Scientific Journal of KubSAU

It is known that not every finite group can be realized over the field of rational numbers as a Galois group of some binomial. In this connection, a more general question arises: suppose that there is given a finite transitive subgroup G of the symmetric group S on n symbols; Can this group G be realized as a Galois group of some trinomial of degree n over the field of rational numbers? In this paper we prove that every transitive subgroup of the group S can be realized in the form of the Galois group of a certain trinomial of the degree n, for the values n = 2, 3, 4. For n = 5 , 6 we give examples that realize concrete Galois groups. In the case n = 7, all the transitive subgroups of the group S are realized, except possibly one group of the isomorphic dihedral group D. Further calculations will be directed to the realization of specific Galois groups for n = 8, 9 ..., however, the number of transitive subgroups of the group S for n = 8, 9 ... grows very fast, so the larger the value of n, the more difficult it is to realize not just everything but the specific subgroup of the group S in the form of a trinomial over Q

The problem of establishing of the factorization of irreducible polynomials with integer coefficients on prime modules p has been long of interest to mathematicians. The quadratic and cubic reciprocity laws solve this problem for quadratic polynomials and binomials of the form x3-a . More general reciprocity laws solve the formulated problem for some classes of polynomials, for example, with Abelian Galois group, but for polynomials with non-Abelian Galois group, the problem is far from its complete solution. Our study shows how using the results of Voronov G.F., Hasse H. and Stickelberger L., one can find conditions that must satisfy prime number p. Gauss received a similar result for binomial x3-2. Specific examples are given, for instance, for the polynomial x3-x - I, also conditions arc formulated for which a quadratic field is immersed in non-Abelian Galois extension of degree 6. Also, conditions are given under which a Diophantine equation: а12a22-4a22-4a13a3- 27a32+18a1a2a3=D has a solution for integer values of D

Traditionally, control decisions are made by solving repeatedly the forecasting problem for different values of control factors and choosing a combination of them that ensures the transfer of the control object to the target state. However, real control objects are affected by hundreds or thousands of control factors, each of which can have dozens of values. A complete search of all possible combinations of values of control factors leads to the need to solve the problem of forecasting tens or hundreds of thousands or even millions of times to make a single decision, and this is completely unacceptable in practice. Therefore, we need a decision-making method that does not require significant computing resources. Thus, there is a contradiction between the actual and the desired, a contradiction between them, which is the problem to be solved in the work. In this work, we propose a developed algorithm for decision-making by solving the inverse forecasting problem once (automated SWOT analysis), using the results of cluster-constructive analysis of the target states of the control object and the values of factors and a single solution of the forecasting problem. This determines the relevance of the topic. The purpose of the work is to solve the problem. By decomposing the goal, we have formulated the following tasks, which are the stages of achieving the goal: cognitive-target structuring of the subject area; formalization of the subject area (development of classification and descriptive scales and gradations and formation of a training sample); synthesis, verification and increasing the reliability of the model of the control object; forecasting, decision-making and research of the control object by studying its model. The study uses the automated system-cognitive analysis and its software tools (the intelligent system called "Eidos") as a method for solving the set tasks. As a result of the work, we propose a developed decision-making algorithm, which is applicable in intelligent control systems. The main conclusion of the work is that the proposed approach has successfully solved the problem

This work continues the series of works written by the author on the application of modern scientific methods in the study of human consciousness. In 1979-1981, two monographs were written devoted to higher forms of consciousness, the prospects of man, technology and society. One of these monographs was two-volume and was called "Theoretical Foundations of the Synthesis of Quasi-Biological Robots." In these monographs the author proposed: 1) criterial periodic classification of 49 forms of consciousness, including higher forms of consciousness (HFC); 2) based on this classification, there were psychological, microsocial and technological methods of transition between various forms of consciousness, including methods of transition from the usual form of consciousness to the HFC; 3) information-functional theory of the development of technology (including the rule of improving the quality of the basis); 4) information theory of value; 5) 11 functional schemes of technical systems of future forms of society, including remote telekinetic (mental) control systems; 6) the concept of development of society in groups of socio-economic formations; 7) the concept of determining the form of human consciousness by the functional level of the technological environment; 8) mathematical and numerical modeling of the dynamics of the probability density of states of human consciousness in evolution using the theory of Markov’s random processes. In this study, we carry out a complete automated system-cognitive analysis (ASC- analysis) of the periodic criteria classification of forms of consciousness proposed by the author in 1978. To this end, the following tasks are solved in the work: cognitive structuring and formalization of the subject area; synthesis and verification of statistical and system-cognitive models (multi-parameter typification of forms of consciousness); systemic identification of forms of consciousness; their typological analysis; investigations of a simulated domain by examining its model. We have also given a detailed numerical example of solving all these problems

