
Isamiddin RakhimovMARA University of Technology | UiTM · Department of Mathematics
Isamiddin Rakhimov
PhD & DSc Professor
About
167
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1,264
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Introduction
Additional affiliations
October 2019 - present
January 2016 - October 2017
Education
December 1982 - December 1985
Publications
Publications (167)
In the paper we utilize a new approach to the classification problem of finite-dimensional algebras. We give a complete classifications of associative and diassociative algebra structures on two-dimensional vector space over any basic field.
In the paper we give complete classification of two classes of two-dimensional PI-algebras over any basic field. The choice of these two classes is predicted by the polynomial identities of the classes: the identity of one of them is given by using the binary operation of the algebra another one involves the bracket operation in the identity. The l...
Classifying Frobenius algebras is a key question that has been addressed in various contexts. The structure of finite-dimensional Frobenius algebras depends on the base field and the dimension of the algebra, leading to different classification results depending on whether the base field has characteristic zero, characteristic $p$, or other propert...
A complete classifications, up to isomorphism, of two-dimensional associative and diassociative algebras over any basic field are given.
In the paper, we consider the class of so-called endo-commutative algebras. From the identity imposed to specify this class, one can easily see that the product in this class preserves the square of elements. We give a complete classification of this class, up to isomorphism, over any basic field in dimension two. This elementary and self-contained...
In the paper we describe the derivations of two N-graded infinite-dimensional Lie algebras n1 and n2 which are the positive parts of the affine Kac-Moody algebras A(1) 1 and A(2) 2 , respectively. Then we construct all pro-solvable Lie algebras whose potential nilpotent ideals are n1 and n2 and compute low-dimensional (co)homology groups of the pro...
In the paper, we propose a series of convolution identities for the sequence of Catalan numbers. The proofs of two of them are provided and the other identities are proposed as conjectures. We include a simple Maple code to generate and confirm the identities.
For a field F, where any polynomial of degree two and three possess a root, we give an algorithm transforming any two-dimensional algebra over F to a "canonical form" in terms of structure constants of the algebra. A system of invariant conditions in form of equalities and non-equalities (inequalities) separating non-isomorphic two-dimensional alge...
The paper is devoted studying solvable Leibniz algebras with a nilradical possessing the codimension equals the number of its generators. We describe this class in non-split nilradical case up to isomorphism. Then the case of split nilradical is worked out. We show that the results obtained earlier on this class of Leibniz algebras come as particul...
The paper is devoted studying solvable Leibniz algebras with a nilradical possessing the codimension equals the number of its generators. We describe this class in non-split nilradical case. Then the case of split nilradical is worked out. We show that the results obtained earlier on this class of Leibniz algebras come as particular cases of the re...
In the paper we describe the class of all solvable extensions of an infinite-dimensional filiform Leibniz algebra. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of a residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial.
In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial.
In the paper we describe the derivations of two $\mathbb{N}$-graded infinity-dimensional Lie algebras $\mathbf{n}_1$ and $\mathbf{n}_1$ what are positive parts of affine Kats-Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$, respectively. Then we construct all pro-solvable Lie algebras whose potential nilpotent ideals are $\mathbf{n}_1$ and $\mathbf{n}_2...
In the paper, we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all two-dimensional algebras with respect to these identities is given.
In the work, we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is given.
In the paper we give a complete classification of $2$-dimensional evolution algebras over algebraically closed fields, describe their groups of automorphisms and derivation algebras.
The descriptions, up to isomorphism, of all two-dimensional left (right) unital algebras over algebraically closed fields and $\mathbb{R}$ are given.
All subalgebras, idempotents, left(right) ideals and left quasiunits of two-dimensional algebras are described. Classifications of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphic classes of all simple two-dimensional algebras is given.
All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classifications of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphism classes of all simple two-dimensional algebras is given. In the study of ideals and subalgebras, the numb...
In this paper, we give a complete algebraic classification of 5-dimensional complex nilpotent associative commutative algebras.
