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Articles 1 - 27 of 27
Full-Text Articles in Physical Sciences and Mathematics
Involutive Automorphisms And Derivations Of The Quaternions, Eyüp Kizil, Adriano Da Silva, Okan Duman
Involutive Automorphisms And Derivations Of The Quaternions, Eyüp Kizil, Adriano Da Silva, Okan Duman
Turkish Journal of Mathematics
Let $Q=(\frac{a,b}{{\Bbb R}})$ denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra $H$ of Hamilton's quaternions. We have presented explicitly in \cite{Kizil-Alagoz} the matrix of a typical derivation of $Q$. Given a derivation $d\in Der(H)$, we show that the matrix $D\in M_{3}({\Bbb R})$ that represents $d$ on the linear subspace $% H_{0}\simeq {\Bbb R}^{3}$ of pure quaternions provides a pair of idempotent matrices $AdjD$ and $-D^{2}$ that correspond bijectively to the involutary matrix $\Sigma $ of a quaternion involution $\sigma $ and present several equations involving these matrices. In particular, …
Local And 2-Local Derivations On Small Dimensional Zinbiel Algebras, Bakhtiyor Yusupov, Sabohat Rozimova
Local And 2-Local Derivations On Small Dimensional Zinbiel Algebras, Bakhtiyor Yusupov, Sabohat Rozimova
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
In the present paper we investigate local and 2-local derivations on small dimensional Zinbiel algebras. We give a description of derivations and local derivations on all three and four-dimensional Zinbiel algebras. Moreover, similar problem concerning 2-local derivations on all three and four-dimensional Zinbiel algebras are investigated.
Jordan Maps And Zero Lie Product Determined Algebras, Matej Bresar
Jordan Maps And Zero Lie Product Determined Algebras, Matej Bresar
Turkish Journal of Mathematics
Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $\Phi(x^2,x)=0 $ for all $x\in A$ implies that $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ for all $x,y,z\in A$. This is applicable to the question of whether $A$ is zero Lie product determined and is also used in proving that a Jordan homomorphism from $A$ onto a semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.
Flow Of Quantum Genetic Lotka-Volterra Algebras On M2(ℂ), Sondos Muhammed Syam
Flow Of Quantum Genetic Lotka-Volterra Algebras On M2(ℂ), Sondos Muhammed Syam
Theses
In this thesis, a class of flow quantum Lotka-Volterra genetic algebras (FQLVG-A) is investigated and its structure is studied. Moreover, the necessary and sufficient conditions for the associativity and alternatively of FQGLV-A are derived. In addition, idempotent elements in FQGLV-A are found. Also, derivations of a class of FQLVG-A are described. Also, the automorphisms of a class of FQLVG-A and their positivity are examined.
Local And 2-Local Derivations On Octonion Algebras, Allayar Allambergenov
Local And 2-Local Derivations On Octonion Algebras, Allayar Allambergenov
Karakalpak Scientific Journal
The present paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra \[{\mathbb{O}_\mathbb{R}}.\] This description implies that the space of all local derivations on \[{\mathbb{O}_\mathbb{R}}\] when equipped with Lie bracket is isomorphic to the Lie algebra \[\mathfrak{s}{\mathfrak{o}_7}(\mathbb{R})\] of all real skew-symmetric \[7 \times 7\]-matrices. We also consider \[2\]-local derivations on an octonion algebra \[{\mathbb{O}_\mathbb{F}}\] over an algebraically closed field \[\mathbb{F}\] of the characteristic zero and prove that every \[2\]-local derivation on \[{\mathbb{O}_\mathbb{F}}\] is a derivation. Further, we apply these results to similar problems …
Local And 2-Local Derivations Of Solvable Leibniz Algebras With Null-Filiform Nilradical, Sardorbek Umrzaqov
Local And 2-Local Derivations Of Solvable Leibniz Algebras With Null-Filiform Nilradical, Sardorbek Umrzaqov
Scientific Bulletin. Physical and Mathematical Research
