• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.111-121
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• 2006

All d-homogeneous $C^*$-algebras $T^d$ over $\prod^{s_4}S^4{\times}\prod^{s_2}S^2{\times}\prod^{s_3}S^3{\times}\prod^{s_1}S^1$ are constructed. It is shown that $T^d$ are strongly Morita equivalent to $C(\prod^{s_4}S^4{\times}\prod^{s_2}S^2{\times}\prod^{s_3}S^3{\times}\prod^{s_1}S^1)$.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 1991.10a
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• pp.7-9
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• 1991

$\prod$-A characteristics of N-docosylquinolium-TCNQ complex were investigated with the following variations. I> $\prod$-A characteristics with a variation of spreading amount. ii> $\prod$-A characteristics with a variation of barrier speed. iii> $\prod$-A characteristics with a variation of subphase temperature. An optimum surface pressure for a deposition of Langmuir-Blodgett(LB) layer was found to be a 40∼50 mN/m in pure water subphase (pH 5.4, $25^{\circ}C$).

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.107-110
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• 1996

Usng the Pochhammer symbol $(\lambda)_n$ given by $$ (1.1) (\lambda)_n = {1, if n = 0 {\lambda(\lambda + 1) \cdots (\lambda + n - 1), if n \in N = {1, 2, 3, \ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;\upsilon}^{p:r;u} [\alpha_1, \ldots, \alpha_p : \gamma_1, \ldots, \gamma_r; \lambda_1, \ldots, \lambda_u;_{x,y}][\beta_1, \ldots, \beta_q : \delta_1, \ldots, \delta_s; \mu_1, \ldots, \mu_v; ] = \sum_{l,m = 0}^{\infty} \frac {\prod_{j=1}^{q} (\beta_j)_{l+m} \prod_{j=1}^{s} (\delta_j)_l \prod_{j=1}^{v} (\mu_j)_m)}{\prod_{j=1}^{p} (\alpha_j)_{l+m} \prod_{j=1}^{r} (\gamma_j)_l \prod_{j=1}^{u} (\lambda_j)_m} \frac{l!}{x^l} \frac{m!}{y^m} $$ provided that the double series converges.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.381-403
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• 2021

Consider the three-dimensional system of difference equations $x_{n+1}=\frac{{\prod_{j=0}^{k}}z_n-3j}{{\prod_{j=1}^{k}}x_n-(3j-1)\;\(a_n+b_n{\prod_{j=0}^{k}}z_n-3j\)}$, $y_{n+1}=\frac{{\prod_{j=0}^{k}}x_n-3j}{{\prod_{j=1}^{k}}y_n-(3j-1)\;\(c_n+d_n{\prod_{j=0}^{k}}x_n-3j\)}$, $z_{n+1}=\frac{{\prod_{j=0}^{k}}y_n-3j}{{\prod_{j=1}^{k}}z_n-(3j-1)\;\(e_n+f_n{\prod_{j=0}^{k}}y_n-3j\)}$, n ∈ ℕ₀, where k ∈ ℕ₀, the sequences $(a_n)_{n{\in}{\mathbb{N}}_0$, $(b_n)_{n{\in}{\mathbb{N}}_0$, $(c_n)_{n{\in}{\mathbb{N}}_0$, $(d_n)_{n{\in}{\mathbb{N}}_0$, $(e_n)_{n{\in}{\mathbb{N}}_0$, $(f_n)_{n{\in}{\mathbb{N}}_0$ and the initial values x_-3k, x_-3k+1, …, x₀, y_-3k, y_-3k+1, …, y₀, z_-3k, z_-3k+1, …, z₀ are real numbers. In this work, we give explicit formulas for the well defined solutions of the above system. Also, the forbidden set of solution of the system is found. For the constant case, a result on the existence of periodic solutions is provided and the asymptotic behavior of the solutions is investigated in detail.

• Honam Mathematical Journal
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• v.1 no.1
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• pp.21-25
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• 1979

Abelian군(群)의 presheaf에 관한 직적(直積)과 직화(直和)를 Category 입장에서 정의(定義)하고 presheaf $F_{\lambda}\;({\lambda}{\epsilon}{\Lambda})$들의 두 직적(直積)(또는 直和)은 서로 동형적(同型的) 관계(關係)에 있으며, 특히 ${\phi}:X{\rightarrow}Y$가 homeomorphism이라 하고 ${\phi}_*F$를 X상(上)의 presheaf F의 direct image이라 하면 (1) $({\phi}_*F, \;{\phi}_*(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직적(直積)일 때 오직 그때 한하여 $(F,\;(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직적(直積)이다. (2) $({\phi}_*F,\;{\phi}_*(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직화(直和)일 때 오직 그때 한하여 $(F,\;(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직화(直和)이다. Let $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ be an indexed set of presheaves of abelian group on topological space X. We can define the cartesian product $$\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda}$$ of $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ by $$(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U)=\prod_{{\lambda}{\epsilon}{\Lambda}}(F_{\lambda}(U))$$ for U open in X $${\rho}_v^u:\;(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U){\rightarrow}(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(V)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}(_{\lambda}{\rho}_v^u(s_{\lambda}))_{{\lambda}{\epsilon}{\Lambda}})$$ for $V{\subseteq}U$ open in X where $_{\lambda}{\rho}^U_V$ is a restriction of $F_{\lambda}$, And we have natural presheaf morphisms ${\pi}_{\lambda}$ and ${\iota}_{\lambda}$ such that ${\pi}_{\lambda}(U):\;({\prod}_\;F_{\lambda})(U){\rightarrow}F_{\lambda}(U)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}s_{\lambda})$ ${\iota}_{\lambda}(U):\;F_{\lambda}(U){\rightarrow}({\prod}\;F_{\lambda})(U)(s_{\lambda}{\rightarrow}(o,o,{\cdots}\;{\cdots}o,s_{\lambda},o,{\cdots}\;{\cdots}o)$ for $(s_{\lambda}){\epsilon}{\prod}_{\lambda}\;F_{\lambda}(U)$ and $(s_{\lambda}){\epsilon}F_{\lambda}(U)$.

