• Journal of the Korean Statistical Society
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• v.22 no.1
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• pp.39-54
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• 1993

In the three-factor nested variance component model with equal numbers in the cells given by $y_{ijkm} = \mu + A_i + B_{ij} + C_{ijk} + \varepsilon_{ijkm}$, the exact confidence intervals of the variance component of $\sigma^2_A, \sigma^2_B, \sigma^2_C, \sigma^2_{\varepsilon}, \sigma^2_A/\sigma^2_{\varepsilon}, \sigma^2_B/\sigma^2_{\varepsilon}, \sigma^2_C/\sigma^2_{\varepsilon}, \sigma^2_A/\sigma^2_C, \sigma^2_B/\sigma^2_C$ and $\sigma^2_A/\sigma^2_B$ are not found out yet. In this paper approximate lower and upper confidence intervals are presented.

• East Asian mathematical journal
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• v.18 no.2
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• pp.271-282
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• 2002

Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

• Journal of Advanced Marine Engineering and Technology
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• v.24 no.6
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• pp.24-33
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• 2000

This paper investigates the relationship between J-integral and crack tip opening displacement, ${\delta}_t$ using Gordens results of numerical analysis. Estimation were carried out for several strength levels such as ultimate, flow, yield, ultimate-flow, flow-yield stress to determine the influence of strain hardening and the ratio of crack length to width on the $J-{\delta}_t$ relationship. It was found that for SE(B) specimens, the $J-{\delta}_t$ relationship can be applied to relate J to ${\delta}_t$ as follows $J=m_j{\times}{\sigma}_i{\times}{\delta}_t$ where $m_j=1.27773+0.8307({\alpha}/W)$, ${\sigma}_i:{\sigma}_U$, ${\sigma}_{U-F}={\frac{1}{2}} ({\sigma}_U+{\sigma}_F$), ${\sigma}_F$, ${\sigma}_F}$ $Y=({\sigma}_F+{\sigma}_Y)$, ${\sigma}_Y$

• Journal of the Korean Mathematical Society
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• v.42 no.1
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• pp.53-63
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• 2005

Let R be a ring R and ${\sigma}$ be an endomorphism of R. R is called ${\sigma}$-rigid (resp. reduced) if $a{\sigma}r(a) = 0 (resp{\cdot}a^2 = 0)$ for any $a{\in}R$ implies a = 0. An ideal I of R is called a ${\sigma}$-ideal if ${\sigma}(I){\subseteq}I$. R is called ${\sigma}$-quasi-Baer (resp. right (or left) ${\sigma}$-p.q.-Baer) if the right annihilator of every ${\sigma}$-ideal (resp. right (or left) principal ${\sigma}$-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[$x;{\sigma}$] of a ring R is investigated as follows: For a ${\sigma}$-rigid ring R, (1) R is ${\sigma}$-quasi-Baer if and only if A is quasi-Baer if and only if A is $\={\sigma}$-quasi-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$ (2) R is right ${\sigma}$-p.q.-Baer if and only if R is ${\sigma}$-p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is $\={\sigma}$-p.q.-Baer if and only if A is right $\={\sigma}$-p.q.-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$.

• Journal of The Korean Society of Grassland and Forage Science
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• v.9 no.1
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• pp.34-42
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• 1989

This pot experiments were conducted to find out the optimal fertilization ratios of ${\Sigma}anion$ : ${\Sigma}cation$, ${\Sigma}A$/${\Sigma}C$, and the optimal application rates of total major nutrients in an orchardgrass/ladino clover mixed sward. The optimum ratios and concentrations in equivalent basis were computed by the Homes systematic variations technique. The results were summarized as follows; 1. The optimum fertilization ratios of ${\Sigma}A$ : ${\Sigma}C$ and the optimum application rates of total nutrients for the high yields by forage species in a mixed sward were obtained (Table 6 in detail); ${\Sigma}A$ : ${\Sigma}C$ = 2 : 1 at 80 and 320 meq/pot, and 3 : 2 at 560 and 800 meq for grass and grass plus legume, and ${\Sigma}A$ : ${\Sigma}C$ = 1 : 2 for legume in general. 2. The optimum application rates of total nutrients for the high yields of grass and grass plus kgum were increased by decreasing the ${\Sigma}A$/ ${\Sigma}C$: ratio, whereas these for legume showed a valible. range without significance. 3. The yields 01 grass and grass plus legume were generally increasing by increasing both the ${\Sigma}A$/ ${\Sigma}C$ ratio and total concentration, but they were significantly higher at the ${\Sigma}A$/ ${\Sigma}C$ = 1.273 than at the 2.125 under the high total ion concentration. The legume yields were proportional to ${\Sigma}C$ ratio and increased by increasing the total ion concentration under the condition of high ${\Sigma}C$ ratio. 4. The efficiencies of ${\Sigma}A$ and ${\Sigma}C$ in relation to the grass and grass plus legume yields were highest with the low ratios of each other and the low rates of total nutrients ${\Sigma}A$ efficiency m the legume yield tended to be similar to that of ${\Sigma}A$ in the grass yield noted above. The ${\Sigma}C$ efficiency in the legume yield, however, was generally proportional to the ${\Sigma}C$ ratio except at the low rate of 80 meqlpot. 5. The yields of grass plus legume, yield components and botanical compositions in a mixed sward were greatly influenced by the ${\Sigma}A$/${\Sigma}C$ ratios, the fertilization rates of total nutrients, and the interaction of ratio and rate noted above. These effects were generally different and opposite accading to grass and legume. In addition, the soil chemical properties and mineral contents of forages were partially influenced by these systematic variations.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.477-484
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• 2005