In the works of Ayupov, Khudoyberdiyev and Yusupov proved that the local and 2-local derivation of solvable Leibniz algebras with model nilradical are derivations. Solvable Leibniz algebras with null-filiform nilradical are the partial case of solvable Leibniz algebras with model nilradical. However, the proof in the paper is different from model nilradical case. The derivation is a fundamental notion in mathematics. Derivations play a prominent role in algebra. There are many generalizations of derivations as antiderivation, δ-derivations, ternary derivations and (α,β,γ)-derivations. One of the important generalizations of derivation is local and 2-local derivations. Local derivations defined by Kadison, Larson and …
On Symmetric Higher (U,R)-N-Derivation Of Prime Rings, Anwar Khaleel Faraj, Marwa Hadi Sapur
On Symmetric Higher (U,R)-N-Derivation Of Prime Rings, Anwar Khaleel Faraj, Marwa Hadi Sapur
Al-Qadisiyah Journal of Pure Science
The main aim of this paper is to define the notions of Symmetric higher (U,R)-n-derivation, (U,R) n-derivation, Jordan(U,R)-n-derivation and higher n-derivation of prime ring to generalize Awtar’s theorem of derivation on Lie ideal of prime ring to symmetric higher(U,R)-n-derivation.
Local And 2-Local Derivation On Solvable Leibniz Algebras Whose Nilradical Is A Quasi-Filiform Leibniz Algebra Of Maximum Length, Shavkat Ayupov, Bakhtiyor Yusupov
Local And 2-Local Derivation On Solvable Leibniz Algebras Whose Nilradical Is A Quasi-Filiform Leibniz Algebra Of Maximum Length, Shavkat Ayupov, Bakhtiyor Yusupov
Karakalpak Scientific Journal
We show that any local derivation on the solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length with the maximal dimension of complementary space to the nilradical is a derivation. Moreover, a similar problem concerning 2-local derivations of such algebras is investigated.
Dual Quaternion Algebra And Its Derivations, Eyüp Kizil, Yasemi̇n Alagöz
Dual Quaternion Algebra And Its Derivations, Eyüp Kizil, Yasemi̇n Alagöz
Turkish Journal of Mathematics
It is well known that the automorphism group $Aut(H)$ of the algebra of real quaternions $H$ consists entirely of inner automorphisms $i_{q}:p\rightarrow q\cdot p\cdot q^{-1}$ for invertible $q\in H$ and is isomorphic to the group of rotations $SO(3)$. Hence, $H$ has only inner derivations $D=ad(x),$ $x\in H$. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra $H_{d}$ of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual …
Linear Mappings Satisfying Some Recursive Sequences, Amin Hosseini, Mehdi Mohammadzadeh Karizaki
Linear Mappings Satisfying Some Recursive Sequences, Amin Hosseini, Mehdi Mohammadzadeh Karizaki
Turkish Journal of Mathematics
Let $\mathcal{A}$ be a unital, complex normed $\ast$-algebra with the identity element $\textbf{e}$ such that the set of all algebraic elements of $\mathcal{A}$ is norm dense in the set of all self-adjoint elements of $\mathcal{A}$ and let $\{D_n\}_{n = 0}^{\infty}$ and $\{\Delta_n\}_{n = 0}^{\infty}$ be sequences of continuous linear mappings on $\mathcal{A}$ satisfying \[ \left\lbrace \begin{array}{c l} D_{n + 1}(p) = \sum_{k = 0}^{n}D_{n - k}(p)D_k(p),\\ \\ \Delta_{n + 1}(p) = \sum_{k = 0}^{n}\Delta_{n - k}(p)D_k(p), \end{array} \right. \] for all projections $p$ of $\mathcal{A}$ and all nonnegative integers $n$. Moreover, suppose that $D_0(p) = D_0(p)^2$ holds for all projections …
2-Local Derivations On Virasoro Algebras, Shavkat Ayupov, Bakhtiyor Yusupov
2-Local Derivations On Virasoro Algebras, Shavkat Ayupov, Bakhtiyor Yusupov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
The present paper is devoted to study 2-local derivations on infinite-dimensional Lie algebras over a field of characteristic zero. We show that every derivation on Virasoro algebra is inner and prove that all 2-local derivations on this algebra is a derivation. We give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation.