The inverse matrix for the square matrix A of order n with coefficients of some field exists, as it is known then and only then, when its determinant is not equal to zero. If the matrix A has a certain type (certain structure), then an inverse matrix A-1 should not have exactly the same structure. Therefore, it is interesting to describe such square matrices A, which have an inverse matrix A-1, having the same structure as the matrix A, under certain conditions. For example, a subdiagonal matrix with nonzero elements on the main diagonal has an inverse matrix over a field of characteristic zero, having also the form of subdiagonal matrix. Similarly, an inverse matrix towards symmetrical or skew-symmetric matrix is also symmetric or skew-symmetric accordingly. Also, the matrix inverse to non-degenerate (nonsingular) circulant will be a circulant itself, and finally, the matrix inverse to nonsingular quasdiagonal matrix D will be quasdiagonal itself, and will have the same partitioned structure as D. Thus, there is a problem of determining these types of nonsingular matrices that have an inverse matrix of the same type as a given matrix. In line with this problem in the present study it is determined such type of matrices for which an inverse matrix has the same type, at that the conditions are identified in explicit form, ensuring the nonsingularity of the matrix. The matrices of three orders are shown in detail. These results allow determining the characteristics of fields over which there are inverse matrices of the considered types

The Euler function is very important in number theory and in Mathematics, however, the range of its values in the natural numbers has not been written off. The greatest value of the Euler function reaches on Prime numbers, furthermore, it is multiplicative. The value of the Euler function is closely associated with the values of the Moebius function and the function values of the sum of the divisors of the given natural number. The Euler function is linked with systems of public key encryption. The individual values of the Euler function behave irregularly because of the irregular distribution of primes in the natural numbers. This tract is illustrated in the article with charts; summatory function for the Euler function and its average value are more predictable. We prove the formula of Martinga and, based on it, we study the approximation accuracy of the average value of the Euler function with corresponding quadratic polynomial. There is a new feature associated with the average value of the Euler function and calculate intervals of its values. We also introduce the concept of density values of the Euler function and calculate its value on the interval of the natural numbers. It can be noted that the results of the behavior of the Euler function are followed by the results in the behavior of functions of sums of divisors of natural numbers and vice versa. We have also given the results of A.Z.Valfish and A.N.Saltykov on this subject

In this article, we discuss various issues related to the formulas approximating the distribution function of prime numbers pi(x). This question has occupied many scholars, but the exact function is well approximated function pi(x) over the number of positive integers not. Based on certain hypotheses, we present a new function s(x) is very well approximated pi(x). The above article hypotheses are so important that their numerical validation and refinement for the lengths of the segments more in 1014 - one of the main areas related to the problem of approximation of the function pi(x) throughout the series of natural numbers. After analyzing the behaviors and constructs many functions, we are building the basis of the function s(x), which is well approximates the function pi(x) throughout the series of natural numbers. We also present a table of values for x, less or equal 1022 for the difference of s(x) - pi(x)

In this article, we present the fundamental theorem of arithmetic and its role. We consider various rings for its performance

The article presents the theorem of Chebyshev on the distribution of primes, considering functions that approximated prime numbers. We have also considered a new function, which is quite good for approximation of prime numbers. A review of the known results on distribution of prime numbers is given as well

The article obtained the explicit form of root polynomials for cyclic polynomials of degree three over fields of characteristic 2. We also give an overview of known results on the root polynomials over arbitrary fields