• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.441-452
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• 2006

We introduce the counting function ${\pi}^*_{2.8}(x)$ of the primes with difference 8 between consecutive primes ($p_n,\;p_{n+l}=p_n+8$) can be approximated by logarithm integral $Li^*_{2.8}$. We calculate the values of ${\pi}^*_{2.8}(x)$ and the sum $C_{2,8}(x)$ of reciprocals of primes with difference 8 between consecutive primes $p_n,\;p_{n+l}=p_n+8$ where x is counted up to $7{\times}10^{10}$. From the results of these calculations. we obtain ${\pi}^*_{2.8}(7{\times}10^{10}$)= 133295081 and $C_{2.8}(7{\times}10^{10}) = 0.3374{\pm}2.6{\times}10^{-4}$.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.24 no.2
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• pp.321-324
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• 2020

In this work, we developed a selective etching process for GaN that is a key process in p-GaN/AlGaN/GaN enhancement-mode (E-mode) power switching field-effect transistor (FET) fabrication. In order to achieve a high current density of p-GaN/AlGaN/GaN E-mode FET, the p-GaN layer beside the gate region must be selectively etched whereas the underneath AlGaN layer should be maintained. A selective etching process was implemented by oxidizing the surface of the AlGaN layer and the GaN layer by adding O2 gas to Cl2/N2 gas which is generally used for GaN etching. A selective etching process was optimized using Cl2/N2/O2 gas mixture and a high selectivity of 53:1 (= GaN/AlGaN) was achieved.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.465-479
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• 2012

For $0<p{\leq}{\infty}$ and a convex body $K$ in $\mathbb{R}^n$, Lutwak, Yang and Zhang defined the concept of dual $L_p$-centroid body ${\Gamma}_{-p}K$ and $L_p$-John ellipsoid $E_pK$. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $K$, there exist an ellipsoid $E$ and a parallelotope $P$ such that for $1{\leq}p{\leq}2$ and $0<q{\leq}{\infty}$, $E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$; For $2{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, $2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$. (ii) For any convex body $K$ whose John point is at the origin, there exists a simplex $T$ such that for $1{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, ${\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT$ and $V(K)=V(T)$.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.245-258
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• 2005

In this paper we prove that the alternating groups A_n, for n = p, p+1, p+2 and symmetric groups $S_n$, for n = p, p+1, where p$\ge$3 is a prime number, can be uniquely determined by their order components. As one of the important consequence of this characterization we show that the simple groups An, where n = p, p+1, P+2 and p$\ge$3 is prime, satisfy in Thompson's conjecture and Shi's conjecture.

• Journal of Gastric Cancer
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• v.6 no.2
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• pp.114-119
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• 2006

Purpose: The purpose of this study is to determine whether it is possible to evaluate patients with pN2 or pN3 early gastric cancer (EGC) as being in an advanced stage before and during the operation. Materials and Methods: 4,430 patients underwent a gastrectomy for cancer from 1990 to 2003. Eight of the 552 patients with EGC included pN2 or pN3. The estimated clinical and surgical stage before and during the operation were compared to the pathological results, and a follow-up of progression was done. Results: The patients were evenly distributed among all age groups with seven men and one woman. The pre-operative estimate of T1 by CT was 25% (2/8). In the main, the cT stage was over estimated. The estimate of over N2 was 50% (4/8). One patient was preoperatively staged as la sT1 during operation was 57.1% (4/7), and the estimate of over N2 was 67% (4/6). Two patients were intraoperatively evaluated as Ia. Only one patient survived over 5 years, and the mean survival of these patients was 15 months $(95%\;Cl:\;0{\sim}35.5)$. Conclusion: It was generally possible to evaluate patients with EGC of over pN2 as being in an advanced stage before and during the operation. Although very rare (2/552, 0.04%), there were EGC patients whose stages were not predictable at all. Therefore, more precise preoperative and intraoperative staging methods are warranted.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.983-991
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• 2013

For each real number $n$ > 6, we prove that there is a sequence $\{pk(n,z)\}^{\infty}_{k=1}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $p_{k+1}(n,z)$ has the largest (in modulus) zero ${\alpha}{\beta}$ where ${\alpha}$ and ${\beta}$ are the first and the second largest (in modulus) zeros of $p_k(n,z)$, respectively. One such sequence is given by $p_k(n,z)$ so that $$p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1$$, where $q_0(n)=1$ and other $q_k(n)^{\prime}s$ are polynomials in n defined by the severely nonlinear recurrence $$4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)$$ for $m{\geq}1$, with the usual empty product conventions, i.e., ${\prod}_{j=0}^{-1}\;b_j=1$.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.375-382
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• 2007