Derivations Of Generalized Quaternion Algebra, Eyüp Kizil, Yasemi̇n Alagöz
Derivations Of Generalized Quaternion Algebra, Eyüp Kizil, Yasemi̇n Alagöz
Turkish Journal of Mathematics
The purpose of this paper is to determine derivations of the algebra $H_{\alpha ,\beta }$ of generalized quaternions over the reals and hence to obtain the algebra $Der(H_{\alpha ,\beta })$ of derivations of $H_{\alpha ,\beta }$. Once we know derivations we might decompose $Der(H_{\alpha ,\beta })$ in terms of its inner and/or central derivations whenever they exist. Apart from $Der(H_{\alpha ,\beta })$ we would also be able to obtain generalized derivations, which have been studied by analysts in the context of algebras of some normed spaces, and of prime and semiprime rings.
An Introduction To Lie Algebra, Amanda Renee Talley
An Introduction To Lie Algebra, Amanda Renee Talley
Electronic Theses, Projects, and Dissertations
An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency, …
Generalized $\Ast$-Lie Ideal Of $\Ast$-Prime Ring, Seli̇n Türkmen, Neşet Aydin
Generalized $\Ast$-Lie Ideal Of $\Ast$-Prime Ring, Seli̇n Türkmen, Neşet Aydin
Turkish Journal of Mathematics
Let $R$ be a $\ast$-prime ring with characteristic not $2,$ $\sigma, \tau:R\rightarrow R$ be two automorphisms, $U$ be a nonzero $\ast$-$\left( \sigma,\tau\right) $-Lie ideal of $R$ such that $\tau~$commutes with $\ast$, and $a,b$ be in $R.$ $\left( i\right) $ If $a\in S_{\ast}\left( R\right) $ and $\left[ U,a\right] =0$, then $a\in Z\left( R\right) $ or $U\subset Z\left( R\right) .$ $\left( ii\right) $ If $a\in S_{\ast}\left( R\right) $ and $\left[ U,a\right] _{\sigma,\tau}\subset$ $C_{\sigma,\tau}$, then $a\in Z\left( R\right) ~$or$~U\subset Z\left( R\right) .$ $\left( iii\right) $ If $U\not \subset Z\left( R\right) $ and $U\not \subset C_{\sigma,\tau}$, then there exists a nonzero $\ast$-ideal $M$ of …
On $*$-Commuting Mappings And Derivations In Rings With Involution, Nadeem Ahmad Dar, Shakir Ali
On $*$-Commuting Mappings And Derivations In Rings With Involution, Nadeem Ahmad Dar, Shakir Ali
Turkish Journal of Mathematics
Let $R$ be a ring with involution $*$. A mapping $f:R\rightarrow R$ is said to be $*$-commuting on $R$ if $[f(x),x^*]=0$ holds for all $x\in R$. The purpose of this paper is to describe the structure of a pair of additive mappings that are $*$-commuting on a semiprime ring with involution. Furthermore, we study the commutativity of prime rings with involution satisfying any one of the following conditions: (i) $[d(x),d(x^*)]=0,$ (ii) $d(x)\circ d(x^*)=0$, (iii) $d([x,x^*])\pm [x,x^*]=0$ (iv) $d(x\circ x^*)\pm (x\circ x^*)=0,$ (v) $d([x,x^*])\pm (x\circ x^*)=0$, (vi) $d(x\circ x^*)\pm [x,x^*]=0$, where $d$ is a nonzero derivation of $R$. Finally, an example …
A Characterization Of Derivations On Uniformly Mean Value Banach Algebras, Amin Hosseini
A Characterization Of Derivations On Uniformly Mean Value Banach Algebras, Amin Hosseini
Turkish Journal of Mathematics
