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실수계의 공리를 이용한 지수 a<sup>r</sup>의 학습과 지도
이병수,Lee, Byung-Soo 영남수학회 2012 East Asian mathematical journal Vol.28 No.2
In all Mathematics I Textbooks(Kim, S. H., 2010; Kim, H. K., 2010; Yang, S. K., 2010; Woo, M. H., 2010; Woo, J. H., 2010; You, H. C., 2010; Youn, J. H., 2010; Lee, K. S., 2010; Lee, D. W., 2010; Lee, M. K., 2010; Lee, J. Y., 2010; Jung, S. K., 2010; Choi, Y. J., 2010; Huang, S. K., 2010; Huang, S. W., 2010) in high schools in Korea these days, it is written and taught that for a positive real number $a$, $a^{\frac{m}{n}}$ is defined as $a^{\frac{m}{n}}={^n}\sqrt{a^m}$, where $m{\in}\mathbb{Z}$ and $n{\in}\mathbb{N}$ have common prime factors. For that situation, the author shows his opinion that the definition is not well-defined and $a^{\frac{m}{n}}$ must be defined as $a^{\frac{m}{n}}=({^n}\sqrt{a})^m$, whenever $^n\sqrt{a}$ is defined, based on the field axiom of the real number system including rational number system and natural number system. And he shows that the following laws of exponents for reals: $$\{a^{r+s}=a^r{\cdot}a^s\\(a^r)^s=a^{rs}\\(ab)^r=a^rb^r$$ for $a$, $b$>0 and $s{\in}\mathbb{R}$ hold by the completeness axiom of the real number system and the laws of exponents for natural numbers, integers, rational numbers and real numbers are logically equivalent.
Weakly Relaxed $\alpha$-semi-pseudomonotone Set-valued Variational-like Inequalities
이병수,Bok-Doo Lee 한국수학교육학회 2004 純粹 및 應用數學 Vol.11 No.3
In this paper, we introduce weakly relaxed ��-pseudomonotonicity and weakly relaxed ��-semi-pseudomonotonicity of set-valued maps. Using the KKM technique, we obtain existence of solutions to the variational-like inequalities with weakly relaxed ��-pseudomonotone set-valued maps in re°exive Banach spaces. We also present the solvability of the variational-like inequalities with weakly relaxed ��- semi-pseudomonotone set-valued maps in arbitrary Banach spaces using Kakutani- Fan-Glicksberg ¯xed point theorem.
HIC용 $RuO_2$ 후막저항체에서 유리의 물리적 성질이 TCR에 미치는 영향
이병수,이준,Lee, B.S.,Lee, J. 한국세라믹학회 1993 한국세라믹학회지 Vol.30 No.11
Glass viscosity effects on the electrical properties and microstructure of RuO2 based thick film resistors (TFR) using alumina modified lead borosilicate glasses were studied. AT 85$0^{\circ}C$, the glass viscosities were increased from 4.24Pa.s to 51.5Pa.s when the alumina was added from none to 14 weight percent to the standard glass of 63% PbO, 25% B2O3 and 12% SiO2. The resistivities of resistors were generally decreased and the microstructure development was retarded as the viscosity of the glass increased. This is contrary to the generally accepted thought that the low resistivity is due to fast microstructure development kinetics in TFR. Even though the glass viscosity retards the microstructure development kinetics, the overall network formations are favored for higher viscosity of glass, such that the sheet resistivities were decreased as the glass viscosity increased.