We Study, firstly, the dynamics of the difference equation $x_{n+1}\;=\;{\alpha}\;+\;{\frac{x^p_n}{x^p_{n-1}}}$, with $p\;{\in}\;(0,\;1)\;and\;{\alpha}\;{\in}\;[0,\;{\infty})$. Then, we generalize our results to the (k + 1)th order difference equation $x_{n+1}\;=\;{\alpha}\;+\;{\frac{x^p_n}{nx^p_{n-k}}$, $k\;=\;2,\;3,\;{\cdots}$ with positive initial conditions.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.977-986
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• 2010

Let $X_1$, $X_2$, $\cdots$ be identically distributed negatively associated random variables with $EX_1\;=\;0$ and $E|X_1|^3$ < $\infty$. In this paper we prove $lim_{{\epsilon\downarrow}0}\;\frac{1}{-\log\;\epsilon}\sum\limits_{n=1}^\infty\frac{1}{n^2}ES_n^2I\{|S_n|\;{\geq}\;{\sigma\epsilon}n\}\;=\;2$ and $lim_{\epsilon\downarrow0}\;\epsilon^{2-p}\sum\limits_{n=1}^\infty\frac{1}{n^p}$ $E|S_n|^pI\{|S_n|\;{\geq}\;{\sigma\epsilon}n\}\;=\;\frac{2}{2-p}$ for 0 < p < 2, where $S_n\;=\;\sum\limits_{i=1}^{n}X_i$ and 0 < $\sigma^2\;=\;EX_1^2\;+\;\sum\limits_{i=2}^{\infty}Cov(X_1,\;X_i)$ < $\infty$. We consider some results of i.i.d. random variables obtained by Liu and Lin(2006) under negative association assumption.

• Korean Journal of Crystallography
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• v.15 no.1
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• pp.35-39
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• 2004

Two new nickel vanadium borophosphate cluster compounds, $(NH_4)_{10}[Ni(H_2O)_5]_4[V_2P_2BO_{12}]_6{\cdot}nH_2O$ (1) and $(NH_4)_{3.5}(C_3H_{12}N_2)_{3.5}[Ni(H_2O)_6]_{1.25}{[Ni(H_2O)_5]_2[V_2P_2BO_{12}]_6{\cdot}nH_2O$ (2) have been synthesized and structurally characterized. Inter-diffusion methods were employed to prepare the compounds. The cluster anion $[(NH_4)\;{\supset}\;V_2P_2BO_{12}]_6$ is used as a building unit in the synthesis of new compounds containing $Ni(H_2O){^{2+}_5}$ in the presence of pyrazine and 1,3-diaminopropane. Compounds contain isolated cluster anions with general composition ${[Ni(H_2O)_5]_n[(NH_4)\;{\supset}\;V_2P_2BO_{12}]_6}^{-(17-2n)}$ (n = 2, 4). Crystal data: $(NH_4)_{10}[Ni(H_2O)_5]_4[V_2P_2BO_{12}]_6{\cdot}nH_2O$, monoclinic, space group C2/m (no. 12), a = 27.538(2) ${\AA}$, b = 20.366(2) ${\AA}$, c = 11.9614(9) ${\AA}$, ${\beta}$ = 112.131(1)$^{\circ}$, Z = 8; $(NH_4)_{3.5}(C_3H_{12}N_2)_b[Ni(H_2O)_6]_{3.5}{[Ni(H_2O)_5]_2[V_2P_2BO_{12}]_6{\cdot}nH_2O$, triclinic, space group P-1 (no. 2), a = 17.7668(9) ${\AA}$, b = 17.881(1) ${\AA}$, c = 20.668(1) ${\AA}$, ${\alpha}$ = 86.729(1)$^{\circ}$, ${\beta}$ \ 65.77(1)$^{\circ}$, ${\gamma}$ = 80.388(1)$^{\circ}$, Z = 2.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.31-49
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• 2002

An R$^{n}$ -geometry (P$^{n}$ , L) is a generalization of the Euclidean geometry on R$^{n}$ (see Def. 1.1). We can consider some topologies (see Def. 2.2) on the line set L such that the join operation V : P$^{n}$ $\times$ P$^{n}$ \ $\Delta$ longrightarrow L is continuous. It is a notable fact that in the case n = 2 the introduced topologies on L are same and the join operation V : P$^2$ $\times$ P$^2$ \ $\Delta$ longrightarrow L is continuous and open [10, 11]. It is a fundamental topological property of plane geometry, but in the cases n $\geq$ 3, it is no longer true. There are counter examples [2]. Hence, it is a fundamental problem to find suitable topologies on the line set L in an R$^{n}$ -geometry (P$^{n}$ , L) such that these topologies are compatible with the incidence structure of (P$^{n}$ , L). Therefore, we need to study the topologies of the line set L in an R$^{n}$ -geometry (P$^{n}$ , L). In this paper, the relations of such topologies on the line set L are studied.