• Korean Journal of Mathematics
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• v.19 no.1
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• pp.17-24
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• 2011

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0, and $$2(_{2d-2}C_{d-1}-_{2d-2}C_d)f\({\sum_{j=1}^{2d}}x_j\)+{\sum_{{\iota}(j)=0,1,{{\small\sum}_{j=1}^{2d}}{\iota}(j)=d}}\;f\({\sum_{j=1}^{2d}}(-1)^{{\iota}(j)}x_j\)=2(_{2d-1}C_d+_{2d-2}C_{d-1}-_{2d-2}C_d){\sum_{j=1}^{2d}}f(x_j)$$ for all $x_1$, ${\cdots}$, $x_{2d}{\in}X$, then the even mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Hyers-Ulam stability of the above functional equation in Banach spaces.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.617-630
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• 2011

Let X, Y be vector spaces, and let r be 2 or 4. It is shown that if an odd mapping $f:X{\rightarrow}Y$ satisfies the functional equation $${\hspace{50}}rf(\frac{\sum_{j=1}^{d}\;x_j} {r})+\;{\sum\limits_{\iota(j)=0,1 \atop {\sum_{j=1}^{d}}\;{\iota}(j)=l}}\;rf(\frac{\sum_{j=1}^{d}{(-1)^{\iota(j)}x_j}}{r}) \\({\ddag}){\hspace{160}}=(_{d-1}C_l-_{d-1}C_{l-1}+1)\;{\sum\limits_{j=1}^{d}\;f(x_j)}$$ then the odd mapping $f:X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation in Banach modules over a unital $C^*$-algebra. As an application, we show that every almost linear bijection $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital $C^*$-algebra ${\mathcal{A}}$ onto a unital $C^*$-algebra ${\mathcal{B}}$ is a $C^*$-algebra isomorphism when $h(2^nuy)=h(2^nu)h(y)$ for all unitaries $u{\in}{\mathcal{A}}$, all $y{\in}{\mathcal{A}}$, and $n=0,1,2,{\cdots}$.

• Journal of Internet Computing and Services
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• v.15 no.6
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• pp.77-84
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• 2014

The purpose of this study is to explore factors affecting the effect of e-learning, and to find out the casual relationship among these factors. Subjects are 2,091 students who have participated in e-learning based class during the period of second semester in 2013. Those of them, 1,732 students response to the survey questions. After gathering data, they are analyzed by using Confirmative Factor Analysis and Structural Equation Modeling. From the result of Confirmative Factor analysis, data have reduced four factors, and are named as four latent variables likes e-learning effect, contents satisfaction, managing assistant factor, and system functional factor. From the result of Structural Equation Modeling, it is known as the relation and impact among factors: (a) "managing assistant factor" affects to "contents satisfaction" directly. (b) "contents satisfaction" affects to "e-learning effect" directly. (c) "system function factor" affects directly to "contents satisfaction", but does not affect directly to "e-learning effect". (d) both "managing assistant factor" and "system function factor" have an indirect effect on "e-learning effect" via "contents satisfaction".

• Journal of Microbiology and Biotechnology
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• v.26 no.8
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• pp.1428-1438
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• 2016

In the present work, Arthrobacter sp. 25, a lead-tolerant bacterium, was assayed to remove lead(II) from aqueous solution. The biosorption process was optimized by response surface methodology (RSM) based on the Box-Behnken design. The relationships between dependent and independent variables were quantitatively determined by second-order polynomial equation and 3D response surface plots. The biosorption mechanism was explored by characterization of the biosorbent before and after biosorption using atomic force microscopy (AFM), scanning electron microscopy, energy dispersive X-ray spectroscopy, X-ray diffraction, and Fourier transform infrared spectroscopy. The results showed that the maximum adsorption capacity of 9.6 mg/g was obtained at the initial lead ion concentration of 108.79 mg/l, pH value of 5.75, and biosorbent dosage of 9.9 g/l (fresh weight), which was close to the theoretically expected value of 9.88 mg/g. Arthrobacter sp. 25 is an ellipsoidal-shaped bacterium covered with extracellular polymeric substances. The biosorption mechanism involved physical adsorption and microprecipitation as well as ion exchange, and functional groups such as phosphoryl, hydroxyl, amino, amide, carbonyl, and phosphate groups played vital roles in adsorption. The results indicate that Arthrobacter sp. 25 may be potentially used as a biosorbent for low-concentration lead(II) removal from wastewater.