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Aleks Kleyn

Publications and source records attributed to Aleks Kleyn.

At least 19 recordsLinked to original sources
arXiv2025

Continuous Group of Transformations

Let $B$ be Banach algebra and $M$ be topological space. If there exists homeomorphism \[ f:M\rightarrow N \] of topological space $M$ into convex set $N$ of the space $B^n$, then homeomorphism $f$ is called chart of the set $M$. The set $M$ is called simple $B$-manifold of class $C^k$ if for any two charts \[ f_1:M\rightarrow N_1\subseteq B^n \] \[ f_2:M\rightarrow N_2\subseteq B^n \] there exists diffeomorphism \[ f: B^n\rightarrow B^n \] of class $C^k$ such that \[ f_1\circ f=f_2 \] Topological space $M$ is called differential $B$-manifold of class $C^k$ if topological space $M$ is a union of simple $B$-manifolds $M_i$, $i\in I$, and intersection $M_i\cap M_j$ of simple $B$-manifolds $M_i$, $M_j$ is also simple $B$-manifold. Differential $B$-manifold $G$ equipped with group structure such that map \[ (f,g)\rightarrow fg^{-1} \] is differentiable is called Lie group. Module $T_eG$ equipped with product \[ [v,w]^c= R_{Ljm}^c\circ(v^m,w^j) -R_{Lmj}^c\circ(w^j,v^m) \in T_eG \] is Lie algebra $g_L$ of Lie group $G$.

math.GM↗
arXiv2024

Introduction into Geometry over Division Ring

Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the notion of tower of representations of F_i-algebras, i=1, ..., n, as the set of coordinated representations of F_i-algebras. I explore the geometry of affine space as example of tower of representations. Exploration of curvilinear coordinates allows us to see how look at the main structures of the manifold with affine connection. I explore Euclidean space over division ring.

math.GM↗
arXiv2024

Introduction into Noncommutative Algebra, Volume 1

I dedicated the volume $1$ of monograph 'Introduction into Noncommutative Algebra' to studying of algebra over commutative ring. The main topics that I covered in this volume: definition of module and algebra over commutative ring; linear map of algebra over commutative ring; vector space over associative division algebra; system of linear equations over associative division algebra; homomorphism of vector space over associative division algebra and covariance theory; eigenvalue and eigenvector over associative division algebra.

math.GM↗
arXiv2024

Linear Equation over Non-Commutative Algebra

I considered solving of the system of linear equations $$a^1_{1s0}x^1a^1_{1s1}+...+a^1_{ns0}x^na^1_{ns1}=b^1$$ $$...$$ $$a^n_{1s0}x^1a^n_{1s1}+...+a^n_{ns0}x^na^n_{ns1}=b^n$$ over non-commutative associative algebra. I considered examples in quaternion algebra. I considered also Newton's method to solve the equation $$f(x)=a$$ over non-commutative associative algebra.

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arXiv2023

Covariance in Non-Commutative Algebra

Consider vector space over non-commutative division algebra. Set of automorphisms of this vector space is group $GL$. Group $GL$ acts on the set of bases of vector space (basis manifold) single transitive and generates active representation. Twin representation on basis manifold is called passive representation. There is no automorphism associated with passive transformation. However passive transformation generates transformation of coordinates of vector with respect to basis. If we consider homomorphism of vector space $V$ into vector space $W$, then we can learn how passive transformation in vector space $V$ generates transformation of coordinates of vector in vector space $W$. Vector in vector space $W$ is called geometric object in vector space $V$. Covariance principle states that geometric object does not depend on the choice of basis. I considered transformation of coordinates of vector and polylinear map.

math.GM↗
arXiv2023

Quadratic Equation over Associative D-Algebra

In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Viete's theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.

math.GM↗
arXiv2022

Differential Equation over Banach Algebra

In the book, I considered differential equations of order $1$ over Banach $D$\Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential equations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.

math.GM↗
arXiv2022

Eigenvector in Non-Commutative Algebra

$\newcommand{\Vector}[1]{\bar{#1}{}}$ $\newcommand{\Basis}[1]{\bar{\bar{#1}}{}}$ $\newcommand{\RC}{{}_*{}^*-}$ Let $\Basis e$ be a basis of vector space $V$ over non-commutative $D$-algebra $A$. Endomorhism $\Vector{\Basis eb}$ of vector space $V$ whose matrix with respect to given basis $\Basis e$ has form $Eb$ where $E$ is identity matrix is called similarity transformation with respect to the basis $\Basis e$. Let $V$ be a left $A$-vector space and $\Basis e$ be basis of left $A$-vector space $V$. The vector $v\in V$ is called eigenvector of the endomorphism \[\Vector f:V\rightarrow V\] with respect to the basis $\Basis e$, if there exists $b\in A$ such that \[ \Vector f\circ{v}= \Vector{\Basis eb} \circ{v} \] $A$-number $b$ is called eigenvalue of the endomorphism $\Vector f$ with respect to the basis $\Basis e$. There are two products of matrices: ${}_*{}^*$ (row column: $(ab)^i_j=a^i_kb^k_j$) and ${}^*{}_*$ (column row: $(ab)^i_j=a^k_jb^i_k$). $A$-number $b$ is called $\RC$ eigenvalue of the matrix $f$ if the matrix $f-bE$ is $\RC$ singular matrix. The $A$-number $b$ is called right $\RC$ eigenvalue if there exists the column vector $u$ which satisfies to the equality \[a{}_*{}^* u=ub\] The column vector $u$ is called eigencolumn for right $\RC$ eigenvalue $b$. The $A$-number $b$ is called left $\RC$ eigenvalue if there exists the row vector $u$ which satisfies to the equality \[u{}_*{}^* a=bu\] The row vector $u$ is called eigenrow for right $\RC$ eigenvalue $b$. The set $\RC$ $\mathrm{spec}(a)$ of all left and right $\RC$ eigenvalues is called $\RC$ spectrum of the matrix a.

