• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.25-27
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• 1994

Suppose that X is a Banach space with continuous dual $X^{**}$, ($\Omega$, $\Sigma$, ${\mu}$) is a finite measure space. f : $\Omega\;{\longrightarrow}$ $X^{*}$ is a weakly measurable function such that $\chi$$^{**}$ f $\in$ $L_1$(${\mu}$) for each $\chi$$^{**}$ $\in$ $X^{**}$ and $T_{f}$ : $X^{**}$ \longrightarrow $L_1$(${\mu}$) is the operator defined by $T_{f}$($\chi$$^{**}$) = $\chi$$^{**}$f. In this paper we study the properties of bounded $X^{*}$ - valued weakly measurable functions and bounded $X^{*}$ - valued weak＊ measurable functions.(omitted)

• ALGAE
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• v.27 no.2
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• pp.83-94
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• 2012

The taxonomic distinctiveness and cosmopolitan distributions of the red algae $Gelidium$ $crinale$ and $G.$ $pusillum$ remain unclear. Both species were first described in Devon in southwestern England; namely in Ilfracome for $G.$ $crinale$ and Sidmouth for $G.$ $pusillum$. We analyzed mitochondrial $cox$1 and plastid $rbc$L sequences from specimens collected in East Asia, Australia, Europe and North America. In all phylogenetic analyses of $cox$1 and $rbc$L sequences, $G.$ $crinale$ was distinct from congeners of the genus. The analyses also revealed a sister relationship with the $G.$ $coulteri$ and $G.$ $capense$ clade. Nineteen $cox$1 haplotypes were identified for $G.$ $crinale$, and they were likely geographically structured. Despite the distinctiveness in both $cox$1 and $rbc$L datasets, the sister relationship of $G.$ $pusillum$ in the genus was not resolved. Our $cox$1 and $rbc$L datasets indicate that $G.$ $crinale$ is a cosmopolitan species, found in East Asia, Australia, Europe and North America, while the distribution of $G.$ $pusillum$ is restricted to Europe and Atlantic North America. Our results suggest that infraspecific classification of $G.$ $pusillum$ may be abandoned.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.473-481
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• 2009

In this paper, we study the uniqueness of entire functions and prove the following theorem. Let n(${\geq}$ 5), k be positive integers, and let $S_1$ = {z : $z^n$ = 1}, $S_2$ = {$a_1$, $a_2$, ${\cdots}$, $a_m$}, where $a_1$, $a_2$, ${\cdots}$, $a_m$ are distinct nonzero constants. If two non-constant entire functions f and g satisfy $E_f(S_1,2)$ = $E_g(S_1,2)$ and $E_{f^{(k)}}(S_2,{\infty})$ = $E_{g^{(k)}}(S_2,{\infty})$, then one of the following cases must occur: (1) f = tg, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = t{$a_1$, $a_2$, ${\cdots}$, $a_m$}, where t is a constant satisfying $t^n$ = 1; (2) f(z) = $de^{cz}$, g(z) = $\frac{t}{d}e^{-cz}$, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = $(-1)^kc^{2k}t\{\frac{1}{a_1},{\cdots},\frac{1}{a_m}\}$, where t, c, d are nonzero constants and $t^n$ = 1. The results in this paper improve the result given by Fang (M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc. 24(2001), 7-16).

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.403-410
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• 2007

Using the belongs to relation (${\in}$) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (${\alpha}$, ${\beta}$)-fuzzy subalgebras where ${\alpha}$ and ${\beta}$ areany two of {${\in}$, q, ${\in}{\vee}q$, ${\in}{\wedge}q$} with ${\alpha}{\neq}{\in}{\wedge}q$ was already introduced, and related properties were investigated (see [3]). In this paper, we give a condition for an (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra to be an (${\in}$, ${\in}$)-fuzzy subalgebra. We provide characterizations of an (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra. We show that a proper (${\in}$, ${\in}$)-fuzzy subalgebra $\mathfrak{A}$ of X with additional conditions can be expressed as the union of two proper non-equivalent (${\in}$, ${\in}$)-fuzzy subalgebras of X. We also prove that if $\mathfrak{A}$ is a proper (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra of a CK/BCI-algebra X such that #($\mathfrak{A}(x){\mid}\mathfrak{A}(x)$ < 0.5} ${\geq}2$, then there exist two prope non-equivalent (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebras of X such that $\mathfrak{A}$ can be expressed as the union of them.

