• Kyungpook Mathematical Journal
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• v.53 no.1
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• pp.117-124
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• 2013

We show that for 1 < $p$ < ${\infty}$, $k$, $m{\in}\mathbb{N}$, $n^{(k)}(l_p)=inf\{n^{(k)}(l^m_p):m{\in}\mathbb{N}\}$ and that for any positive measure ${\mu}$, $n^{(k)}(L_p({\mu})){\geq}n^{(k)}(l_p)$. We also prove that for every $Q{\in}P(^kl_p:l_p)$ (1 < $p$ < ${\infty}$), if $v(Q)=0$, then ${\parallel}Q{\parallel}=0$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.657-662
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• 2001

It is proved that if for each n, $1\leqp_n\leq2 \;and \;the(p_n, p’_n)$ Clarkson inequality holds in each Banach space X$_{n}$ then the (t, t’) Clarkson inequality holds in ($\sum^\infty_{n=1}\; X_n)_r, \;the \ell^r-sum \;of\; X_n’s,\; where\; 1\leqr<\infty,\; t=min{p, r, r’} \;and \;p = \;inf{p_n}.$ The (p, p’) Clarkson inequality is preserved by quotient maps and a new proof of a Takahashi-Kato theorem stating that the (p, p’) Clarkson inequality holds in a Banach space X if and only if it holds in its dual space $X_*$ is given.n.

• Honam Mathematical Journal
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• v.1 no.1
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• pp.1-9
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• 1979

Positive Local Coordmate($(x^1,x^2,{\cdots}x^n)$)을 갖는 Oriented Manifold M을 생각한다. M이 Compact Carrier를 갖는 경우, M위의 n차(次) Differential Form ${\phi}^{(n)}$의 적분(積分)을 $${\int}{\phi}^{(n)}=\sum_{\alpha}{\int}_{-{\infty}}^{\infty}{\cdots}{\int}_{-{\infty}}^{\infty}f_{\alpha}{\phi}^{(n)}dx^1{\cdots}dx^n$$로 정의(定義)하며 (정의(定義) 7), M위의 p 차(次)의 Differential form $\beta^{(p)}$와 Differential simplex $S^{(p)}=(S^{(p)},\;{\pi},\;{\varepsilon})$에 대하여 $S^{(p)}$위의 $\beta^{(p)}$의 적분(積分)을 $${\int}_{^{(p)}S}{\beta}^{(p)}={\int}_{S^{(p)}}{\varepsilon}{\pi}^*{\beta}^{(p)}={\int}_{E^p}f{\cdot}{\varepsilon}{\cdot}{\pi}^*{\beta}^{(p)}$$로 정의(定義)한다 (정의(定義) 9). 단(但) $\bar{S}^{(p)}$는 $S^{(p)}=(p_0{\cdot}p_1{\cdots}p_p)$에 의(依)하여 Spanning 되는 $E^p$의 Subspace이고 f는 $\bar{S}^{(p)}$의 특성함수(特性函數)이다. 이때 (n-1)차(次)의 Differential Form ${\beta}^{(n-1)}$이 Compart인 Carrier를 가지면 ${\int}d{\beta}^{(n-1)}=0$이 됨을 고찰(考察)하며(정리(定理 8), (p-1)차(次) Differential Form ${\beta}^{(p-1)$과 p차(次) Differential Chain $C^{(p)}$에 관(關)하여 $${\int}_{C^{(p)}}d{\beta}^{(p-1)}={\int}_{{\partial}C^{(p)}}{\beta}^{(p-1)}$$이 성립(成立)함을 구명(究明)하려 한다(정리(定理) 10).

• Journal of Digital Convergence
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• v.12 no.3
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• pp.7-16
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• 2014

This study aims to analyze the key logic of the current C-P-N-D ICT ecological system, to find out the shortcomings of the current system, and then to offer policy suggestions for the establishment of a new creative contents industry ecological system; that is, ICCT (Information, Communication, Contents and Technology) System.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.945-959
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• 2008

n-ary $H_v$-structures is a generalisation of both n-ary structures and $H_v$-structures. A wide class of n-ary $P-H_v$-groups is the n-ary $P-H_v$-groups that is concidered in this paper. In this paper the notion of a normal subgroup of an n-ary $P-H_v$-groups is introduced and the isomorphism theorems for n-ary $P-H_v$-groups are stated and proved. Also some examples and related properties are investigated.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.26 no.6
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• pp.1401-1411
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• 2016

The Multi-Prime RSA and the Prime Power RSA are the variants of the RSA cryptosystem, where the Multi-Prime RSA uses the modulus $N=p_1p_2{\cdots}p_r$ for distinct primes $p_1,p_2,{\cdots},p_r$ (r>2) and the Prime Power RSA uses the modulus $N=p^rq$ for two distinct primes p, q and a positive integer r(>1). This paper analyzes the security of these systems by using the technique given by Heninger and Shacham. More specifically, this paper shows that if the $2-2^{1/r}$ random portion of bits of $p_1,p_2,{\cdots},p_r$ is given, then $N=p_1p_2{\cdots}p_r$ can be factorized in the expected polynomial time and if the $2-{\sqrt{2}}$ random fraction of bits of p, q is given, then $N=p^rq$ can be factorized in the expected polynomial time. The analysis is then validated with experimental results for $N=p_1p_2p_3$, $N=p^2q$ and $N=p^3q$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1279-1300
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• 2021

