Scientific Journal of KubSAU

Polythematic online scientific journal
of Kuban State Agrarian University
ISSN 1990-4665
AGRIS logo UlrichsWeb logo DOAJ logo

Name

Sergeev Alexander Eduardovich

Scholastic degree


Academic rank

associated professor

Honorary rank

Organization, job position

Kuban State University
   

Web site url

Email

alexander2000@mail.kubsu.ru


Articles count: 10

Sort by: Date Title Views
469 kb

ABOUT EULER FUNCTION

abstract 1271703004 issue 127 pp. 113 – 125 31.03.2017 ru 319
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
108 kb

FUNDAMENTAL THEOREM OF ARITHMETIC AND SOME OF ITS ASPECTS

abstract 1131509010 issue 113 pp. 127 – 132 30.11.2015 ru 355
In this article, we present the fundamental theorem of arithmetic and its role. We consider various rings for its performance
144 kb

GENERIC POLYNOMIALS FOR THE CYCLIC 2-GROUPS OVER FIELDS WITH CHARACTERISTIC TWO

abstract 0761202072 issue 76 pp. 898 – 908 29.02.2012 ru 1076
In this article, the generic polynomials for cyclic groups of order 4, 8 and 16 over fields with characteristic two are constructed. With this construction, the generic polynomials for all cyclic 2-groups over fields with characteristic two can be obtained. We also give survey of known results of generic polynomials for the cyclic groups.
395 kb

HADAMARD MATRICES

abstract 1261702033 issue 126 pp. 471 – 483 28.02.2017 ru 214
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
162 kb

ON THE NUMERATIONS OF THE FINITE PARTIALLY ORDERED SETS

abstract 1181604047 issue 118 pp. 805 – 816 29.04.2016 ru 336
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)
130 kb

PARAMETRIC TRINOMIALS WITH ALTERNATING GALOIS GROUPS

abstract 0761202071 issue 76 pp. 888 – 897 29.02.2012 ru 1106
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
116 kb

ROOT POLYNOMIALS

abstract 0781204067 issue 78 pp. 842 – 851 30.04.2012 ru 916
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
227 kb

SPECIAL CASES OF INVERSE MATRICES

abstract 1301706071 issue 130 pp. 975 – 981 30.06.2017 ru 144
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
572 kb

THE REALIZATION OF GALOIS GROUPS BY TRINOMIALS OVER THE FIELD OF RATIONAL NUMBERS Q

abstract 1311707124 issue 131 pp. 1497 – 1524 29.09.2017 ru 23
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
154 kb

THE THEOREMS OF CHEBYSHEV ABOUT THE DISTRIBUTION OF PRIME NUMBERS AND SOME PROBLEMS, CONNECTED WITH PRIME NUMBERS

abstract 1131509009 issue 113 pp. 115 – 126 30.11.2015 ru 445
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
.