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arXiv2021

Polynomial in Non-Commutative Algebra

I considered definition and properties of polynomial in no-commutative algebra. There exists polynomial which has finite, infinite or empty set of roots. For instance, the polynomial $$p_1(x)=ix-xi-1$$ have no root and the polynomial $$p_k(x)=ix-xi-k$$ has the set of roots $$x=C_1+C_2i+\frac 12 j$$ I considered division of polynomials with remainder.

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arXiv2019

Lorentz Transformation and General Covariance Principle

I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometrical object, concept of reference frame, geometry of metric-affinne manifold. Using this concept I learn few physical applications: dynamics and Lorentz transformation in gravitational fields, Doppler shift. A reference frame in event space is a smooth field of orthonormal bases. Every reference frame is equipped by anholonomic coordinates. Using anholonomic coordinates allows to find out relative speed of two observers and appropriate Lorentz transformation. Synchronization of a reference frame is an anholonomic time coordinate. Simple calculations show how synchronization influences time measurement in the vicinity of the Earth. Measurement of Doppler shift from the star orbiting the black hole helps to determine mass of the black hole. We call a manifold with torsion and nonmetricity the metric\hyph affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable. Learning how torsion influences on tidal force reveals similarity between tidal equation for geodesic and the Killing equation of second type.

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arXiv2019

Diagram of Representations of Universal Algebras

Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation.

math.GM↗
arXiv2017

Introduction into Calculus over Banach algebra

Let $A$, $B$ be Banach $D$-algebras. The map $f:A\rightarrow B$ is called differentiable on the set $U\subset A$, if at every point $x\in U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =\frac{d f(x)}{d x}\circ dx +o(dx)$$ where $$\frac{d f(x)}{d x}:A\rightarrow B$$ is linear map and $o:A\rightarrow B$ is such continuous map that $$\lim_{a\rightarrow 0}\frac{\|o(a)\|_B}{\|a\|_A}=0$$ Linear map $\displaystyle\frac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $ω\in\mathcal{LA}(D;A\rightarrow B)$ is defined by map $g:A\rightarrow B\otimes B$, $ω=g\circ dx$. If the map $g$, is derivative of the map $f:A\rightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=\int g(x)\circ dx=\intω$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$\int_a^bω=\int_γω$$ for any path $γ$ from $a$ to $b$.

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arXiv2016

Linear Map of $D$-Algebra

Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general point of view I started the book from consideration of Cartesian product of representations. Polymorphism of representations is a map of Cartesian product of representations which is a morphism of representations with respect to each separate independent variable. Reduced morphism of representations allows us to simplify the study of morphisms of representations. However a representation has to satisfy specific requirements for existence of reduced polymomorphism of representations. It is possible that Abelian group is only $Ω$-algebra, such that representation in this algebra admits polymorphism of representations. However, today, this statement has not been proved. Multiplicative $Ω$-group is $Ω$-algebra in which product is defined. The definition of tensor product of representations of Abelian multiplicative $Ω$-group is based on properties of reduced polymorphism of representations of Abelian multiplicative $Ω$-group. Since an algebra is a module in which the product is defined, then we can use this theory to study linear map of algebra. For instance, we can study the set of linear transformations of $D$-algebra $A$ as representation of algebra $A\otimes A$ in algebra $A$.

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arXiv2015

English Russian Scientific Dictionary

English Russian and Russian English dictionaries presented in this paper are dedicated to help translate a scientific text from one language to another. I also included the bilingual name index into this book.

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arXiv2015

Derivative of Map of Banach algebra

Let $A$ be Banach algebra over commutative ring $D$. The map $f:A\rightarrow A\ $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=\partial f(x)\circ a+o(a)$$ where the Gateaux derivative $\partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ \lim_{a\rightarrow 0}\frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $\partial^{n-1} f(x)$ of order $n-1$, we define $$ \partial^n f(x)\circ(a_1\otimes...\otimes a_n) =\partial(\partial^{n-1} f(x)\circ(a_1\otimes...\otimes a_{n-1}))\circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=\sum_{n=0}^{\infty}(n!)^{-1}\partial^n f(x_0)\circ(x-x_0)^n $$

math.GM↗
arXiv2015

Polynomial over Associative D-Algebra

In the paper I considered algebra of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation of polynomial. I considered questions related to divisibility of polynomial of any power over polynomial of power 1.

math.GM↗
arXiv2015

Representation of Universal Algebra

Theory of representations of universal algebra is a natural development of the theory of universal algebra. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the notion of tower of representations of $F_i$-algebras, i=1 ..., n, as the set of coordinated representations of $F_i$-algebras.

math.GM↗