• Journal of KIISE:Computer Systems and Theory
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• v.29 no.12
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• pp.665-679
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• 2002

The fault diameter of a graph G is the maximum of lengths of the shortest paths between all two vertices when there are $\chi$(G) - 1 or less faulty vertices, where $\chi$(G) is connectivity of G. In this paper, we analyze the fault diameter of recursive circulant $G(2^m,2^k)$ with $k{\geq}3$. Let $ dia_{m.k}$ denote the diameter of $G(2^m,2^k)$. We show that if $2{\leq}m,2{\leq}k, the fault diameter of $G(2{\leq}m,2{\leq}k)$ is equal to $2^m-2$, and if m=k+1, it is equal to $2^k-1$. It is also shown that for m>k+1, the fault diameter is equal to di a_$m{\neq}1$(mod 2k); otherwise, it is less than or equal to$dia_{m.k+2}$.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.89-96
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• 2009

In this article, we show a generalization of Clement's theorem on the pair of primes. For any integers n and k, integers n and n + 2k are a pair of primes if and only if 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) whenever (n, (2k)!) = (n + 2k, (2k)!) = 1. Especially, n or n + 2k is a composite number, a pair (n, n + 2k), for which 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) is called a pair of pseudoprimes for any positive integer k. We have pairs of pseudorimes (n, n + 2k) with $n{\leq}5{\times}10^4$ for each positive integer $k(4{\leq}k{\leq}10)$.

• The KIPS Transactions:PartA
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• v.14A no.6
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• pp.327-332
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• 2007

In this paper, we will analysis embedding between $2^{2n-k}{\times}2^k$ torus and interconnection networks HFN(n,n), HCN(n,n). First, we will prove that $2^{2n-k}{\times}2^k$ torus can be embedded into HFN(n,n) with dilation 3, congestion 4 and the average dilation is less than 2. And we will show that $2^{2n-k}{\times}2^k$ torus can be embedded into HCN(n,n) with dilation 3 and the average dilation is less than 2. Also, we will prove that interconnection networks HFN(n,n) and HCN(n,n) can be embedded into $2^{2n-k}{\times}2^k$ torus with dilation O(n). These results mean so many developed algorithms in torus can be used efficiently in HFN(n,n) and HCN(n,n).

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.263-275
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• 2010

Let {$X_n;n\;\geq\;1$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n\;=\;{\sum}^n_{k=1}X_k$, $M_n\;=\;max_{k{\leq}n}|S_k|$, $n\;{\geq}\;1$. Suppose $\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k$ (0 < $\sigma$ < $\infty$). We prove that for any b > -1/2, if $E|X|^{2+\delta}$(0<$\delta$$\leq$1), then $$lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}$$ and for any b > -1/2, $$lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2'}}$$, where $\Gamma(\cdot)$ is the Gamma function and N stands for the standard normal random variable.

• Honam Mathematical Journal
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• v.33 no.1
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• pp.115-120
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• 2011

Given vectors x and y in a separable complex Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$\cal{L}$ be a tridiagonal algebra on a separable complex Hilbert space H and let x = ($x_i$) and y = ($y_i$) be vectors in H. Then the following are equivalent: (1) There exists an invertible operator A = ($a_{kj}$) in Alg$\cal{L}$ such that Ax = y. (2) There exist bounded sequences $\{{\alpha}_n\}$ and $\{{{\beta}}_n\}$ in $\mathbb{C}$ such that for all $k\in\mathbb{N}$, ${\alpha}_{2k-1}\neq0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=\frac{\alpha_{2k}}{{\alpha}_{2k-1}\alpha_{2k+1}}$ and $$y_1={\alpha}_1x_1+{\alpha}_2x_2$$ $$y_{2k}={\alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}$$.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.243-250
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• 2005

Given operators X and Y acting on a separable Hilbert space ${\mathcal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. We show the following: Let ${\mathcal{L}}$ be a subspace lattice acting on a separable complex Hilbert space ${\mathcal{H}}$. and let $X=(x_{ij})$ and $Y=(y_{ij})$ be operators acting on ${\mathcal{H}}$. Then the following are equivalent: (1) There exists an invertible operator $A=(a_{ij})$ in $Alg{\mathcal{L}}$ such that AX = Y. (2) There exist bounded sequences {${\alpha}_n$} and {${\beta}_n$} in ${\mathbb{C}}$ such that $${\alpha}_{2k-1}{\neq}0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=-\frac{{\alpha}_{2k}}{{\alpha}_{2k-1}{\alpha}_{2k+1}}$$ and $$y_{i1}={\alpha}_1x_{i1}+{\alpha}_2x_{i2}$$ $$y_{i\;2k}={\alpha}_{4k-1}x_{i\;2k}$$ $$y_{i\;2k+1}={\alpha}_{4k}x_{i\;2k}+{\alpha}_{4k+1}x_{i\;2k+1}+{\alpha}_{4k+2}x_{i\;2k+2}$$ for $$k{\in}N$$.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.40 no.3
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• pp.491-496
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• 2015

