• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.967-986
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• 2017

We give the necessary condition for an operator defined on a cartesian product of $c_0(\mathcal{X})$ spaces to be summing or dominated and we show that for the multiplication operators this condition is also sufficient. By using these results, we show that ${\Pi}_s(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_s(l^m_2{\mid}m{\in}\mathbb{N})$ for s > 2 or a copy of $1_s(l^m_1{\mid}{\in}\mathbb{N})$, for any $l{\leq}S$ < ${\infty}$. Also ${\Delta}_{s_1,{\ldots},s_n}(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}$ if ${\upsilon}_n(s_1,{\ldots},s_n){\leq}2$ or a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}(l^m_2{\mid}m{\in}\mathbb{N})$ if 2 < ${\upsilon}_n(s_1,{\ldots},s_n)$, where ${\frac{1}{{\upsilon}_n(s_1,{\ldots},s_n})}={\frac{1}{s_1}}+{\cdots}+{\frac{1}{s_n}}$. We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be 1-summing or 2-dominated.

• Korean Journal of Materials Research
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• v.25 no.3
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• pp.138-143
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• 2015

Aluminum nitride, a compound semiconductor, has a Wurtzite structure; good material properties such as high thermal conductivity, great electric conductivity, high dielectric breakdown strength, a wide energy band gap (6.2eV), a fast elastic wave speed; and excellent in thermal and chemical stability. Furthermore, the thermal expansion coefficient of the aluminum nitride is similar to those of Si and GaAs. Due to these characteristics, aluminum nitride can be applied to electric packaging components, dielectric materials, SAW (surface acoustic wave) devices, and photoelectric devices. In this study, we surveyed the crystallization and preferred orientation of AlN thin films with an X-ray diffractometer. To fabricate the AlN thin film, we used the magnetron sputtering method with $N_2$, NH3 and Ar. According to an increase in the partial pressures of $N_2$ and $NH_3$, Al was nitrified and deposited onto a substrate in a molecular form. When AlN was fabricated with $N_2$, it showed a c-axis orientation and tended toward a high orientation with an increase in the temperature. On the other hand, when AlN was fabricated with $NH_3$, it showed a-axis orientation. This result is coincident with the proposed mechanism. We fabricated AlN thin films with an a-axis orientation by controlling the sputtering electric power, $NH_3$ pressure, deposition speed, and substrate temperature. According to the proposed mechanism, we also fabricated AlN thin films which demonstrated high a-axis and c-axis orientations.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.119-128
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• 2010

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_n}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1], [4], [2], [3], [5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_n}$, in cases when the $N_n$, $n\geq1$ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.341-349
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• 2003

Let {$x_{nk}\;$\mid$1\;\leq\;k\;\leq\;n,\;n\;\geq\;1$} be an array of random varianbles and $\{a_n$\mid$n\;\geq\;1\}\;and\;\{b_n$\mid$n\;\geq\;1} be a sequence of constants with $a_n\;>\;0,\;b_n\;>\;0,\;n\;\geq\;1. In this paper, for array of row negatively associated(NA) random variables, we establish a general weak law of large numbers (WLLA) of the form (${\sum_{\kappa=1}}^n\;a_{\kappa}X_{n\kappa}\;-\;\nu_{n\kappa})\;/b_n$ converges in probability to zero, as $n\;\rightarrow\;\infty$, where {$\nu_{n\kappa}$\mid$1\;\leq\;\kappa\;\leq\;n,\;n\;\geq\;1$} is a suitable array of constants.

• Archives of Pharmacal Research
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• v.17 no.1
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• pp.39-41
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• 1994

Various N,N'-disubstituted ureas 5 were easily prepared from the corresponding primary amide 1 by tratment with N-Bromophthalimide $(NBP)-AgOAc-RNH$_{2}$ 4 in dry N,N-dimrthylformamide (DMF). This reaction envolved the intemediate formation of isocyanate 3 from amide 1 via Hofmann rearrangement by treatment with AgOAc and NBP and nucleophilic addition of amine 4 to this isocyante 3. This method is simple enough to be applied to the synthesis of various N,N'-disubstututed ureas scale conviently.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.279-286
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• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.8 no.2
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• pp.1-5
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• 2004

A multiplicative plane partition is a two-dimensional array of positive integers larger than 1 that are nonincreasing both from left to right and top to bottom and whose multiple is a given number n. For a natural number n, let $f_2(n)$ be the number of multiplicative plane partitions of n. In this paper, we prove $f_2(n)\;{\leq}\;n^2$ and a table of them up to $10^5$ is provided.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.377-384
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• 1998

Let ${X_n,n \geq 1}$ be a sequence of stochatically dominated pairwise independent random variables. Let ${a_n, n \geq 1}$ and ${b_n, n \geq 1}$ be seqence of constants such that $a_n \neq 0$ and $0 < b_n \uparrow \infty$. A strong law large numbers of the form $\sum^{n}_{j=1}{a_j X_i//b_n \to 0$ almost surely is obtained.

• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.61-65
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• 2008

In this paper we prove that every positive solution of the third order rational difference equation $$x_{n+1}\;=\;\frac{{\beta}x_n\;+\;x_{n-2}}{A\;+\;Bx_n\;+\;x_{n-2}}$ converges to the positive equilibrium point $$\bar{x}\;=\;\frac{{\beta}\;+\;1\;-\;A}{B\;+\;1}$, where $0\;<\;{\beta}\;{\leq}\;B$, $1\;<\;A\;<\;{\beta}\;+\;1$

• Proceedings of the Korean Society of Machine Tool Engineers Conference
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• 2002.04a
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• pp.655-659
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• 2002

This study is performed to develop the Si$_3$N$_4$ based ceramic inserts. Si$_3$N$_4$with addition of SiC and A1$_2$O$_3$ is investigated to determine the possibility to be a new tool. The tool life of Si$_3$N$_4$ insert with more than 20wt% SiC is shorter than commercial Si$_3$N$_4$ insert during machining both heat treated SCM440 and gray cast iron. Even though SiC has higher hardness than Si$_3$N$_4$, its chemical affinity to the iron on high temperature may causes deteriorat ion of tool life. To the contrary, Si$_3$N$_4$insert with A1$_2$O$_3$ shows increase of tool life up to 300% compared to the commercial Si$_3$N$_4$insert. It may attribute to the high temperature stability of A1$_2$O$_3$. Further study will be focused on the optimization of ceramic inserts with the composition of Si$_3$N$_4$and A1$_2$O$_3$.