• Journal of the Korean Ceramic Society
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• v.38 no.5
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• pp.431-437
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• 2001

본 연구에서는 양이온 교환반응에 의해 Na-Mont와 R$_{11}$SO$_4$로부터 R$_{11}$SO$_4$-Mont 층간화합물을 합성한 후, 이 R$_{11}$SO$_4$-Mont 층간화합물을 다시 제 4차 유기 양이온인 (Et)$_4$N$^{+}$ 이온 및 (Bu)$_4$N$^{+}$ 이온과 반응시켜 그 거동을 살펴보았다. R$_{11}$SO$_4$-Mont를 (Et)$_4$N$^{+}$이온과 반응시켰을 대보다 (Bu)$_4$N$^{+}$ 이온과 반응시킨 경우에 좀 더 큰 층간거리가 얻어졌다. R$_{11}$SO$_4$-Mont를 (Et)$_4$N$^{+}$이온 및 (Bu)$_4$N$^{+}$과 반응시켜 얻은 층간화합물을 다시 아세토니트릴, 에탄올 및 디옥산과 팽윤반응을 수행한 결과 층간거리가 확장되었으나, 이는 Na-Mont를 R$_{11}$SO$_4$와 48시간 교환 반응시킨 후에 교환용액 하에서 얻은 R$_{11}$SO$_4$-Mont의 층간거리에 불과했다. 원소분석결과, R$_{11}$SO$_4$-Mont와 (Bu)$_4$N$^{+}$를 반응시킬 경우 반응이 반응식 b에 의거 진행되고, (Bu)$_4$N-Mont를 R$_{11}$SO$_4$와 반응시킬 경우에는 반응식 c에 의거 진행됨을 예측할 수 있었다.의거 진행됨을 예측할 수 있었다.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 1997.11a
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• pp.60-63
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• 1997

Dieletric ailed piezoelectric properties of Pb[$Zr_{0.45}$/ $Ti_{0.5-x}$/L $u_{x}$ (M $n_{1}$3//S $b_{2}$3)$_{0.05}$] $O_3$(0$\leq$x $\leq$0.03) were investigated. The partial substitution of $Ti^{4+}$ by a L $u^{3+}$ permitted improvement of the piezoelectric constant( $d_{33}$ ), electromechanical coupling factor ( $k_{p}$ ) and dielectric constant($\varepsilon$$_{33}$ /Sup T/). The dielectric loss(tan $\delta$ ) increased and mechanical quality factor(Qm) decreased with an increase of x were observed. A new piezoelectric material for actuator application was developed at x=0.02 with $d_{33}$ =370$\times$10$^{-2}$ /C/N, $k_{p}$ =58.5%, $\varepsilon$$_{33}$ $^{T}$ =1321, $Q_{m}$ =714 and tan $\delta$ =0.98%.%..%.%.

• Applied Chemistry for Engineering
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• v.9 no.1
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• pp.38-43
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• 1998

It was realized that the nucleation rate in the synthesis of M-ZSM-5 using various alkali cations such as $Li^+$, $Na^+$, $K^+$ and $Cs^+$ at low temperature and atmospheric pressure was decreased in the order of $Na^+>K^+>Li^+>Cs^+$. Unlike conventional synthesis method at high temperature and pressure, the results showed that at low temperature and atmospheric pressure, the higher the nucleation rate is, the larger the crystal size of M-ZSM-5 obtained ; that is, the crystal size in the order of $K^+>Na^+>Cs^+>Li^+$. In addition, it also suggests that regardless of alkali cations to be used, the current synthesis method can produce M-ZSM-5 with BET surface area greater than $300m^2/g$ within 52hrs. of reaction time, in particular greater than $400m^2/g$ within 32hrs, for $Na^+$ cation.

