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Sergey Malev

Publications and source records attributed to Sergey Malev.

16 recordsLinked to original sources
arXiv2026

Exponential stability of second order delay differential equations through Floquet theory

In this paper, we obtain results on exponential stability of second order delay differential equations, which are based on a version of the Floquet theory for delay differential equations of the second order we proposed. Our version allows researchers to preserve the order of equation and to obtain analogues of the classical results of the Floquet theory known for ordinary differential equations. On the basis of our version of the Floquet theory, new original unexpected results on the exponential stability are proposed. We demonstrate that choosing period of coefficients and delays of the gain in corresponding intervals allows to achieve the exponential stabilization in the cases considered as impossible when the standard technique was applied.

math.DS↗
arXiv2023

Algebro-geometric proof of Christoph's Theorem

In this paper an algebraic proof of Christoph's theorem is provided. This theorem from algebraic-geometry is about the existence of a finite automaton for computing coefficient of a series for an algebraic function.

math.AG↗
arXiv2022

On the Distribution function of area and perimeter for planar poisson line process

The challenges of examining random partitions of space are a significant class of problems in the theory of geometric transformations. Richard Miles calculated moments of areas and perimeters of any order (including expectation) of the random division of space in 1972. In the paper we calculate whole distribution function of random divisions of plane by poisson line process. The idea is to interpret a random polygon as the evolution of a segment along a moving straight line. In the plane example, the issue connected with an infinite number of parameters is overcome by considering a secant line. We shall take into account the following tasks: {\textbf 1.} On the plane, a random set of straight lines is provided, all shifts are equally likely, and the distribution law is of the form $F(\varphi).$ What is the area distribution of the partition's components? {\textbf 2.} On the plane, a random set of points is marked. Each point $A$ has an associated area of attraction, which is the collection of points in the plane to which the point $A$ is the nearest of the designated ones. In the first problem, the density of moved sections adjacent to the line allows for the expression of the balancing ratio in kinetic form. Similarly, you can write the perimeters kinetic equations. We will demonstrate how to reduce these equations to the Riccati equation using the Laplace transformation in this paper.

math.PR↗
arXiv2022

The images of multilinear and semihomogeneous polynomials on the algebra of octonions

The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra $A$ the image of a multilinear polynomial on $A$ is a vector space. In this paper we prove it for the algebra of octonions $\mathbb{O}$ over a field satisfying certain specified conditions (in particular, we prove it for quadratically closed field and for field $\mathbb{R}$). In fact, we prove that the image set must be either $\{0\}$, $F$, the space of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the corresponding Malcev algebra.

math.AG↗
arXiv2021

Evaluations of multilinear polynomials on low rank Jordan algebras

In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of multilinear polynomial must be either $\{0\}$, $\mathbb{R}$, the space of pure elements $V$, or $J_n$.

math.RA↗
arXiv2020

Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In Section 2.1 we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin's problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. Section 2.2 is dedicated to particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last two sections deal with algorithmic problems for noncommutative and commutative algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. Section 3.2 is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

math.RA↗
arXiv2020

Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${\rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $n\geq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

math.RA↗
arXiv2020

The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak M}$ of every non-associative multilinear polynomial over ${\mathbb F}$ is a vector space. The same question we consider for two subalgebras: an algebra of the rock, the paper and the scissors without unit, and an algebra of trace zero elements with zero scalar part. Moreover in this paper we consider the questions of possible evaluations of homogeneous polynomials on these algebras.

math.RA↗
arXiv2019

The images of non-commutative polynomials evaluated on the Quaternion algebra

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.

math.RA↗
arXiv2017

Finite Gröbner basis algebra with unsolvable nilpotency problem and zero divisors problem

This work presents a sample constructions of two algebras both with the ideal of relations defined by a finite Gröbner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

math.RA↗
arXiv2015

The images of Lie polynomials evaluated on matrices

Kaplansky asked about the possible images of a polynomial $f$ in several noncommuting variables. In this paper we consider the case of $f$ a Lie polynomial. We describe all the possible images of $f$ in $M_2(K)$ and provide an example of $f$ whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial $s_k$ is not a Lie polynomial, for $k>2.$

math.AG↗
arXiv2014

The images of multilinear polynomials evaluated on $3\times 3$ matrices

Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in a algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$ matrices. The image is one of the following: \begin{itemize} \item \{0\}, \item the set of scalar matrices, \item a (Zariski) dense subset of $\sl_3(K)$, the matrices of trace 0, \item a dense subset of $M_3(K)$, \item the set of $3-$scalar matrices (i.e., matrices having eigenvalues $(β, β\varepsilon, β\varepsilon^2)$ where $\varepsilon$ is a cube root of 1), or \item the set of scalars plus $3-$scalar matrices.

math.AG↗
arXiv2013

The images of non-commutative polynomials evaluated on $2\times 2$ matrices over an arbitrary field

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, or all of $M_n(K)$. This conjecture was proved for $n=2$ when $K$ is closed under quadratic extensions. In this paper the conjecture is verified for $K=\mathbb{R}$ and $n=2$, also for semi-homogeneous polynomials $p$, with a partial solution for an arbitrary field $K$.

math.AG↗
arXiv2013

Power-central polynomials on matrices

Any multilinear non-central polynomial $p$ (in several noncommuting variables) takes on values of degree $n$ in the matrix algebra $M_n(F)$ over an infinite field $F$. The polynomial $p$ is called {\it $ν$-central} for $M_n(F)$ if $p^ν$ takes on only scalar values, with $k$ minimal such. Multilinear $ν$-central polynomials do not exist for any $ν$ with $n>3$, thereby answering a question of Drensky. Saltman proved that an arbitrary polynomial $p$ cannot be $ν$-central for $M_n(F)$ for $n$ odd unless $n$ is prime; we show for $n$ even, that $ν$ must be 2.

math.AG↗
arXiv2011

The images of non-commutative polynomials evaluated on $2\times 2$ matrices

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$.

math.AG↗