In this paper we describe all power-associative algebra structures on a two-dimensional vector space over algebraically closed fields and ℝ. The list of all two-dimensional left(right) unital power-associative algebras, along with their unit elements, is specified. Also we compare the result of the paper with that results obtained earlier.
In this paper we describe all power-associative algebra structures on a two-dimensional vector space over algebraically closed fields and ℝ. The list of all two-dimensional left(right) unital power-associative algebras, along with their unit elements, is specified. Also we compare the result of the paper with that results obtained earlier.
In this paper, we propose a method for processing image forgery
detection using the Krawtchouk moments. The proposed method
detects copy move forged regions of images by using Krawtchouk
moments. Krawtchouk moments are used to extract features from the
images to be used to identify the similar patches.
In the paper we describe all (not necessarily commutative) Jordan algebra structures on a two-dimensional vector space over R in terms of their matrices of structure constants.
All subalgebras, idempotents, left(right) ideals and left quasiunits of two-dimensional algebras are described. Classifications of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphic classes of all simple two-dimensional algebras is given.
All subalgebras, idempotents, left(right) ideals and left quasiunits of two-dimensional algebras are described. Classifications of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphic classes of all simple two-dimensional algebras is given.
The paper consists of three parts. In the first part we discuss on extensions of Lie algebras and their importance in Physics. Then we deal with the extensions of some classes of algebras with one binary operation. The third part is devoted to the study of extensions of two classes of algebras, possessing two algebraic operations, called dialgebras...
In this paper we study the geometric moments invariants. We describe an
image in terms of features which are invariant to some sort of transformations i.e mentioned
translation, rotation and scaling change in exposure, brightness etc. Our aim is to check the
performance of components for feature vectors.
In the paper we describe all (not necessarily commutative) Jordan algebra structures on a two-dimensional vector space over R in terms of their matrices of structure constants.
The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.
This paper is devoted to the study of structure of Leibniz algebras. The classical methods to obtain classifications are essentially to solve a system of equations given by the identities of specified classes of algebras. In this work, some invariants were employed to obtain a complete list of three-dimensional Leibniz algebras over an arbitrary fi...
In the paper, the face recognition problem is analyzed. Chebyshev and Krawtchouk polynomial features are examined for the task. It is shown that proper image preprocessing, a specific polynomial features extraction and matching metrics can be better classification outcomes. Experimental results obtained on AT and T faces database are described.
In the paper the feature extraction capability of Krawtchouk and Chebyshev polynomials are tested using Kylberg texture set. The original feature extraction technique based on the combination of either Chebyshev or Krawtchouk moments and low order statis- tical moments is proposed. Among possible configurations of statistical moments, the most suit...
In this paper we describe all, up to isomorphism, left unital, right unital and unital algebra structures on two-dimensional vector space over any algebraically closed field and $\mathbb{R}$. We tabulate the algebras with the units.
In this paper, we introduce the concept of inner derivations of finite dimensional associative algebras and study their properties. The main objective of this study is to propose an algorithm to describe the inner derivations of any n-dimensional associative algebras in a matrix form. In addition, we apply the algorithm to two- and three-dimensiona...
A classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over any algebraically closed field is presented in terms of their matrices of structure constants.
In the paper we give a complete classification of $2$-dimensional evolution algebras over algebraically closed fields, describe their groups of automorphisms and derivation algebras.
In this paper, we extend some properties of algebraic semigroups from the algebraic setting to the framework of the bornological sets. More specially, the concept of bornological semigroup (BSG) is introduced and some constructions in the class of bornological semigroups are discussed.
In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basi...
A classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over any alge-braically closed field is presented in terms of their matrices of structure constants
This paper is dedicated to the study extensions of Leibniz algebras using the annihilator approach. The extensions methods have been used earlier to classify certain classes of algebras. In the paper we first review and adjust theoretical background of the method for Leibniz algebras then apply it to classify four-dimensional Leibniz algebras over...
