• Journal of applied mathematics & informatics
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• 제39권1_2호
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• pp.73-81
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• 2021

Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If �� be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that (��(f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).

• 대한수학회논문집
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• 제39권3호
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• pp.585-593
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• 2024

Let 𝒦 be a ring. An additive map 𝖚^⋄ → 𝖚 is called Jordan involution on 𝒦 if (𝖚^⋄)^⋄ = 𝖚 and (𝖚𝖛+𝖛𝖚)^⋄ = 𝖚^⋄𝖛^⋄+𝖛^⋄𝖚^⋄ for all 𝖚, 𝖛 ∈ 𝒦. If Θ is a (non-zero) 𝜂-generalized derivation on 𝒦 associated with a derivation Ω on 𝒦, then it is shown that Θ(𝖚) = 𝛄𝖚 for all u ∈ 𝒦 such that 𝛄 ∈ Ξ and 𝛄² = 1, whenever Θ possesses [Θ(𝖚), Θ(𝖚^⋄)] = [𝖚, 𝖚^⋄] for all 𝖚 ∈ 𝒦.

• 대한수학회논문집
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• 제33권4호
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• pp.1123-1123
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• 2018

• Kyungpook Mathematical Journal
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• 제53권4호
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• pp.565-571
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• 2013

In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) $F(x){\circ}D(y)=[x,y]$, (ii) $D(x){\circ}F(y)=F[x,y]$, (iii) $D(x){\circ}F(y)=xy$, (iv) $F(x{\circ}y)=[F(x) y]+[D(y),x]$, and (v) $F[x,y]=F(x){\circ}y-D(y){\circ}x$ for all x, y in some appropriate subsets of R.

• 대한수학회논문집
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• 제34권1호
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• pp.279-286
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• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.1-11
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• 2019

Let R be a prime ring with involution of the second kind and centre Z(R). Suppose R admits a generalized derivation $F:R{\rightarrow}R$ associated with a derivation $d:R{\rightarrow}R$. The purpose of this paper is to study the commutativity of a prime ring R satisfying any one of the following identities: (i) $F(x){\circ}x^*{\in}Z(R)$ (ii) $F([x,x^*]){\pm}x{\circ}x^*{\in}Z(R)$ (iii) $F(x{\circ}x^*){\pm}[x,x^*]{\in}Z(R)$ (iv) $F(x){\circ}d(x^*){\pm}x{\circ}x^*{\in}Z(R)$ (v) $[F(x),d(x^*)]{\pm}x{\circ}x^*{\in}Z(R)$ (vi) $F(x){\pm}x{\circ}x^*{\in}Z(R)$ (vii) $F(x){\pm}[x,x^*]{\in}Z(R)$ (viii) $[F(x),x^*]{\mp}F(x){\circ}x^*{\in}Z(R)$ (ix) $F(x{\circ}x^*){\in}Z(R)$ for all $x{\in}R$.

• 대한수학회논문집
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• 제31권1호
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• pp.101-113
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• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• 대한수학회보
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• 제42권4호
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• pp.679-690
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• 2005

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, f($x_1,{\cdots},\;x_n$) a multilinear polynomial in n non-commuting variables over K, a $\in$ R. Supppose that, for any $x_1,{\cdots},\;x_n\;\in\;I,\;a[d(f(x_1,{\cdots},\;x_n)),\;f(x_1,{\cdots},\;x_n)]$ = 0. If $[f(x_1,{\cdots},\;x_n),\;x_{n+1}]x_{n+2}$ is not an identity for I and $$S_4(I,\;I,\;I,\;I)\;I\;\neq\;0$$, then aI = ad(I) = 0.

• East Asian mathematical journal
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• 제18권1호
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• pp.111-125
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• 2002

Many different families of infinite series were recently observed to be summable in closed forms by means of certain operators of fractional calculus(that is, calculus of integrals and derivatives of any arbitrary real or complex order). In this sequel to some of these recent investigations, the authors present yet another instance of applications of certain fractional calculus operators. Alternative derivations without using these fractional calculus operators are shown to lead naturally a family of analogous infinite sums involving hypergeometric functions.

• Structural Engineering and Mechanics
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• 제73권5호
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• pp.565-584
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• 2020

The nonlinear thermo-electro-elastic buckling behavior of viscoelastic nanoplates under magnetic field is investigated based on nonlocal elasticity theory. Employing nonlinear strain-displacement relations, the geometrical nonlinearity is modeled while governing equations are derived through Hamilton's principle and they are solved applying semi-analytical generalized differential quadrature (GDQ) method. Eringen's nonlocal elasticity theory considers the effect of small size, which enables the present model to become effective in the analysis and design of nano-sensors and nano actuators. Based on Kelvin-Voigt model, the influence of the viscoelastic coefficient is also discussed. It is demonstrated that the GDQ method has high precision and computational efficiency in the buckling analysis of viscoelastic nanoplates. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as electric voltage, small scale effects, elastomeric medium, magnetic field, temperature effects, the viscidity and aspect ratio of the nanoplate on its nonlinear buckling characteristics. It is explicitly shown that the thermo-electro-elastic nonlinear buckling behavior of viscoelastic nanoplates is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of viscoelastic nanoplates as fundamental elements in nanoelectromechanical systems.