In this paper, a uniformly mean value Banach algebra (briefly UMV-Banach algebra) is defined as a new class of Banach algebras, and we characterize derivations on this class of Banach algebras. Indeed, it is proved that if $\mathcal{A}$ is a unital UMV-Banach algebra such that either $a = 0$ or $b = 0$ whenever $ab = 0$ in $\mathcal{A}$, and if $\delta:\mathcal{A} \rightarrow \mathcal{A}$ is a derivation such that $a \delta(a) = \delta(a)a$ for all $a \in \mathcal{A}$, then the following assertions are equivalent:\\ (i) $\delta$ is continuous; \\(ii) $\delta(e^a) = e^a\delta(a)$ for all $a \in \mathcal{A}$; \\(iii) $\delta$ is …
Generalized Derivations Of Prime Rings On Multilinear Polynomials With Annihilator Conditions, Nurcan Argaç, Çağri Demi̇r
Generalized Derivations Of Prime Rings On Multilinear Polynomials With Annihilator Conditions, Nurcan Argaç, Çağri Demi̇r
Turkish Journal of Mathematics
Let K be a commutative ring with unity, R be a prime K-algebra with characteristic not 2, U be the right Utumi quotient ring of R, C the extended centroid of R, I a nonzero right ideal of R and a a fixed element of R. Let g be a generalized derivation of R and f(X_1,..., X_n) a multilinear polynomial over K. If ag(f(x_1,...,x_n))f(x_1,...,x_n)=0 for all x_1,...,x_n \in I, then one of the following holds: (1) aI=ag(I)=0; (2) g(x)=bx+[c,x] for all x\in R, where b,c\in U. In this case either [c,I]I=0=abI or aI=0=a(b+c)I; (3) [f(X_1,...,X_n),X_{n+1}]X_{n+2} is an identity for I.
On Generalized (\Alpha,\Beta)-Derivations Of Semiprime Rings, Faisal Ali, Muhammad Anwar Chaudhry
On Generalized (\Alpha,\Beta)-Derivations Of Semiprime Rings, Faisal Ali, Muhammad Anwar Chaudhry
Turkish Journal of Mathematics
We investigate some properties of generalized (\alpha,\beta)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (\alpha,\beta)-derivation, with associated (\alpha,\beta)-derivation \delta, on a semiprime ring R such that [g(x),\alpha(x)]=0 for all x\in R, then \delta(x)[y,z]=0 for all x,y,z\in R and \delta is central. We also show that if \alpha,\nu,\tau are endomorphisms and \beta,\mu are automorphisms of a semiprime ring R and if R has a generalized (\alpha,\beta)-derivation g, with associated (\alpha,\beta)-derivation \delta, such that g([\mu(x),w(y)])=[\nu(x),w(y)]_{\alpha,\tau}, where w:R\rightarrow R is commutativity preserving, then [y,z]\delta(w(p))=0 for all y,z,p\in R.
Whitehead Products In Function Spaces: Quillen Model Formulae, Gregory Lupton, Samuel Bruce Smith
Whitehead Products In Function Spaces: Quillen Model Formulae, Gregory Lupton, Samuel Bruce Smith
Mathematics and Statistics Faculty Publications
We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f : X→Y, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components.
Note On Generalized Jordan Derivations Associate With Hochschild 2-Cocycles Of Rings, Atsushi Nakajima
Note On Generalized Jordan Derivations Associate With Hochschild 2-Cocycles Of Rings, Atsushi Nakajima
Turkish Journal of Mathematics
We introduce a new type of generalized derivations associate with Hochschild 2-cocycles and prove that every generalized Jordan derivation of this type is a generalized derivation under certain conditions. This result contains the results of I. N. Herstein [6, Theorem 3.1] and M. Ashraf and N-U. Rehman [1, Theorem].