In this paper, we introduce two classes of optimal codes, [$2^k-1$, k, $2^{k-1}$] simplex codes and [$2^k-1+k$, k, $2^{k-1}+1$] codes, attaining Griesmer bound with equality. We further present and compare the locality of them. The [$2^k-1+k$, k, $2^{k-1}+1$] codes have good locality property as well as optimal code length with given code dimension and minimum distance. Therefore, we expect that [$2^k-1+k$, k, $2^{k-1}+1$] codes can be applied to various distributed storage systems.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1341-1354
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• 2021

Let r ≥ 2, and let k_i ≥ 2 for 1 ≤ i ≤ r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k₁, k₂, k₃, …, k_r; r) such that for any n ≥ w, every r-colouring of [1, n] admits a k_i-term arithmetic progression with colour i for some i ∈ [1, r]. For k ≥ 3 and r ≥ 2, the mixed van der Waerden number w(k, 2, 2, …, 2; r) is denoted by w₂(k; r). B. Landman and A. Robertson [9] showed that for k < r < $\frac{3}{2}$(k - 1) and r ≥ 2k + 2, the inequality w₂(k; r) ≤ r(k - 1) holds. In this note, we establish some results on w₂(k; r) for 2 ≤ r ≤ k.

• Clean Technology
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• v.23 no.1
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• pp.104-112
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• 2017

Hydrogen production via steam reforming of liquefaction liquid from marine algae over hydrothermal liquefaction was carried out at 873 ~ 1073 K with a commercial catalyst and Ni based $K_2Ti_xO_y$ added catalysts. Liquefaction liquid obtained by hydrothermal liquefaction (503 K, 2 h) was used as a reactant and comparison studies for catalytic activity over different catalysts (FCR-4-02, $Ni/K_2Ti_xO_y-Al_2O_3$, $Ni/K_2Ti_xO_y-SiO_2$, $Ni/K_2Ti_xO_y-ZrO_2/CeO_2$ and Ni/$K_2Ti_xO_y$-MgO), reaction temperature were performed. Experimental results showed Ni/$K_2Ti_xO_y$ based catalysts ($Ni/K_2Ti_xO_y-Al_2O_3$, $Ni/K_2Ti_xO_y-SiO_2$, Ni/$K_2Ti_xO_y-ZrO_2$/ $CeO_2$ and Ni/$K_2Ti_xO_y$-MgO) have a higher activity than commercial catalyst (FCR-4-02) and In particular, a product composition was different depending on support materials. An acidic support ($Al_2O_3$) and a basic support (MgO) led to a higher selectivity for CO while a neutral support ($SiO_2$) and a reducing support ($ZrO_2/CeO_2$) resulted in a higher $CO_2$ selectivity due to water gas shift reaction.

• Journal of the Korean Chemical Society
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• v.39 no.1
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• pp.7-13
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• 1995

The crystal sructures of $X(Ca_{46}Al_{92}Si_{100}O_{384})$ and $Ca_{32}K_{28}-X(Ca_{32}K_{28}Al_{92}Si_{100}O_{384})$ dehydrated at $360^{\circ}C$ and $2{\times}10^{-6}$ Torr have been determined by single-crystal X-ray diffraction techniques in the cubic space group Fd3 at $21(1)^{\circ}C.$ Their structures were refined to the final error indices, R_1=0.096,\;and\;R_2=0.068$ with 166 reflections, and R_1=0.078\;and\;R_2=0.056$ with 130 reflections, respectively, for which I > $3\sigma(I).$ In dehydrated $Ca_{48}-X,\;Ca^{2+}$ ions are located at two different sites opf high occupancies. Sixteen $Ca^{2+}$ ions are located at site I, the centers of the double six rings $(Ca(1)-O(3)=2.51(2)\AA$ and thirty $Ca^{2+}$ ions are located at site II, the six-membered ring faces of sodalite units in the supercage. Latter $Ca^{2+}$ ions are recessed $0.44\AA$ into the supercage from the three O(2) oxygen plane (Ca(2)-O(2)= $2.24(2)\AA$ and $O(2)-Ca(2)-O(2)=119(l)^{\circ}).$ In the structure of $Ca_{32}K_{28}-X$, all $Ca^{2+}$ ions and $K^+$ ions are located at the four different crystallographic sites: 16 $Ca^{2+}$ ions are located in the centers of the double six rings, another sixteen $Ca^{2+}$ ions and sixteen $K^+$ ions are located at the site II in the supercage. These $Ca^{2+}$ ions adn $K^+$ ions are recessed $0.56\AA$ and $1.54\AA$, respectively, into the supercage from their three O(2) oxygen planes $(Ca(2)-O(2)=2.29(2)\AA$, $O(2)-Ca(2)-O(2)=119(1)^{\circ}$, $K(1)-O(2)=2.59(2)\AA$, and $O(2)-K(1)-O(2)=99.2(8)^{\circ}).$ Twelve $K^+$ ions lie at the site III, twofold axis of edge of the four-membered ring ladders inside the supercage $(K(2)-O(4)=3.11(6)\AA$ and $O(1)-K(2)-O(1)=128(2)^{\circ}).$