• Journal of the Korean Mathematical Society
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• v.43 no.2
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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 $JC^*-algebras$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.1
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• pp.49-57
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• 2010

It is shown that for a derivation $$f(x_1{\cdots}x_{j-1}x_jx_{j+1}{\cdots}x_k)=\sum_{j=1}^{k}x_{1}{\cdots}x_{j-1}x_{j+1}{\cdots}x_kf(x_j)$$ on a unital $C^*$-algebra $\mathcal{B}$, there exists a unique $\mathbb{C}$-linear *-derivation $D:{\mathcal{B}}{\rightarrow}{\mathcal{B}}$ near the derivation, by using the Hyers-Ulam-Rassias stability of functional equations. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.1-12
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• 2018

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that $D^2(x)[D(x),x]=0$ or $[D(x),x]D^2(x)=0$ for all $x{\in}R$. In this case we have $f(x)^5=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D^2(x)[D(x),x]{\in}rad(A)$ or $[D(x),x]D^2(x){\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Economic and Environmental Geology
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• v.30 no.4
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• pp.289-301
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• 1997

The Boguk cobalt mine is located within the Cretaceous Gyeongsang Sedimentary Basin. Major ore minerals including cobalt-bearing minerals (loellingite, cobaltite, and glaucodot) and Co-bearing arsenopyrite occur together with base-metal sulfides (pyrrhotite, chalcopyrite, pyrite, sphalerite, etc.) and minor amounts of oxides (magnetite and hematite) within fracture-filling $quartz{\pm}actinolite{\pm}carbonate$ veins. These veins are developed within an epicrustal micrographic granite stock which intrudes the Konchonri Formation (mainly of shale). Radiometric date of the granite (85.98 Ma) indicates a Late Cretaceous age for granite emplacement and associated cobalt mineralization. The vein mineralogy is relatively complex and changes with time: cobalt-bearing minerals with actinolite, carbonates, and quartz gangues (stages I and II) ${\rightarrow}$ base-metal sulfides, gold, and Fe oxides with quartz gangues (stage III) ${\rightarrow}$ barren carbonates (stages IV and V). The common occurrence of high-temperature minerals (cobalt-bearing minerals, molybdenite and actinolite) with low-temperature minerals (base-metal sulfides, gold and carbonates) in veins indicates a xenothermal condition of the hydrothermal mineralization. High enrichment of Co in the granite (avg. 50.90 ppm) indicates the magmatic hydrothermal derivation of cobalt from this cooling granite stock, whereas higher amounts of Cu and Zn in the Konchonri Formation shale suggest their derivations largely from shale. The decrease in temperature of hydrothermal fluids with a concomitant increase in fugacity of oxygen with time (for cobalt deposition in stages I and II, $T=560^{\circ}C-390^{\circ}C$ and log $fO_2=$ >-32.7 to -30.7 atm at $350^{\circ}C$; for base-metal sulfide deposition in stage III, $T=380^{\circ}-345^{\circ}C$ and log $fO_2={\geq}-30.7$ atm at $350^{\circ}C$) indicates a transition of the hydrothermal system from a magmatic-water domination toward a less-evolved meteoric-water domination. Sulfur isotope data of stage II sulfide minerals evidence that early, Co-bearing hydrothermal fluids derived originally from an igneous source with a ${\delta}^{34}S_{{\Sigma}S}$ value near 3 to 5‰. The remarkable increase in ${\delta}^{34}S_{H2S}$ values of hydrothermal fluids with time from cobalt deposition in stage II (3-5‰) to base-metal sulfide deposition in stage III (up to about 20‰) also indicates the change of the hydrothermal system toward the meteoric water domination, which resulted in the leaching-out and concentration of isotopically heavier sulfur (sedimentary sulfates), base metals (Cu, Zn, etc.) and gold from surrounding sedimentary rocks during the huge, meteoric water circulation. We suggest that without the formation of the later, meteoric water circulation extensively through surrounding sedimentary rocks the Boguk cobalt deposits would be simple veins only with actinolite + quartz + cobalt-bearing minerals. Furthermore, the formation of the meteoric water circulation after the culmination of a magmatic hydrothermal system resulted in the common occurrence of high-temperature minerals with later, lower-temperature minerals, resulting in a xenothermal feature of the mineralization.