• 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.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.205-217
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• 2006

In this paper, we consider the problem of testing for a change of the parameter function ${\theta}(t)$ that may have a discontinuity at some unknown point ${\tau}$. We introduce a varying-h moving estimate to test the null hypothesis that ${\theta}(t)$ is continuous against the alternative that ${\theta}({\tau}-){\neq}{\theta}({\tau}+)$. Simulation results are provided for illustration.

• Transactions of the Korean Society of Mechanical Engineers
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• v.9 no.5
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• pp.548-554
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• 1985

This research is to find a means of reducing diesel engine combustion noise with none or minimum sacrifice of engine performance by investigating the influence of Cylinder Pressure Level(CPL). For this purpose, modified Wiebe's combustion function, considering the heat release curve as a combustion of both premixed and diffusive combustion portion, was exclusively used to obtain the indicator diagram and computer coeds were developed for the numerical analysis. Following are the results of this research. (1) CPL increases almostly with lag of ignition timing increasing .alpha. and decreasing. theta.$_{d}$, but at the crank angle with the maximal efficiency, CPL is independent of .alpha. and .theta.$_{d}$ with constant value of 200 dB especially at the low frequency. (2) For the constant ignition timing, the effects of .alpha. and .theta.$_{d}$ on CPL were the most significant at the frequency of about 1KHz and 300Hz respectively. (3) For the constant value of .alpha. and .theta.$_{d}$, CPL increases linearly with load but thermal efficiency increase very rapidly with maximum value of load Q$_{T}$=30-40 MJ/Kmol, then starts to decrease slowly. (4) The most effective way of reducing combustion noise without sacrificing thermal efficiency, is to decrease .alpha.. In the case of constant .alpha., there always exists a optimum value of .theta.$_{d}$ with respect to the various compression ratio.o..atio.o..

• Korean Journal of Mathematics
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• v.12 no.1
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• pp.15-22
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• 2004

In this paper we establish some recurrence relations satisfied by quotient moments of upper record values from the power distribution. Let {$X_n$, $n{\geq}1$} be a sequence of independent an identically distributed random variables with a common continuous distribution function(cdf) $F(x)$ and probability density function(pdf) $f(x)$. Let $Y_n=max\{X_1,X_2,{\cdots},X_n\}$ 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 $u(n)=min\{j{\mid}j>u(n-1),X_j>X_{u(n-1)},n{\geq}2\}$ and $u(1)=1$. Suppose $X{\in}POW(0,1,{\theta})$ then $$E\left(\frac{X^r_{u(m)}}{X^{s+1}_{u(n)}}\right)=\frac{\theta}{s}E\left(\frac{X^r_{u(m)}}{X^s_{u(n-1)}}\right)+\frac{(s-\theta)}{s}E\left(\frac{X^r_{u(m)}}{X^s_{u(n)}\right)\;and\;E\left(\frac{X^{r+1}_{u(m)}}{X^s_{u(n)}}\right)=\frac{\theta}{n+1}\left[E\left(\frac{X^{r+1}_{u(m-1)}}{X^s_{u(n+1)}}\right)-E\left(\frac{X^{r+1}_{u(m)}}{X^s_{u(n-1)}}\right)+\frac{r+1}{\theta}E\left(\frac{X^r_{u(m)}}{X^s_{u(n)}}\right)\right]$$.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.333-347
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• 2015

In this paper, by using the modular forms of weight nk ($2{\leq}n{\in}\mathbb{N}$ and $k{\in}\mathbb{Z}$), we construct a formula which generates modular forms of weight 2nk+4. This formula consist of some known results in [14] and [4]. Moreover, we obtain Fourier expansion of these modular forms. We also give some properties of an operator related to the derivative formula. Finally, by using the function $j_4$, we obtain the Fourier coefficients of modular forms with weight 4.

