• 대한수학회보
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• 제42권4호
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• pp.777-787
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• 2005

In this paper we study (n + 1)-dimensional compact contact CR-submanifolds of (n - 1) contact CR-dimension immersed in an odd-dimensional unit sphere $S^{2m+1}$. Especially we provide necessary conditions in order for such a sub manifold to be the generalized Clifford surface $$S^{2n_1+1}(((2n_1+1)/(n+1))^{\frac{1}{2}})\;{\times}\;S^{2n_2+1}(((2n_2+1)/(n+1)^{\frac{1}{2}})$$ for some portion (n1, n2) of (n - 1)/2 in terms with sectional curvature.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.441-446
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• 2003

Let { $X_{n}$ , n $\geq$ 1} be a sequence of independent and identically distributed random variables with a common continuous distribution function F(x) and probability density function f(x). Let $Y_{n}$ = max{ $X_1$, $X_2$, …, $X_{n}$ } for n $\geq$ 1. We say $X_{j}$ is an upper record value of { $X_{n}$ , n$\geq$1} if $Y_{j}$ > $Y_{j-1}$, j > 1. The indices at which the upper record values occur are given by the record times {u(n)}, n$\geq$1, where u(n) = min{j｜j>u(n-1), $X_{j}$ > $X_{u}$ (n-1), n$\geq$2} and u(1) = 1. We call the random variable X $\in$ Beta (1, c) if the corresponding probability cumulative function F(x) of x is of the form F(x) = 1-(1-x)$^{c}$ , c>0, 0$\leq$x$\leq$1. In this paper, we will give a characterization of the beta distribution of the first kind by considering conditional expectations of record values.s.

• Elastomers and Composites
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• 제39권4호
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• pp.309-317
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• 2004

풀러렌[$C_{70}$]을 3-chloroperoxy benzoic acid, chromium (VI) oxide, benzoyl peroxide, trichloroisocyanuric acid 등의 산화제를 사용하여 마이크로파 조건에서 반응시켜 풀러렌 난화물[$C_{70}O_n$] ($n=1{\sim}3$ or n=1)을 합성하였다. 동일한 마이크로파 조건에서 여러가지 산화제와 풀러렌[$C_{70}$]의 고체상태 반응성은 3-chloroperoxy benzoic acid > chromium(VI) oxide > trichloroisocyanuric acid $\simeq$ benzoyl peroxide 순으로 증가함을 나타냈다. MALDI-TOF-MS, UV-visible, 그리고 HPLC를 사용하여 분석한 결과 생성된 풀러렌 산화물은 [$C_{70}O_n$] ($n=1{\sim}3$ or n=1)임을 알 수 있었다.

• 한국토양비료학회지
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• 제46권4호
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• pp.253-259
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• 2013

지중관비를 이용한 시설오이 재배에서 돈분 액비의 공급이 오이의 생육, 수량 및 토양 중 질소 분포에 미치는 영향을 평가하고자 오이 반촉성 및 억제작형에 걸쳐서 수행하였다. 초장과 마디수 등 오이의 생육은 정식 후 50일에서는 처리간의 차이는 보이지 않았지만 정식 후 85일에서는 무시비의 초장 및 마디수가 각각 388.8 cm와 40.1 ea $plant^{-1}$를 보여 생육량의 차이를 나타내었다. 오이의 수량에서도 무시비를 제외하고 액비 1.0N 및 액비 0.5N + 요소 0.5N처리와 요소 1.0N간에는 통계적으로 차이가 없이 동일하였다. 오이 재배기간에 토양 중 질산태질소의 분포는 관비의 영향으로 지중 15~30 cm 깊이에서 무시비구를 제외하고 질산태 질소의 농도가 가장 높았으며 오이 재배가 끝난 작기 후에는 0~15 cm 깊이에서 가장 높은 질소 양분량을 나타내었다. 액비 1.0N과 액비 0.5N + 요소 0.5N의 질소이용효율은 촉성작형에서 0.34와 0.37를 억제작형에서는 0.29와 0.26를 각각 보여 요소 1.0N의 0.32와 0.21과 비교하여 약간 높은 값을 나타내었다. 액비 1.0N과 액비 0.5N + 요소 0.5N는 요소 1.0N과 동일한 수량을 만족하지만 2기작에 걸친 액비 1.0N 처리는 칼륨이 176 kg $ha^{-1}$이 추가적으로 토양에 공급되므로 장기간에 걸친 액비의 시용 시에는 칼륨이 과잉으로 집적될 수 있다. 따라서 지중관비를 이용한 시설오이 재배에서 질소 추천량에 대해서 반량은 액비로 공급하고 나머지 반량에 있어서는 화학비료로 시비하는 방법이 추천된다.