Let P be a finite point set in ℝ² with the set of distance n-chains defined as ∆_n(P) = {(|p₁ - p₂|, |p₂ - p₃|, …, |p_n - p_n+1|) : p_i ∈ P}. We show that for 2 ⩽ n = O_|P|(1) we have ${\mid}{\Delta}_n(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^n}{{\log}^{\frac{13}{2}(n-1)}{\mid}P{\mid}}}$. Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in [28], which allows one to iterate efficiently on intersecting nested subsets of Guth-Katz lines. Let G is a simple connected graph on m = O(1) vertices with m ⩾ 2. Define the graph-distance set ∆_G(P) as ∆_G(P) = {(|p_i - p_j|)_{i,j}∈E(G) : p_i, p_j ∈ P}. Combining with results of Guth and Katz [17] and Rudnev [28] with the above, if G has a Hamiltonian path we have ${\mid}{\Delta}_G(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^{m-1}}{\text{polylog}{\mid}P{\mid}}}$.

• East Asian mathematical journal
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• v.19 no.1
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• pp.73-80
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• 2003

Let D be a bounded domain in $\mathbb{C}^n$ with $C^2$ boundary. In this paper, we prove the following inequality $${\parallel}u{\parallel}_{p_2,{\alpha}_2}{\lesssim}{\parallel}u{\parallel}_{p_1,{\alpha}_1}+{\parallel}\bar{\partial}u{\parallel}_{p_1,{\alpha}_1+p_1}/2$$, where $1{\leq}p_1{\leq}p_2<\infty,\;{\alpha}_j>0,(n+{\alpha}_1)/p_1=(n+{\alpha}_1)/p_1=(n+{\alpha}_2)/p_2$, and $1/p_2{\geq}1/p_1-1/2n$.

• Asian-Australasian Journal of Animal Sciences
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• v.22 no.9
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• pp.1248-1255
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• 2009

This study was carried out to evaluate the fatty acid composition in Nellore cows supplemented with either linseed (n-3) or canola grains (n-6 and n-9). Fifteen Nellore cows, aged five years and bodyweight 550 kg${\pm}$48 kg, were randomly distributed to the following treatments: CON (control), LIN (linseed) and CAN (canola grains). The cows were fed for 80 days. The concentrations of C18:0, C18:2 n-6 and C20:3 n-6 fatty acid were higher (p<0.10) in CON blood plasma in comparison to follicular liquid. Likewise, PUFA, n-6 contents, PUFA:SFA and n-6:n-3 ratios were higher (p<0.10) in blood plasma. On the other hand, C18:1 n-9, C22:5 n-3, MUFA and n-3 contents were lower (p<0.10) in blood plasma. C18:0, C18:2 n-6, C18:3 n-3, C22:5 n-3, PUFA, n-6, n-3 contents and PUFA:SFA ratio were higher (p<0.10) in LIN blood plasma than in the follicular liquid. Nevertheless, C14:0, C16:0, C16:1 n-7, PUFA, C16:0, C18:1 n-9 and MUFA contents were lower (p<0.10) in LIN blood plasma. On treatment CAN, the C18:0 and SFA contents, and n-6:n-3 ratios were higher (p<0.10) in blood plasma. However, C20:3 n-6, C22:5 n-3, PUFA and n-3 contents were lower (p<0.10) in blood plasma. C16:0, C18:0, PUFA, SFA contents and PUFA:SFA ratio did not differ (p>0.10) among the treatments. C14:0, C16:1 n-7, C18:2 n-6 and n-6 contents were higher (p<0.10) for CON and CAN than LIN. C17:1 n-7, C20:4 n-6 and C 22:0 contents were higher (p<0.10) for CAN than CON and LIN. C18:1 n-9, C18:3 n-3, MUFA and n-3 contents were higher (p<0.10) for LIN and CAN than CON. C20:3 n-6 content and n-6:n-3 ratio were higher (p<0.10) for CON than LIN and CAN. C22:5 n-3 content were higher (p<0.10) for CON and LIN than CAN. The concentrations of fatty acids in blood plasma and follicular liquid were not correlated for any fatty acid, independent of the treatment studied. Canola grain added to the diet of Nellore cows resulted in increased concentrations of fatty acids n-6 and n-3 in follicular liquid.

• Journal of Korean Society of Environmental Engineers
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• v.36 no.9
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• pp.609-613
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• 2014

The aim of this study was effect of N/P ratio on the nutrient removal in the wastewater using microalgae. For this experiment, 1 to 70 various N/P ratio was prepared and used microalgae as Botryococcus braunii in the wastewater. The results of this study were that 1 to 30 of N/P ratio was need for biomass productivity in the wastewater. TN removal was measured 82% for 1 to 30 N/P ratio and 73-78% for 31 to 70 N/P ratio. TP removal in 1 to 20 N/P ratio was determined up to 80%, but over 21 N/P ratio was decreased significantly and was not changed around 22% of TP removal in the 50 to 70 N/P ratio. Therefore, the optimum N/P ratio in the wastewater was 1 to 30 for biomass productivity, TN and TP removal. The correlation ($R^2$) of TP removal and biomass productivity was 0.9126. However, the relationship between TN removal and biomass productivity was not found. The P content in the wastewater was influenced more than that of TN content.