the same time is proved. The information obtained will allow maintenance personnel and the available automation and telecommunications facilities to make the right and timely decisions for the effective management of electric drives. A mathematical model has been compiled for analyzing the modes of group operation of asynchronous motors powered by a common source in order to determine the parameters of such work. When creating the model, the voltage of the power source, as well as the parameters and design features of induction motors, were chosen as the initial parameters. The developed mathematical model contains a system of differential equations for the analysis of group operation of induction motors and demonstrates the possibilities of developing the theory of group operation of asynchronous motors. As an auxiliary mathematical technique for recording the physical properties of asynchronous motors, the concept of the inverse submatrix was used, and when composing differential equations, the asymmetry of the voltage at the load nodes was taken into account. The developed mathematical model makes it possible to determine the principles for diagnosing the operation of groups of induction motors connected to a common source by registering current ripples arising in the supply network and changing the consumed power and thus indirectly increase the reliability of asynchronous motors in electric drives of technological mechanisms under various load conditions.

SUBSTITUTION CHARACTERISTICS OF FACTORIAL SETS AND CRITERIA FOR CHOOSING THE SINGLE SUBSTITUTIONS

The substitution and permutation function, which are presented in the article as elements of a number of factorial sets, are the key functions of cryptographic systems that provide diffusion and mixing of information. A new scale of notation is proposed while analyzing this problem. This is the notation scale of a number of factorial sets. This scale of notation helps to index the elements of a number of factorial sets and establish a one-to-one correspondence between the number and a specific type of substitution. This allows analyzing substitutions characteristics systematically. This paper presents the basic concepts of a number of the factorial sets. It is noted that the permutations of the factorial sets form symmetric permutation groups, and specific permutations (when raised to a power) form cyclic groups. The group axioms are fulfilled for the permutations of a number of factorial sets. Also, the definition domain, the group operation of multiplication, and identical and inverse substitutions are given for them. The number of independent cycles, decrement, inverse, parity and sign are common characteristics of the substitutions of a number of factorial sets. The criteria for choosing single substitutions with the best characteristics are proposed.