On Orthogonal Generalized Derivations Of Semiprime Rings, Nurcan Argaç, Atsushi Nakajima, Emi̇ne Albaş
On Orthogonal Generalized Derivations Of Semiprime Rings, Nurcan Argaç, Atsushi Nakajima, Emi̇ne Albaş
Turkish Journal of Mathematics
In this paper, we present some results concerning two generalized derivations on a semiprime ring. These results are a generalization of results of M. Bre\u{s}ar and J. Vukman in [2], which are related to a theorem of E. Posner for the product of derivations on a prime ring.
On Derivations Of Prime Gamma Rings, Mehmet Ali̇ Öztürk, Young Bae Jun, Kyung Ho Kim
On Derivations Of Prime Gamma Rings, Mehmet Ali̇ Öztürk, Young Bae Jun, Kyung Ho Kim
Turkish Journal of Mathematics
We consider some results in a \Gamma-ring M with derivation which is related to Q, and the quotient \Gamma-ring of M.
Some Results On Derivation Groups, Murat Alp
Some Results On Derivation Groups, Murat Alp
Turkish Journal of Mathematics
In this paper we describe a share package XMOD of functions for computing with finite, permutation crossed modules, their morphisms and derivations; cat$^1$-groups, their morphisms and their sections, written using the GAP \cite{GAP} group theory programming language. We also give some mathematical results for derivations. These results are suggested by the output produced by the XMOD package.
Lifts Of Derivations To The Semitangent Bundle, Ari̇f A. Salimov, Ekrem Kadioğlu
Lifts Of Derivations To The Semitangent Bundle, Ari̇f A. Salimov, Ekrem Kadioğlu
Turkish Journal of Mathematics
The main purpose of this paper is to investigate the complete lifts of derivations for semitangent bundle and to discuss relations between these and lifts already known.
The K-Derivation Of A Gamma-Ring, Hati̇ce Kandamar
The K-Derivation Of A Gamma-Ring, Hati̇ce Kandamar
Turkish Journal of Mathematics
In this paper, the $k$-derivation is defined on a $\Gamma$-ring $M$ (that is, if $M$ is a $\Gamma$-ring, $d:M\to M$ and $k:\Gamma\to \Gamma$ are to additive maps such that $d(a\beta b )= d(a)\beta b + ak(\beta)b + a\beta d(b) $ for all $a,b\in M, \quad \beta \in \Gamma$, then $d$ is called a $k$-derivation of $M$) and the following results are proved. (1) Let $R$ be a ring of characteristic not equal to 2 such that if $xry=0$ for all $x, y\in R$ then $r=0$. If $d$ is a $k$-derivation of the $(R=)\Gamma$-ring $R$ with $k=d$, then $d$ is the …
Anticommuting Derivations, Steen Pedersen
Anticommuting Derivations, Steen Pedersen
Mathematics and Statistics Faculty Publications
We show that the re are no non-trivial closable derivations of a C*-algebra anticommuting with an ergodic action of a compact group, supposing that the set of squares is dense in the group. We also show that the re are no non-trivial closable densely defined rank one derivations on any C*-algebra.
On The Product Of Two Generalized Derivations, Mohamed Barraa, Steen Pedersen
On The Product Of Two Generalized Derivations, Mohamed Barraa, Steen Pedersen
Mathematics and Statistics Faculty Publications
Two elements A and B in a ring R determine a generalized derivation deltaA,B on R by setting δA,B(X) = AX - XA for any X in R. We characterize when the product δC,DδA,B is a generalized derivation in the cases when the ring R is the algebra of all bounded operators on a Banach space epsilon, and when R is a C*-algebra U. We use the se characterizations to compute the commutant of the range of δA,B.