• 한국통신학회논문지
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• 제32권10C호
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• pp.959-964
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• 2007

이 논문에서는 $M|p^n-1$를 만족하는 양의 정수 M과 소수 p에 대해서 주기가 $p^n-1$인 M-진 Sidel'nikov 수열을 사용해서 M-진 수열 군을 생성하였다. 이 수열군은 상관 값의 최대간이 $3\sqrt{p^{n}}+6$을 상한으로 갖고 수열군의 크기는 p=2일 때 $(M-1)^2(2^{n-1}-1)$+M-1 이거나 p가 홀수인 소수일 때는 $(M-1)^2(p^n-3)/2+M(M-1)/2$가 된다.

• 대한화학회지
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• 제36권2호
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• pp.318-323
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• 1992

Vinylsulfilimine 유도체(H, p-$CH_3$, m-TEX>$CH_3$, p-Cl, p-Br, p-$OCH_3$, 및 p-$NO_2$)에 1-methyl-5-mercapto-1,2,3,4-tetrazole을 반응시켜 다음 7가지의 새로운 호합물을 합성하였다. S-Phenyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine, S-p-tolyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine, S-m-tolyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine, S-p-chlorophenyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine, S-p-bromophenyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine, S-p-methoxyphenyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine and S-p-nitrophenyl-S-2-(1-methyl-1,2,3,4-tetrazole-5-thio)-ethyl-N-p-tosylsulfilimine. 이 화합물들의 구조는 원소분석, MP, UV, IR 및 NMR 스펙트럼에 의해 확인되었다.

• 호남수학학술지
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• 제30권1호
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• pp.9-20
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• 2008

Let ${X_i{\mid}i{\geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}$ with 0 < ${\sigma}^2$ < ${\infty}$. Set $S_n={\sum\limits^n_{i=1}^\{X_i}$, the precise asymptotics for ${\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})$,${\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ and ${\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})$ as ${\varepsilon}{\searrow}0$ are established under the suitable conditions.

• Kyungpook Mathematical Journal
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• 제46권2호
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• pp.255-260
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• 2006

Let $m$ and $k$ be any positive integers, let $\mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(\mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $\mathbb{Z}_k$ and let ${\phi}_m(k)$ the order of $G_m(\mathbb{Z}_k)$. In this paper, ${\phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(\mathbb{Z}_{st})$ is isomorphic to $G_m(\mathbb{Z}_s){\times}G_m(\mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${\phi}_m(p^n)=p^{m^2}{\cdot}{\phi}_m(p^{n-1})=p{^{2m}}^2{\cdot}{\phi}_m(p^{n-2})={\cdots}=p^{{(n-1)m}^2}{\cdot}{\phi}_m(p)$, and so ${\phi}_m(k)={\phi}_m(p_1^n1){\cdot}{\phi}_m(p_2^{n2}){\cdots}{\phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• 대한수학회보
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• 제51권4호
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• pp.1041-1054
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• 2014

Let $P{\geq}3$ be an integer and let ($U_n$) and ($V_n$) denote generalized Fibonacci and Lucas sequences defined by $U_0=0$, $U_1=1$; $V_0= 2$, $V_1=P$, and $U_{n+1}=PU_n-U_{n-1}$, $V_{n+1}=PV_n-V_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equations $V_n=kx^2$ and $V_n=2kx^2$ with k | P and k > 1. Then, when k | P and k > 1, we solve some other equations such as $U_n=kx^2$, $U_n=2kx^2$, $U_n=3kx^2$, $V_n=kx^2{\mp}1$, $V_n=2kx^2{\mp}1$, and $U_n=kx^2{\mp}1$. Moreover, when P is odd, we solve the equations $V_n=wx^2+1$ and $V_n=wx^2-1$ for w = 2, 3, 6. After that, we solve some Diophantine equations.

• 대한수학회논문집
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• 제35권1호
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• pp.301-319
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• 2020

In this paper, we show that the system of difference equations $x_n={\frac{ay^p_{n-1}+b(x_{n-2}y_{n-1})^{p-1}}{cy_{n-1}+dx^{p-1}_{n-2}}}$, $y_n={\frac{{\alpha}x^p_{n-1}+{\beta}(y_{n-2}x_{n-1})^{p-1}}{{\gamma}x_{n-1}+{\delta}y^{p-1}_{n-2}}}$, n ∈ ℕ₀ where the parameters a, b, c, d, α, β, γ, δ, p and the initial values x-2, x-1, y-2, y-1 are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.

• 대한전자공학회논문지
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• 제19권6호
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• pp.52-54
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• 1982

p·Si-전해질 접합을 전해질로 6N H2SO4, 6N H2SO4(Ti3+), 6N H2SO4(Ti4+), 6N H2SO4(Ti4+/Ti3+)을 사용하여 만들었다. 이들 전해질중 6N H2SO4(Ti4-/Ti3+)을 사용할 때 p·Si 광음극이 안정하게 동학하며 높은 광전 감도를 가지고 있었다. p·Si-electrolyte junction are prepared by using p·Si photocatode in four different electrolytes such as 6N H2SO4, 6N H2SO4(Ti3+), 6N H2SO4(Ti4+), 6N H2SO4(Ti4+/Ti3+) respectively. Among those electrolytes 6N H2SO4(Ti4-/Ti3+) shows very good results, in which p·Si photocathode is stable.

• 한국환경과학회지
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• 제17권5호
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• pp.509-516
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• 2008

New functional surfactant, N,N-dimethyl-N-dodecyl-N-(2-methyl benzimidazoyl) ammonium chloride(DDBAC) having benzimidazole(BI) functional group have been synthesized and the critical micellar concentration of DDBAC measured by surface tentiometry and electric conductivity method was $8.9{\times}10^{-4}M$. Micellar effects in DDBAC functional surfactant solution on the hydrolysis of p-nitrophenylacetate(p-NPA), p-nitro-phenylpropionate(p-NPP) and p-nitrophenylvalerate(p-NPV) were observed with change of various pH (Tris-buffer). The pseudo first rate constants of hydrolysis of p-NPA, p-NPP and p-NPV in optimum concentration of DDBAC solution increase to about 160, 280 and 600 times, respectively, as compared with those of aqueous solution at pH 8.00(Tris-buffer). It is considered that benzimidazole functional moiety accelerates the reaction rates of hydrolysis because they act as nucleophile or general base. In optimum concentration of DDBAC solution, the rate constants of hydrolysis of p-NPP and p-NPV increase to about 1.5 and 3.0 times, respectively, as compared with that of p-NPA. It means that the more the carbon numbers of alkyl group of substrates, the larger the binding constants between DDBAC micelle and substrates are. To know the hydrolysis mechanism of p-NPCE(p-NPA, p-NPP and p-NPV), the deuterium kinetic isotope effects were measured in $D_2O$ solutions. Consequently the pseudo first order rate constant ratios in $H_2O$ and $D_2O$ solution, $k_{H_2O}/k_{D_2O}$, were about $2.8{\sim}3.0$ range. It means that the mechanism of hydrolysis were proceeded by nucleophile and general base attack in approximately same value.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.23-34
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• 2019

In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

• 대한수학회보
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• 제58권5호
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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 ${\mid}{\Delta}_G(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^{m-1}}{\text{polylog}{\mid}P{\mid}}}$.