Let R be a ring with an automorphism 17. An ideal [ of R is ($\sigma$-ideal of R if $\sigma$(I).= I. A proper ideal P of R is ($\sigma$-prime ideal of R if P is a $\sigma$-ideal of R and for $\sigma$-ideals I and J of R, IJ $\subseteq$ P implies that I $\subseteq$ P or J $\subseteq$ P. A proper ideal Q of R is $\sigma$-semiprime ideal of Q if Q is a $\sigma$-ideal and for a $\sigma$-ideal I of R, I$^{2}$ $\subseteq$ Q implies that I $\subseteq$ Q. The $\sigma$-prime radical is defined by the intersection of all $\sigma$-prime ideals of R and is denoted by P$_{(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$_{(R) is the smallest $\sigma$-semiprime ideal of R; (2) For any ring R with an automorphism $\sigma$ and for a skew Laurent polynomial ring R[x, x$^{-1}$; $\sigma$], the prime radical of R[x, x$^{-1}$; $\sigma$] is equal to P$_{(R)[x, x$^{-1}$; $\sigma$ ].

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.385-399
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• 2010

Let $\sigma$ be an endomorphism and I a $\sigma$-ideal of a ring R. Pearson and Stephenson called I a $\sigma$-semiprime ideal if whenever A is an ideal of R and m is an integer such that $A{\sigma}^t(A)\;{\subseteq}\;I$ for all $t\;{\geq}\;m$, then $A\;{\subseteq}\;I$, where $\sigma$ is an automorphism, and Hong et al. called I a $\sigma$-rigid ideal if $a{\sigma}(a)\;{\in}\;I$ implies a $a\;{\in}\;I$ for $a\;{\in}\;R$. Notice that R is called a $\sigma$-semiprime ring (resp., a $\sigma$-rigid ring) if the zero ideal of R is a $\sigma$-semiprime ideal (resp., a $\sigma$-rigid ideal). Every $\sigma$-rigid ideal is a $\sigma$-semiprime ideal for an automorphism $\sigma$, but the converse does not hold, in general. We, in this paper, introduce the quasi $\sigma$-rigidness of ideals and rings for an automorphism $\sigma$ which is in between the $\sigma$-rigidness and the $\sigma$-semiprimeness, and study their related properties. A number of connections between the quasi $\sigma$-rigidness of a ring R and one of the Ore extension $R[x;\;{\sigma},\;{\delta}]$ of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if $R[x;\;{\sigma},\;{\delta}]$ is a (principally) quasi-Baer ring, when R is a quasi $\sigma$-rigid ring.

• Journal of Korean Society for Quality Management
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• v.31 no.2
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• pp.194-204
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• 2003

It is very important to understand and interpret the meaning of the sigma level correctly through the Six Sigma project. Especially, the confusion over the relation between sigma level from the short-term point of view and defective proportion or DPMO from the long-term point of view may make a big gap between expected results of the Six Sigma project and real results in the field. The one-tail approximation is commonly used to calculate the sigma level both in most literatures introducing Six Sigma and actual cases of the Six Sigma project. Since the one-tail approximation undervalues the sigma level of the fields such as business and service of which the sigma level is generally low, however. there can be misleading results of the explanation of the sigma level and inappropriate project evaluation. This paper describes the relation between sigma level and defective proportion in detail and clears the difference between the one-tail and two-tail approximation.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.61-71
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• 2001

We introduce a new class of ${\sigma}$-coherent frames and show that HA is a ${\sigma}$-coherent frame if A is a ${\sigma}$-frame. Based on this, it is shown that a frame is ${\sigma}$-coherent iff it is isomorphic to the frame of ${\sigma}$-ideals of a ${\sigma}$-frame. Finally we show that ${\sigma}$-COhFrm and ${\sigma}$Frm are equivalent.

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.85-101
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• 2004

Let A be a Hopf algebra with a linear form ${\sigma}:k{\rightarrow}A{\otimes}A$, which is convolution invertible, such that ${\sigma}_{21}({\Delta}{\otimes}id){\tau}({\sigma}(1))={\sigma}_{32}(id{\otimes}{\Delta}){\tau}({\sigma}(1))$. We define Hopf algebras, ($A_{\sigma}$, m, u, ${\Delta}_{\sigma}$, ${\varepsilon}$, $S_{\sigma}$). If B and C are opposite skew copaired Hopf algebras and $A=B{\otimes}_kC$ then we find Hopf algebras, ($A_{[{\sigma}]}$, $m_B{\otimes}m_C$, $u_B{\otimes}u_C$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}B{\otimes}{\varepsilon}_C$, $S_{[{\sigma}]}$). Let H be a finite dimensional commutative Hopf algebra with dual basis $\{h_i\}$ and $\{h_i^*\}$, and let $A=H^{op}{\otimes}H^*$. We show that if we define ${\sigma}:k{\rightarrow}H^{op}{\otimes}H^*$ by ${\sigma}(1)={\sum}h_i{\otimes}h_i^*$ then ($A_{[{\sigma}]}$, $m_A$, $u_A$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}_A$, $S_{[{\sigma}]}$) is the dual space of Drinfeld double, $D(H)^*$, as Hopf algebra.