Cohomology groups have a broad range of applications in algebraic and geometric classification problems of associative algebras. We focus on the second order cohomology groups of associative algebras. The main objectives of this study are precisely formulated two algorithms for describing the 2-cocycles and 2-coboundaries of an n-dimensional associ...
In this paper we introduce the concept of centroid and derivation of Leibniz algebras. By using the classification results of Leibniz algebras obtained earlier, we describe the centroids and derivations of low-dimensional Leibniz algebras. We also study some properties of centroids of Leibniz algebras and use these properties to categorize the alge...
In this talk we propose a graphical representation of some classes of Leibniz algebras. With each of algebra from these classes we associate a graph. This assignment enables us to reformulate some structural properties of the Leibniz algebras in terms of some conditions for the graphs. The talk focuses on a so-called filiform Leibniz algebras. It i...
The automorphisms groups and derivation algebras of all two-dimensional algebras over algebraically closed fields are described.
In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are di...
Our main focus in this work is to generalize the theory of algebraic semi rings from the algebraic setting to the framework of bornological sets. More specifically, the concept of a new structure bornological semi ring (BSR) is introduced and some constructions in the class of bornological semi rings are discussed. In particular, the existence of a...
This book brings together the main ideas of introductory algebra. They grew out of lectures given at the university to junior students of mathematics and is designed for a one or two- semester course in algebra and presumes some prior knowledge of mathematics at the pre-university level. The authors goal is to present the subject of algebra from i...
The paper is devoted to applications of some computer programs to study structural determination of Loday algebras. We present how these computer programs can be applied in computations of various invariants of Loday algebras and provide several computer programs in Maple to verify Loday algebras’ identities, the isomorphisms between the algebras,...
In this paper, we calculate cohomology groups of low-dimensional complex associative algebras. The calculations are based on a classification result and description of derivations of low-dimensional associative algebras obtained earlier. For the first cohomology group, we give basic cocycles up to inner derivations. We also provide basic coboundari...
The first goal of the present paper is to provide an introduction to the concept of semi bounded set and it is variations in bornological sets (BS). We study semi boundedness properties and keep track behaviour of semi boundedness under some set operations. Then, the concept of bornological ideal (BI) is given. It is provoked by intention to give t...
A complete classification of two-dimensional algebras over algebraically closed fields is provided.
A complete classification of two-dimensional algebras over algebraically closed fields is provided
In this paper, we focus on derivations and centroids of four dimensional associative algebras. Using an existing classification result of low dimensional associative algebras, we describe the derivations and centroids of four dimensional associative algebras. We also identify algebra(s) that belong to the characteristically nilpotent class among th...
The main purpose of the study is to propose an algebraic method to obtain the set of all independence models of I × J two-way contingency tables with the same row sums and column sums which is called fiber in algebraic statistics. This method involves solving a system of linear algebraic equations that only rely on row sums and column sums of the I...
Contraction is one of the most important concepts that play an important role from the mathematical and physical point of view. In this work, the contractions of complex associative algebras are considered. We focus on the variety A2(ℂ) that consisting of all associative algebras of dimension two over the complex numbers ℂ (including nonunital). Va...
This paper gives a graphical representation of a subclass of complex filiform Leibniz algebras. This class is split into three subclasses called first, second and third class denoted, in dimension n over a field of complex numbers ℂ, by FLbn(ℂ), SLbn(ℂ) and TLbn(ℂ), respectively. Here, the combinatorial structures associated with FLbn(ℂ) and SLbn(ℂ...
Let M be a 2-torsion free δ-prime T-ring satisfying the condition abc = abc for all a, b, c ϵ M and, ϵ T, I a δ-prime ideal of M and d a semiderivation associated with a function g which is surjective on I. In the paper we show some conditions on d, such that d = 0 or M is commutative.
Contraction is one of the most important concepts that motivated by numerous applications in different fields of physics and mathematics. In this work, the contractions of complex associative algebras are considered. We focus on the variety A3() of all complex associative algebras of dimension three (including nonunital). Various contractions crite...