• 대한수학회보
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• 제51권6호
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• pp.1841-1850
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• 2014

It is known that the ranks of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $X_n={1,2,{\ldots},n}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\mathcal{SOP}_n$ and $\mathcal{SSPOP}_n$ are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$. Firstly, we characterize the structure of the minimal generating sets of $\mathcal{SOP}_n$. As applications, we obtain that the number of distinct minimal generating sets is $(n-1)^nn!$. Secondly, we show that, for $1{\leq}m{\leq}n-1$, the m-potent ranks of the semigroups $\mathcal{SOP}_n$ and $\mathcal{SPOP}_n$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\mathcal{SSPOP}_n$ is n + 1.

• 대한수학회보
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• 제49권4호
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• pp.693-704
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• 2012

We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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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.

• 충청수학회지
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• 제7권1호
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• pp.117-122
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• 1994

In this paper we show that if {$H_n$} is a continuous strong higher derivation of order n on an ultraprime Banach algebra with a constant c, then $c||H_1||^2{\leq}4||H_2||$ and for each $1{\leq}l$ < n $$c^2||H_1||\;||H_{n-l}{\leq}6||H_n||+\frac{3}{2}\sum_{\array{i+j+k=n\\i,j,k{\geq}1}}||H_i||\;||H_j||\;||H_k||+\frac{3}{2}\sum_{\array{i+k=n\\i{\neq}l,\;n-1}}||H_i||\;||H_k|| $$ and for a strong higher derivation {$H_n$} of order n on a prime ring A we also show that if [$H_n$(x),x]=0 for all $x{\in}A$ and for every $n{\geq}1$, then A is commutative or $H_n=0$ for every $n{\geq}1$.

• 대한수학회논문집
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• 제12권1호
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• pp.157-163
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• 1997

Based on a n-regular polygon $P_n$, we show that $r_n = 1/(2 \sum^{[(n-4)/4]+1}_{j=0}{cos 2j\pi/n)}$ is the ratio of contractions $f_i(1 \leq i \leq n)$ at each vertex of $P_n$ yielding a symmetric gasket $G_n$ associated with the just-touching I.F.S. $g_n = {f_i $\mid$ 1 \leq i \leq n}$. Moreover we see that for any odd n, the ratio $r_n$ is still valid for just-touching I.F.S $H_n = {f_i \circ R $\mid$ 1 \leq i \leq n}$ yielding another symmetric gasket $H_n$ where R is the $\pi/n$-rotation with respect to the center of $P_n$.

• 대한수학회지
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• 제53권3호
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• pp.549-582
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• 2016

All rings are commutative with $1{\neq}0$ and n is a positive integer. Let ${\phi}:{\Im}(R){\rightarrow}{\Im}(R){\cup}\{{\emptyset}\}$ be a function where ${\Im}(R)$ denotes the set of all ideals of R. We say that a proper ideal I of R is ${\phi}$-n-absorbing primary if whenever $a_1,a_2,{\cdots},a_{n+1}{\in}R$ and $a_1,a_2,{\cdots},a_{n+1}{\in}I{\backslash}{\phi}(I)$, either $a_1,a_2,{\cdots},a_n{\in}I$ or the product of $a_{n+1}$ with (n-1) of $a_1,{\cdots},a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of ${\phi}$-n-absorbing primary ideals.