Vol. 56 No. 1 (257) (2022)

DOI: https://doi.org/10.46991/PYSUA.2022.56.1
Published: 2022-04-04

Mathematics

  • Mathematics

    ON SCHUR MULTIPLIERS OF SOME RELATIVELY FREE GROUPS

    Grigor G. Gevorgyan
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    Abstract

    In this paper central extensions of free groups of infinitely based varieties of S.I. Adian are constructed. Using this extensions we prove that the Schur multipliers of mentioned relatively free groups are free Abelian groups of infinite rank. It is well-known that these varieties are given by identities in two variables. For a fixed rank $m$, the set of free groups of rank $m$ of these varieties has the cardinality of continuum.

    References
  • Mathematics

    ON NONTRIVIAL SOLVABILITY OF ONE CLASS OF NONLINEAR INTEGRAL EQUATIONS WITH CONSERVATIVE KERNEL ON THE POSITIVE SEMI-AXIS

    Khachatur A. Khachatryan, Alexander R. Hakobyan
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    Abstract

    The work is devoted to a special class of nonlinear integral equations on the positive semi-axis with conservative kernel that corresponds to a nonlinear operator, for which the property of complete continuity in the space of bounded functions fails. In different special cases this class of equations has applications in particular branches of mathematical physics. In particular, this kind of equations can be met in the radiative transfer theory, kinetic theory of gases, kinetic theory of plasma and in the $p$-adic open-closed string theory. Using a combination of special iterations with the monotonic operator theory methods, that work in defined conical segments it is possible to prove a constructive existence theorem of nonnegative nontrivial bounded solution that has finite limit at infinity. The asymptotics of the constructed solution will also be studied. It is also given an example of nonlinear equation, for which the uniqueness of the solution in the space of bounded functions fails. At the end of the paper will consider some classes of equations both applied and pure theoretical character.

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  • Mathematics

    ON INTERVAL EDGE-COLORINGS OF COMPLETE MULTIPARTITE GRAPHS

    Levon N. Muradyan
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    Abstract

    A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in a way that each vertex in $V_i$ is adjacent to all the other vertices in $V_j$ for $1\leq i<j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote a complete $r$-partite graph with independent sets $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an \emph{interval $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In this paper we have obtained some results on the existence and construction of interval edge-colorings of complete $r$-partite graphs. Moreover, we have also derived an upper bound on the number of colors in interval colorings of complete multipartite graphs.

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  • Mathematics

    PERFECTLY STABLE AND NORMAL SUBGROUPS

    Shant Kh. Navasardyan
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    Abstract

    The equivalence between the concepts of normal and perfectly stable  subgroups is shown. The proof of the main theorem is based on a novel concept of hypergroups over a group.

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  • Mathematics

    RELATION BETWEEN THE COVARIOGRAM AND DISTRIBUTION FUNCTION OF THE DISTANCE BETWEEN TWO UNIFORM AND INDEPENDENT POINTS

    Victor K. Ohanyan, Vigen H. Khalatyan
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    Abstract

    In the present paper we obtain a relationship between the covariogram and distribution function of the distance between two uniformly and independently distributed points. Additionally, we calculate the distribution function of the distance between these two points in a disk, a ball and a triangle.

    References