#### Name

Sergeev Alexander Eduardovich

#### Scholastic degree

•

#### Academic rank

associated professor

#### Honorary rank

—

#### Organization, job position

Kuban State University

#### Web site url

—

## Articles count: 23

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

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

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

In the article, we develop the methodology of strategic planning and management of a holding, on the theoretical basis of automated system-cognitive analysis (ASC-analysis). This methodology provides scientific research of any holding by creating and researching its model. The methodology includes both the synthesis, adaptation and verification of system-cognitive models of the holding, and the use of these models for strategic planning and decision support for the management of the holding, as a complex, multiparametric, nonlinear system. The relevance of the research is due to the special role of holdings and other corporate integrated structures both in Russia as a whole and, in particular, in the Krasnodar region. Despite obvious system advantages, holdings face a wide range of problems related to management efficiency, ensuring their sustainable functioning, etc. The proposed methodology offers ways to solve these problems and can be successfully applied in holdings and other corporate integrated structures of various regions, volumes and areas of activity, which determines the relevance of the research topic. The level of significance and scientific novelty of the Research consists in the development of conceptual and theoretical and methodological provisions aimed at managing the development of holdings. The expected results and their significance are that the methodology developed as a result of the Research can be applied by holding companies and other corporate integrated structures and will significantly improve the quality of their management

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

One of the key problems facing the credit institution is the late payment of the loan. Firstly, it is a deeper analysis - in order to be carried out “manually” it is not even required several days, but weeks. Secondly, it allows you to work with clients much faster. And most importantly scoring allows you to negate the influence of the human factor. An automated system, no matter how you look, cannot be liked or not. Data analysis is only based on facts. Scoring is beneficial to all. The bank is able to work faster and reduce the risk of loan defaults. Clients, in turn, can apply for a loan on terms that are more favorable

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

In this article, we construct polynomials of third, fourth and fifth degrees with Galois groups as and respectively. In addition, we give examples of polynomials different degrees with Galois groups isomorphic transitive subgroup of group , but calculations with help Maple show that Galois groups of this polynomials is . Also Polynomials with as Galois groups are shown

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 1893, the French mathematician J. Adamar
raised the question: given a matrix of fixed order
with coefficients not exceeding modulo this value,
then what is the maximum modulo value can take
the determinant of this matrix? Adamar fully
decided this question in the case when the
coefficients of the matrix are complex numbers and
put forward the corresponding hypothesis in the
case when the matrix coefficients are real numbers
modulo equal to one. Such matrices satisfying the
Hadamard conjecture were called Hadamard
matrices, their order is four and it is unknown
whether this condition is sufficient for their
existence. The article examines a natural
generalization of the Hadamard matrices over the
field of real numbers, they are there for any order.
This paper proposes an algorithm for the
construction of generalized Hadamard matrices,
and it is illustrated by numerical examples. Also
introduces the concept of constants for the natural
numbers are computed values of this constant for
some natural numbers and shown some
applications of Hadamard constants for estimates
on the top and bottom of the module of the
determinant of this order with arbitrary real
coefficients, and these estimates are in some cases
better than the known estimates of Hadamard. The
results of the article are associated with the results
of the con on the value of determinants of matrices
with real coefficients, not exceeding modulo units