Contraction is one of the most important concepts that play an important role from the mathematical and physical point
of view. In this work, the contractions of complex associative algebras are considered. We focus on the variety A2(C) that consisting
of all associative algebras of dimension two over the complex numbers C (including nonunital). Va...
. In this paper, we calculate cohomology groups of low-dimensional complex associative algebras. The calculations are
based on a classification result and description of derivations of low-dimensional associative algebras obtained earlier. For the first
cohomology group, we give basic cocycles up to inner derivations. We also provide basic cobounda...
In this research we introduce a generalized derivations of diassociative algebras and study its properties. This generalization depends on some parameters. In this paper we specify all possible values of the parameters.
We also provide all the generalized derivations of low-dimensional complex diassociative algebras.
In this paper, we focus on derivations and centroids of four dimensional associative algebras. Using an existing classification result of low dimensional associative algebras, we describe the derivations and centroids of four dimensional associative algebras. We also identify algebra(s) that belong to the characteristically nilpotent class among th...
Our main focus in this work is to introduce new structure bornological semi rings. This generalizes the theory of algebraic semi rings from the algebraic setting to the framework of bornological sets. We give basic properties for this new structure. As well as, We study the fundamental construction of bornological semi ring as product, inductive li...
The main objective of this work is to study Markov bases and toric ideals for
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- contingency tables that has fixed two-dimensional marginal when p is a multiple of v and greater than or equal to 2v. Moreover, the connected bipartite graph is also constructed by using elements of Markov basis. This work is an...
The concept of central extensions plays an important in constructing extensions of algebras. This technique has been successfully used in the classification problem of certain classes of algebras. In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of automorph...
In this paper we study centroids and derivations of associative algebras. Using a classification result of associative algebras, we describe the centroids and derivations of low-dimensional associative algebras. We review some properties of the centroids in the light of associative algebras and use these properties to categorize the algebras into h...
We introduce the concept of centroid for associative dialgebras and study its some properties. An algorithm to find centroids of algebras is given. We apply the algorithm to determine the centroids of lowdimensional dialgebras.
Smoking problem is considered as one of the hot topic for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different...
In this paper we introduce and study the concept of a bornological semigroup. This generalizes the theory of algebraic semigroup from the algebraic setting to the framework of bornological set. Working with bornological set allows to extend the scope of the latter theory considerably. In this paper we develop and introduce the concept of bornologic...
The paper is devoted to structural properties of diassociative algebras. We introduce the notions of nilpotency,
solvability of the diassociative algebras and study their properties. The list of all possible nilpotent diassociative
algebra structures on four-dimensional complex vector spaces is given
In the paper we describe derivations of some classes of Leibniz algebras. It
is shown that any derivation of a simple Leibniz algebra can be written as a
combination of three derivations. Two of these ingredients are a Lie algebra
derivations and the third one can be explicitly described. Then we show that
the similar description can found as well...
A pure algebraic approach to differential invariants of curves and surfaces is presented. By the use of this approach the explicit formulae for the generating differential invariants and invariant differential operators for the generic curves in n-dimensional classical geometries are constructed.
We prove that every $2$-local
derivation on a finite-dimensional semi-simple Lie algebra
$\mathcal{L}$ over an algebraically closed field of characteristic
zero is a derivation. We also show that a finite-dimensional
nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$
admits a $2$-local derivation which is not a derivation.
Generalized derivation is an extension of the classical derivation and having a broad range of applications in many different scientific and engineering disciplines. Generalized derivation is a highly useful tool in the modelling of many sorts of scientific phenomena such as image processing, earthquake engineering, biomedical engineering and physi...
The paper concerns the derivations of diassociative algebras. We introduce one important class of diassociative algebras, give simple properties of the right and left multiplication operators in diassociative algebras. Then we describe the derivations of complex diassociative algebras in dimension two and three.