• Journal of the Korean Mathematical Society
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• v.42 no.2
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• pp.203-222
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• 2005

Thompson series is a Hauptmodul for a genus zero group which lies between $\Gamma$o(N) and its normalizer in PSL2(R) ([1]). We construct explicit ring class fields over an imaginary quadratic field K from the Thompson series $T_g$($\alpha$) (Theorem 4), which would be an extension of [3], Theorem 3.7.5 (2) by using the Shimura theory and the standard results of complex multiplication. Also we construct various class fields over K, over a CM-field K (${\zeta}N + {\zeta}_N^{-1}$), and over a field K (${\zeta}N$). Furthermore, we find an explicit formula for the conjugates of Tg ($\alpha$) to calculate its minimal polynomial where $\alpha$ (${\in}{\eta}$) is the quotient of a basis of an integral ideal in K.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.47-59
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• 2007

We find the uniformizers of modular curves $X_{1}(N)\;(N=2,3)$ and explore the relationship with Thompson series and number theoretic property.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.707-723
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• 1999

Since the modular curve $X(4)=\Gamma(4)/{\mathfrak{}}^*$ has genus 0, we have a field isomorphism K(X(4)){\approx}\mathcal{C}(j_{4})$ where $j_{4}(z)={\theta}_{3}(\frac{z}{2})/{\theta}_{4}(\frac{z}{2})$ is a quotient of Jacobi theta series ([9]). We derive recursion formulas for the Fourier coefficients of $j_4$ and $N(j_{4})$ (=the normalized generator), respectively. And we apply these modular functions to Thompson series and the construction of class fields.

• Proceedings of the Korean Statistical Society Conference
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• 2004.11a
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• pp.155-160
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• 2004

불균등 확률 계통추출에서는 모집단 총합에 대한 Horvitz-Thompson 추정량의 대안적 분산 추정량들을 사용하게 된다. 이와 같은 모총합에 관한 분산 추정량들의 설계와 관련한 일반적인 방법은 균등 확률 계통추출에 대한 분산 추정량들에서 시작하고 비율 $y_i,/P_i$에 의한 추정량의 정의에서 $y_i$를 재배치하게 한다. 비선형 조사 통계학에서 추정량들 중의 하나로 테일러 급수 공식을 적용한다. 불균등 확률 계통추출에서의 분산은 8가지 방법으로 추정이 가능하므로 이를 이용한 분산추정량을 구해보고, 비복원 불균등 확률에서의 jackknife방법을 살펴보고자 한다. 또한 이들 분산추정량들에 대한 비교를 몇 가지 방법을 이용하여 알아보도록 한다.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.641-645
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• 2013

In this article, we show that the modular equation ${\Phi}_p^{T_q}(X,Y)$ of Thompson series $T_q$ corresponding to ${\Gamma}_0(q)+q$ gives an affine model of a modular curve $X_0(pq)+q$.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.709-715
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• 2008

We construct certain ring class fields over an imaginary quadratic field by making use of Shimura's canonical models and extend the result of Chen-Yui ([1] Theorem 3.7.5(2)) to the case where (a, b, N) $\neq$ N or (a/N, N) $\neq$ 1 for a positive integer N > 1.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.343-371
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• 2007

We find the super-replication formulae which would be a generalization of replication formulae. And we apply the formulae to derive periodically vanishing property in the Fourier coefficients of the Hauptmodul $\aleph(j_{1,N})$ as a super-replicable function.

• Journal of Biosystems Engineering
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• v.21 no.1
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• pp.72-83
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• 1996

This study was performed to find out drying characteristics and develop drying model for the design of an efficient dryer or drying system of garlic. The basic model which describes drying phenomenon of garlic was first established. A series of drying test were conducted with two varieties of garlic(Uiseong, Namdo) at 9-different drying conditions (drying temperatures ; $40^{\circ}C$, $50^{\circ}C$, $60^{\circ}C$, relative humidities ; 20%, 35%, 50%) and statistical analysis was made to fit the data with exponential equation, approximated diffusion equation, page equation, thompson equation and wang equation, respectively. In this test, the effects of drying air temperature and relative humidity on the drying rate were undertaken. Finally, new drying model based on these experimental results was developed to describe the drying characteristics of garlic. Also, the volatile components of garlic extracts were investigated. For experiment both Uisoeng and Namdo garlic were dried by heated-air-drying, followed by ether extraction. The extracts were analysed by Gas chromatography/Mass spectrometer.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.379-383
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• 2007

We show that the modular equation ${\phi}^{T_n}_m$ (X, Y) for the Thompson series $T_n$ corresponding to ${\Gamma}_0$(n) gives an affine model of the modular curve $X_0$(mn) which has better properties than the one derived from the modular j invariant. Here, m and n are relative prime. As an application, we construct a ring class field over an imaginary quadratic field K by singular values of $T_n(z)\;and\;T_n$(mz).

• Applied Microscopy
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• v.33 no.2
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• pp.93-104
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• 2003

Since the Ptolemaeos' discovery that glass has magnifying power, human desire to see the unseen with naked eyes has lead to the inventions of a series of microscopes. Since the Janssen's first compound microscope in 1595, through the Abbe's non-aberration microscopy, various microscopes using different principles are now being used in various biomedical researches. The discovery of electron by Thompson in 1897 has lead to the first invention of microscope using electron as an illumination source, the electron microscope, in 1931. Now we can see the objects as close as 0.05 nm using 1 MV FE-TEM constructed in 2000. In this review, the authors reviewed the predecessors efforts to develop better microscopes.