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Ural Bekbaev

Publications and source records attributed to Ural Bekbaev.

11 recordsLinked to original sources
arXiv2024

An open subset of the variety of $n$- dimensional algebras, classification and automorphisms

Classification, up to isomorphism, of algebras from a non-empty subset of the variety of $n$- dimensional algebras is presented. It is shown that these algebras have only trivial automorphism and if the basic field is algebraically closed then it is an open dense subset of the variety of $n$- dimensional algebras. Properties of these sets, depending on $n$, with respect to the direct sum and tensor product are considered. Moreover one more way of construction of such subsets with similar properties is considered as well.

math.RA↗
arXiv2015

On classification of finite dimensional algebras

Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with structural constants from this set are classified and a basis to the field of invariant rational functions of structural constants is provided.

math.RA↗
arXiv2015

On relation between rational and differential rational invariants of surfaces with respect to the motion groups

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any invariant of the surface is a function of these invariants and their invariant derivatives, is shown. The offered method is applicable in more general settings than the "Moving Frame Method" does in differential geometry.

math.DG↗
arXiv2010

Matrix Representations for Symmetric and Antisymmetric Multi-Linear Maps

In this paper the main results in arXiv:0901.3179v3, related to the matrix representation of polynomial maps, are restated in traditional way of linear algebra assuming that variable vectors are presented as column vectors. Some new results related to that subject are also included. Here one can find the behavior of the matrices of polynomial maps with respect to the change of variables (coordinate system), matrix representations for symmetric and antisymmetric multi-linear maps. It is shown also that the offered representations are in good concordance with known operations over such multi-linear maps.

math.RA↗
arXiv2009

A matrix representation of composition of polynomial maps

In this paper polynomial maps are represented by the use of matrices whose entries are numbered by pair of multiindices and a new product of such matrices is introduced. A matrix representation of composition of polynomial maps is given. In the case of real and complex numbers different kind of norms of such matrices are introduced. Properties of these norms with respect to the ordinary and new products are investigated. A generalization of Bombieri's inequality is offered.

math.AC↗
arXiv2006

On the field of differential rational invariants of a subgroup of affine group (Partial differential case)

An differential field $(F;\partial_1,...,\partial_m)$ of characteristic zero, a subgroup $H$ of affine group $ GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, \partial >^H=\{f^{\partial}< x> \in C< x, \partial> : f^{\partial}< hx+ h_0> = f^{\partial}< x> {whenever} (h,h_0)\in H \},$$ $$C< x, \partial>^{(GL^{\partial}(m,F),H)}=\{f^{\partial}< x> \in C< x, \partial> : f^{g^{-1}\partial}< hx+ h_0> = f^{\partial}< x> {whenever} g\in GL^{\partial}(m,F) {and} (h,h_0)\in H \}$$ are considered, where $C$ is the constant field of $(F,\partial)$, $C< x, \partial>$ is the field of $\partial$-differential rational functions in $x_1,x_2,...,x_n$ over $C$ and $$GL^{\partial}(m,F)= \{g=(g_{jk})_{j,k=\bar{1,m}}\in GL(m,F): \partial_ig_{jk}= \partial_jg_{ik} {for} i,j,k=\bar{1,m}\}$$, $\partial$ stands for the column-vector with the "coordinates" $\partial_1,. . >.,\partial_m$. In the paper these two fields are described.

math.AG↗
arXiv2006

On the field of differential rational invariants of a subgroup of affine group (Ordinary differential case)

An ordinary differential field $(F,d)$ of characteristic zero, a subgroup $H$ of affine group $ GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, d>^H=\{f^d< x> \in C< x, d> : f^d< hx+ h_0> = f^d< x> {for any} (h,h_0)\in H \},$$ $$C< x, d>^{(F^*,H)}=\{f^d< x> \in C< x, d> : f^{g^{-1}d}< hx+ h_0> = f^d< x> {for any} g\in F^* {and} (h,h_0)\in H \}$$ are considered, where $C$ is the constant field of $(F,d)$ and $C< x, d>$ is the field of differential rational functions in $x_1,x_2,...,x_n$ over $C$. The field $C< x, d>^H$ ($C< x, d>^{(F^*,H)}$) is an important tool in the equivalence problem of paths(respect. curves) in Differential Geometry with respect to the motion group $H$. In this paper an pure algebraic approach is offered to describe these fields. The field $C< x, d>^{(F^*,H)}$ and its relation with $C< x, d>^H$ are investigated. It is shown also that $C< x, d>^H$ can be derived from some algebraic (without derivatives) invariants of $H$. Key words: Differential field, differential rational function, invariant, differential transcendent degree. 2000 Mathematics Subject Classification: 12H05, 53A04, 53A55

math.AG↗
arXiv2006

Ergodicity of power series-map on the simplex of group algebra of a finite group

A finite group $G$, its group algebra $R[G]$ over the field of real numbers, any power series $p(t)= a_0+a_1t+ a_{2}t^{2}+ ...$, where $ a_i \geq 0$, and $a_0+a_1+ a_{2}+...= 1$, and simplex $$ S= \{x=\sum_{g\in G}x_gg\in R[G]: \sum_{g\in G}x_g=1, x_g\geq 0 {for any}\quad g\in G \}$$ are considered. Ergodicity of the map $p: S\to S$, where $p(x)= a_0+a_1x+ a_{2}x^{2}+ >...$ for $x\in S$, on $S$ is shown. The regularity of this map at a given point $x\in S $ is investigated as well.

math.DS↗