• The Mathematical Education
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• v.26 no.1
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• pp.41-45
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• 1987

In this paper, we give several fixed point theorems in a complete metric space for two multi-valued mappings commuting with two single-valued mappings. In fact, our main theorems show the existence of solutions of functional equations f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ and $\chi$=f($\chi$)=g($\chi$)$\in$S$\chi$∩T$\chi$ under certain conditions. We also answer an open question proposed by Rhoades-Singh-Kulsherestha. Throughout this paper, let (X, d) be a complete metric space. We shall follow the following notations : CL(X) = {A; A is a nonempty closed subset of X}, CB(X)={A; A is a nonempty closed and founded subset of X}, C(X)={A; A is a nonempty compact subset of X}, For each A, B$\in$CL(X) and $\varepsilon$>0, N($\varepsilon$, A) = {$\chi$$\in$X; d($\chi$, ${\alpha}$) < $\varepsilon$ for some ${\alpha}$$\in$A}, E$\sub$A, B/={$\varepsilon$ > 0; A⊂N($\varepsilon$ B) and B⊂N($\varepsilon$, A)}, and (equation omitted). Then H is called the generalized Hausdorff distance function fot CL(X) induced by a metric d and H defined CB(X) is said to be the Hausdorff metric induced by d. D($\chi$, A) will denote the ordinary distance between $\chi$$\in$X and a nonempty subset A of X. Let R$\^$+/ and II$\^$+/ denote the sets of nonnegative real numbers and positive integers, respectively, and G the family of functions ${\Phi}$ from (R$\^$+/)$\^$s/ into R$\^$+/ satisfying the following conditions: (1) ${\Phi}$ is nondecreasing and upper semicontinuous in each coordinate variable, and (2) for each t>0, $\psi$(t)=max{$\psi$(t, 0, 0, t, t), ${\Phi}$(t, t, t, 2t, 0), ${\Phi}$(0, t, 0, 0, t)} $\psi$: R$\^$+/ \longrightarrow R$\^$+/ is a nondecreasing upper semicontinuous function from the right. Before sating and proving our main theorems, we give the following lemmas:

• Journal of Bio-Environment Control
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• v.7 no.3
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• pp.246-252
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• 1998

This study was carried out to investigate the quality of ground water for hydroponics in Kyeongnam area in 1995. Water samples were collected and analyzed from 77 wells in green houses throughout Kyeongnam area. The values of several components in well water were as follows ; 7.4 in pH, 0.46dS/m in EC, 0.3mg/L in N $H_{4}$-N, 25.4mg/L in $Ca^{2＋}$, 42.6mg/L in C $l^{[-10]}$ and 72.5mg/L in S $O_{4}$$^{2-}$. The pH value showed high positive significance of correlationships with $Ca^{2＋}$ and EC. Also, the EC value showed high positive significance with N $a^{＋}$, $Ca^{2＋}$, S $O_{4}$$^{2-}$, $Mg^{2＋}$ and C $l^{[-10]}$ .

• Journal of Sericultural and Entomological Science
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• v.15 no.2
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• pp.29-33
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• 1973

This experimental work was carried out to examine the relation between the filament characteristic and the tensile property of various cocoon varieties. Cocoons of 21 varieties were used as materials, from which, bave denier (X$_1$), sericin solubility (X$_2$), tenacity (Y$_1$), elongation (Y$_2$), and Young's modulus (Y$_3$) were tested. The formulae estimating tensile properties with denier and sericin solubility are as follows: A) In case of denier. Y$_1$= -0.4X$_1$＋5.03 Y$_2$=1.2X$_1$＋2.7 Y$_3$= -25.2X$_1$＋140.95 B) In case of sericin solubility Y$_1$= -2.4X$_2$＋3.43 Y$_2$=20X$_2$＋3.7 Y$_3$= -120X$_2$＋78.37.