• Bulletin of the Korean Chemical Society
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• v.35 no.5
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• pp.1397-1402
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• 2014

The complete active space self-consist field method followed by the internally contracted multireference configuration interaction method has been used to compute the potential energy curves of $X^2\prod_g$, $a^4\prod_u$, $A^2\prod_u$, $b^4\sum_{g}^{-}$, and $B^2\sum_{g}^{-}$ states of $S{_2}^+$ cation with large correlation-consistent basis sets. Utilizing the potential energy curves computed with different basis sets, the spectroscopic parameters of these states were evaluated. Finally, the transition dipole moment and the Franck-Condon factors of the transition from $A^2\prod_u$ to $X^2\prod_g$ were evaluated. The radiative lifetime of $A^2\prod_u$ is calculated to be 887 ns, which is in good agreement with experimental value of $805{\pm}10$ ns.

• Proceedings of the KIEE Conference
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• 1991.11a
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• pp.333-335
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• 1991

${\prod}-A$ characteristics and UV-visible spectrum of N-docosylquinolium-TCNQ monolayers were investigated. ${\prod}-A$ characteristics of N-docosylquinolium-TCNQ monolayer were measured following variation. i > ${\prod}-A$ characteristics with a variation of spreading amount. ii > ${\prod}-A$ characteristics with a variation of subphase temperature. An optium surface pressure for a LB Film deposition was found to be a 30, 45 mN/m in pure water subphase(pH 5.4, $25^{\circ}C$). UV-visible spectrum Analysis of these LB film were investigated. As a result, we think two kinds of surface pressure had no effect the monolayer aggregation but monolayer density.

• Bulletin of the Korean Chemical Society
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• v.35 no.5
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• pp.1422-1432
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• 2014

Convergent all-electron multi-reference configuration-interaction (MRCI) calculations are performed for a lithium dimer with Kaufmann's Rydberg basis functions. A comparison of the results of these calculations with those of the effective core potential/core polarization potential (ECP/CPP) method and experimental data reveals the deficiency of the all-electron ab initio method. The deficiency is related to the mere 51.9% attainment of electron correlation for the ground state. The percent attainment of electron correlation for the first excited state is slightly better than that for the ground state, preventing us from obtaining better agreements with experimental data by means of increasing the size of basis sets. The Kaufmann basis functions are then used with the ECP/CPP method to obtain the accurate convergent potential energy curves for the $^1\prod_u$ states correlated to Li(2p) + Li(2p) and Li(2s) + Li(n = 2, 3, 4). Quantum defect curves (QDCs) calculated for both the $X^2\sum_g$ and 1 $^2\prod_u$ states of the $Li{_2}^+$ ion and the Lu-Fano plot reveal a strong series-series interaction between the two $2snp{\pi}$ and $2pnp{\pi}$ Rydberg series. The QDCs are then used to resolve assignment problems in the literature. The reassignments, performed by Jedrzejewski-Szemek et al., of the dissociation product of the D $^1\prod$ state from (2s+3d) to (2s+3p) and that of the 6 $^1\prod_u$ from (2s+4d) to (2s+4p) are found to be incorrect. It may be more natural to assign their $2snp{\pi}$ Rydberg series as a $2snd{\pi}$ series. The state, assigned as 5p $^1\prod_u$ by Ross et al. and 4d $^1\prod$ by Jedrzejewski-Szemek et al., is assigned as the 7 $^1\prod_u$ state, correlated to the Li(2s) + Li(4f) limit.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.21-28
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• 2016

It is known that no two of the roots of the polynomial equation (1) $$\prod\limits_{l=1}^{n}(x-r_l)+\prod\limits_{l=1}^{n}(x+r_l)=0$$, where 0 < $r_1{\leq}r_2{\leq}{\cdots}{\leq}r_n$, can be equal and all of its roots lie on the imaginary axis. In this paper we show that for 0 < h < $r_k$, the roots of $$(x-r_k+h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x-r_l)+(x+r_k-h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x+r_l)=0$$ and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis.

• Bulletin of the Korean Chemical Society
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• v.23 no.2
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• pp.179-183
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• 2002

Resonance-enhanced multiphoton ionization with time-of-flight product imaging of the $N(^2D)$ atoms has been used to study the $N_2O$ photodissociation at 118.2 nm and the two-photon dissociation at 268.9 nm. These imaging experiments allowed the determination of the total kinetic energy distribution of the $NO(X^2{\prod})$ and $N(^2D_{5/2})$ products. The $NO(X^2{\prod})$ fragments resulting from the photodissociation processes are produced in highly vibrationally excited states. The two-photon photodissociation process yields a broad $NO(X^2{\prod})$ vibrational energy distribution, while the 118.2 nm dissociation appears to produce a vibrational distribution sharply peaked at $NO(X^2{\prod},\;{\nu}=14)$.