• Honam Mathematical Journal
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• v.16 no.1
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• pp.119-135
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• 1994

Let $Q_{n+p-1}(\alpha)$ denote the- dass of functions $$f(z)=z^{P}-\sum_{n=0}^\infty{a_{(p+k)}z^{p+k}$$ ($a_{p+k}{\geq}0$, $p{\in}N=\left{1,2,{\cdots}\right}$) which are analytic and p-valent in the unit disc $U=\left{z:{\mid}z:{\mid}<1\right}$ and satisfying $Re\left{\frac{D^{n+p-1}f(\approx))^{\prime}}{pz^{p-a}\right}>{\alpha},0{\leq}{\alpha}<1,n>-p,z{\in}U.$ In this paper we obtain sharp results concerning coefficient estimates, distortion theorem, closure theorems and radii of p-valent close-to- convexity, starlikeness and convexity for the class $Q_{n+p-1}$ ($\alpha$). We also obtain class preserving integral operators of the form $F(z)=\frac{c+p}{z^{c}}\int_{o}^{z}t^{c-1}f(t)dt.$ c>-p $F\left(z\right)=\frac{c+p}{z^{c}}\int_{0}^{z} t^{c-1}f\left(t \right)dt. \qquad c>-p$ for the class $Q_{n+p-1}$ ($\alpha$). Conversely when $F(z){\in}Q_{n+p-1}(\alpha)$, radius of p-valence of f(z) has been determined.

• Korean Journal of Mathematics
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• v.17 no.2
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• pp.169-179
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• 2009

Let P be a (n, n − 1, j)-poset, which is a partially ordered set of cardinality n with n − 1 maximal elements and $j(1{\leq}j{\leq}n-1)$ minimal elements, and $P^*$ the dual poset of P. In this paper, we obtain two types of incidence codes of nonempty proper subset S of P and $P^*$, respectively, by using Bogart's method [1] (see Theorem 3.3).

• 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.

• Communications for Statistical Applications and Methods
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• v.17 no.4
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• pp.507-513
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• 2010

Let {$X_i,-{\infty}$ < 1 < $\infty$} be a doubly infinite sequence of identically distributed and negatively associated random variables with mean zero and finite variance and {$a_i,\;-{\infty}$ < i < ${\infty}$} be an absolutely summable sequence of real numbers. Define a moving average process as $Y_n={\sum}_{i=-\infty}^{\infty}a_{i+n}X_i$, n $\geq$ 1 and $S_n=Y_1+{\cdots}+Y_n$. In this paper we prove that E|$X_1$|$^rh$($|X_1|^p$) < $\infty$ implies ${\sum}_{n=1}^{\infty}n^{r/p-2-q/p}h(n)E{max_{1{\leq}k{\leq}n}|S_k|-{\epsilon}n^{1/p}}{_+^q}<{\infty}$ and ${\sum}_{n=1}^{\infty}n^{r/p-2}h(n)E{sup_{k{\leq}n}|k^{-1/p}S_k|-{\epsilon}}{_+^q}<{\infty}$ for all ${\epsilon}$ > 0 and all q > 0, where h(x) > 0 (x > 0) is a slowly varying function, 1 ${\leq}$ p < 2 and r > 1 + p/2.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1469-1483
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• 2018

We study a subclass of p-ary functions in n variables, denoted by ${\mathcal{A}}_n$, which is a collection of p-ary functions in n variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all p-ary (n - 1)-plateaued functions in n variables by proving that every (n - 1)-plateaued function should be contained in ${\mathcal{A}}_n$. Secondly, we prove that if f is a p-ary r-plateaued function contained in ${\mathcal{A}}_n$ with deg f > $1+{\frac{n-r}{4}}(p-1)$, then the highest degree term of f is only a single term. Furthermore, we prove that there is no p-ary r-plateaued function in ${\mathcal{A}}_n$ with maximum degree $(p-1){\frac{n-4}{2}}+1$. As application, we partially classify all (n - 2)-plateaued functions in ${\mathcal{A}}_n$ when p = 3, 5, and 7, and p-ary bent functions in ${\mathcal{A}}_2$ are completely classified for the cases p = 3 and 5.

• Kyungpook Mathematical Journal
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• v.46 no.2
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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 $k$, $k=p_1^{n1}{\cdot}p_2^{n2}{\cdots}p_s^{ns}$.

• 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 ${\parallel}Q{\parallel}=0$.

• Journal of Korean Society of Forest Science
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• v.108 no.1
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• pp.21-28
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• 2019

This study was conducted to determine the carbon (C) and nitrogen (N) response of litterfall components as affected by N addition in compound fertilizer in a Korean red pine (Pinus densiflora S. et Z.) stand in southern Korea. Litterfall in a mature red pine stand was collected for two years following compound fertilizer application ($N_3P_4K_1$; $P_4K_1$) and no fertilization (control). The C concentration of litterfall components was not significantly (P > 0.05) different between the $N_3P_4K_1$ and the control plots, whereas the N concentration of the litterfall components was significantly higher in the $N_3P_4K_1$ plot than in the control plot. The $N_3P_4K_1$ and $P_4K_1$ additions induced a lower C/N ratio of litterfall components compared with the control plot. Annual C and N fluxes via litterfall components were not affected by the $N_3P_4K_1$ addition over the study period, except for reproduction litter. Annual N fluxes via reproduction litter were significantly higher in the $N_3P_4K_1$ plot than in the control plot. Thus, the $N_3P_4K_1$ and $P_4K_1$ additions could modify differently nutrient distribution of the forest floor and mineral soils in a red pine stand. These results indicate that N concentration and C/N ratio in litterfall components are more susceptible to fertilizer application than the C response in litterfall components.

• Journal of the Korean Chemical Society
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• v.37 no.6
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• pp.618-625
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• 1993

Rate constants of hydrolysis of N-(benzoyl)-C-(N-methylanilino)imidoylchlorides were determined by UV spectrophotometry in 50% (v/v) aqueous methanol at 25$^{\circ}C$. On the basis of rate equation, substituent effect, solvent effect, salt effect, thermodynamic parameters and hydrolysis product analysis, it may be concluded that the hydrolysis of N-(benzoyl)-C-(N-methylanilino)imidoylchlorides proceed through $S_N$1 mechanism via azocarbonium ion intermediate in the range of from pH 3.0 to pH 10.0, while above pH 10.0 and below pH 3.0 the hydrolysis proceeds through nucleophilic addition-elimination (A$d_{N-E}$) mechanism.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.483-488
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• 2012

In this paper we introduce the notion of P-strongly regular near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completely semiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) $Na$ + P is an ideal of N for any $a{\in}N$. (ii) Every P-prime ideal of N containing P is maximal. (iii) Every ideal I of N fulfills I + P = $I^2$ + P.