• 대한수학회논문집
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• 제34권1호
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• pp.83-93
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• 2019

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R and f($x_1,{\ldots},x_n$) a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all the evaluations of f($x_1,{\ldots},x_n$) on R. If d is a nonzero derivation of R and G a nonzero generalized derivation of R such that $$d(G(u)u){\in}Z(R)$$ for all $u{\in}f(R)$, then $f(x_1,{\ldots},x_n)^2$ is central-valued on R and there exists $b{\in}U$ such that G(x) = bx for all $x{\in}R$ with $d(b){\in}C$. As an application of this result, we investigate the commutator $[F(u)u,G(v)v]{\in}Z(R)$ for all $u,v{\in}f(R)$, where F and G are two nonzero generalized derivations of R.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권1호
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• pp.99-106
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• 2009

In this paper, we improve the generalized Hyers-Ulam stability and the superstability of module left derivations due to the results of [7].

• 대한수학회보
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• 제61권2호
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• pp.335-353
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• 2024

The objective of this paper is to study some central identities involving generalized derivations and anti-automorphisms in prime rings. Using the tools of the theory of functional identities, several known results have been generalized as well as improved.

• 대한수학회논문집
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• 제32권4호
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• pp.827-833
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• 2017

In this paper, we define a set including of all $f_a$ with $a{\in}R$ generalized derivations of R and is denoted by $f_R$. It is proved that (i) the mapping $g:L(R){\rightarrow}f_R$ given by g (a) = f-a for all $a{\in}R$ is a Lie epimorphism with kernel $N_{{\sigma},{\tau}}$ ; (ii) if R is a semiprime ring and ${\sigma}$ is an epimorphism of R, the mapping $h:f_R{\rightarrow}I(R)$ given by $h(f_a)=i_{{\sigma}(-a)}$ is a Lie epimorphism with kernel $l(f_R)$ ; (iii) if $f_R$ is a prime Lie ring and A, B are Lie ideals of R, then $[f_A,f_B]=(0)$ implies that either $f_A=(0)$ or $f_B=(0)$.

• Kyungpook Mathematical Journal
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• 제55권1호
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• pp.63-71
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• 2015

Let H be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on H into itself. Let $A,B{\in}\mathcal{L}(H)$ we define the generalized derivation ${\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)$ by ${\delta}_{A,B}(X)=AX-XB$, we note ${\delta}_{A,A}={\delta}_A$. If the inequality ${\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}$ holds for all $X{\in}\mathcal{L}(H)$ and for all $T{\in}ker{\delta}_A$, then we say that the range of ${\delta}_A$ is orthogonal to the kernel of ${\delta}_A$ in the sense of Birkhoff. The operator $A{\in}\mathcal{L}(H)$ is said to be finite [22] if ${\parallel}I-(AX-XA){\parallel}{\geq}1(*)$ for all $X{\in}\mathcal{L}(H)$, where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.

• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.887-902
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• 2023

In this article, we investigate an approximate quadratic Lie *-derivation of a quadratic functional equation f(ax + by) + abf(x - y) = (a + b)(af(x) + bf(y)), where ab ≠ 0, a, b ∈ ℕ, associated with the identity f([x, y]) = [f(x), y²] + [x², f(y)] on a 𝜌-complete convex modular *-algebra χ_𝜌 by using ∆₂-condition via convex modular 𝜌.

• 대한수학회지
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• 제41권3호
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• pp.461-477
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• 2004

It is shown that every almost linear mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping h : A\longrightarrowB of a unital $C^{*}$ -algebra A of real rank zero to a unital $C^{*}$ -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of *-homomorphisms between unital $C^{*}$ -algebras, and of C-linear *-derivations on unital $C^{*}$ -algebras./ -algebras.

• 대한수학회지
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• 제43권2호
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• pp.323-356
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• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• ETRI Journal
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• 제17권3호
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• pp.17-43
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• 1995

We present a comprehensive derivation of the transport of holes involving an interacting two-valence-band system in terms of a generalized relaxation time approach. We sole a pair of semiclassical Boltzmann equations in a general way first, and then employ the conventional relaxation time concept to simplify the results. For polar optical phonon scattering, we develop a simple method th compensate for the inherent deficiencies in the relaxation time concept and apply it to calculate effective relaxation times separately for each band. Also, formulas for scattering rates and momentum relaxation times for the two-band model are presented for all the major scattering mechanisms for p-type GaAs for simple, practical mobility calculations. Finally, in the newly proposed theoretical frame-work, first-principles calculations for the Hall mobility and Hall factor of p-type GaAs at room temperature are carried out with no adjustable parameters in order to obtain a direct comparison between the theory and recent available experimental results, which would stimulate further analysis toward better understanding of the complex transport properties of the valence band. The calculated Hall mobilities show a general agreement with our experimental data for carbon doped p-GaAs samples in a range of degenerate hole densities. The calculated Hall factors show $r_H$=1.25~1.75 over all hole densities($2{\times}10^{17}{\sim}1{\times}10^{20}cm^{-3}$ considered in the calculations.

• 한국해양공학회지
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• 제27권2호
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• pp.1-7
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• 2013

This paper aims at contributing to the field of ship design by introducing new systematic variation methods for ship hull forms. Hull form design is generally carried out in two stages. The first is the global variation considering the sectional area curve. Because the geometric properties of a sectional area curve have a decisive effect on the global hydrodynamic properties of ships, the design of a sectional area curve that satisfies various global design conditions, e.g., the displacement, longitudinal center of buoyancy, etc., is important in the initial hull form design stage. The second stage involves the local design of section forms. Section forms affect the local hydrodynamic properties, e.g., the local pressure in the fore- and aftbody. This paper deals with a new method for the systematic variation of sectional area curves. The longitudinal volume distribution of a ship depends on the sectional area curve, which can geometrically be controlled using parametric variation and a variation that uses the modification function. Based on these methods, we suggest a more generalized method in connection with the derivation of the lines for a new design compared to those for similar ships. This is the so-called the volumetric balanced variation (VOB) method for ship forms using a B-spline modification function and an optimization technique. In this paper the global geometric properties of hull forms are totally controlled by the form parameters. We describe the new method and some application examples in detail.