• Communications for Statistical Applications and Methods
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• v.5 no.3
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• pp.855-868
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• 1998

For a random sample of size n from general linear model, $Y_i= heta(X_i)+varepsilon_i,;let Y_{in}$ denote the ith oder statistics of the Y sample values. The X-value associated with $Y_{in}$ is denoted by $X_{[in]}$ and is called the concomitant of ith order statistics. The estimator of the location of a maximum of a regression function, $ heta$($\chi$), was proposed by (equation omitted) and was found the convergence rate of it under certain weak assumptions on $ heta$. We will discuss the asymptotic distributions of both $ heta(X_{〔n-r+1〕}$) and (equation omitted) when r is fixed as nolongrightarrow$\infty$(i.e. extreme case) on the basis of the theorem of the concomitants of order statistics. And the will investigate the asymptotic behavior of Max｛$\theta$( $X_{〔n-r+1:n〕/}$ ), . , $\theta$( $X_{〔n:n〕}$)｝as an estimator for the peak of a regression function.

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.471-487
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• 2020

We find congruences modulo 32, 64 and 128 for the partition function ${\overline{PP}_o}(n)$, the number of overpartition pairs of n into odd parts, with the aid of Ramamnujan's theta function identities and some known identities of t_k(n), for k = 6, 7, where t_k(n) denotes the number of representations of n as a sum of k triangular numbers. We also find two Ramanujan-like congruences for ${\overline{PP}_o}(n)$ modulo 128.

• International Journal of Reliability and Applications
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• v.14 no.2
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• pp.79-96
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• 2013

Exponential distribution plays a key role in engineering reliability and its applications. The exponential failure model has been studied for years. This article introduces two new preliminary test estimators for the reliability function (R(t)) in complete and censored samples from the exponential model with the use of a prior estimation (${\theta}_0$) of the mean (${\theta}$). The proposed preliminary test estimators are studied and compared numerically with the existing estimators. Computer-intensive calculations for bias and relative efficiency show that for, different values of levels of significance and for varying constants involved in the proposed estimators, the proposed estimators are far better than classical and existing estimators.

• The Korean Journal of Ecology
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• v.13 no.2
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• pp.83-92
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• 1990

Two-dimensional analysis provides a comprehensive description of the structure, scales of pattern and directional components in a spatial data set. In spectral analysisi, four functions are illustrated,; the autocorrelation, the periodogram, the R-spectrum and the $\theta$ -spectrum. The R-spectrum and $\theta$ -spectrum function respectively summarize the periodogram in term of scale of pattern and directional components. Sampling is measured in the Naejang National Park area where the Daphniphyllum trees grow. 320 contiguous (15$\times$15)m plots are located along the transect and density of all trees over DBH 3 cm recorded respectively. 12 species of vascular plant are recorded in this survey area. The trend surface of density of all plant are estimated using polynomial regression and are exhibited in 3-dimensional graph and density contour map. Transformation to the corresponding polar spectrum from the periodogram emphasized the directional components and the scales to pattern. R-spectrum corresponding to the scale of pattern of periodogram showed a large peak 15.47 in the interval 9$\theta$-spectrum corresponding to directional components have two peaks 8.28 and 11.05 in the interval $35^{\circ}\theta <45^{\circ}and 125^{\circ}\theta< <135^{\circ}, respectively. Programs to compute all the analyses described in this study was obtained from Dr. Ranshow and was translated to BASIC by the author.

• Journal of Microbiology and Biotechnology
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• v.3 no.4
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• pp.292-297
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• 1993

Aspergillus niger No. PFST-38 was grown on complex media in 30L agitated fermentors at various aeration rates and stirrer speeds. We could correlate the mixing time as a function of the Reynolds number and the apparent viscosity, as follows. ${\theta}_M=2.95\;\NRe^{-0.52},\;{\theta}_M=1.88\;{\eta_a}^{0.57}$ Also, the effects of the apparent viscosity (${\theta}_a$), the impeller rotational speed (N), the air flow rate ($V_s$), and the mixing time (${\theta}_M$) on the oxygen transfer coefficient, $K_L a$ were determined experimentally, and equated as follows. $K_La=12.04N^{0.88}Vs^{0.71}{n_a}^{-0.83},\;K_La=30.2N^{0.88}Vs^{0.71}{\theta_M}^{-1.45}$ $K_La$ increased as the agitation speed and the air flow rate increased. The rate of $K_La$ increase was dependent more on the rotational speed of impeller than on the air flow rate. The glucoamylase production increased with the increase of the agitation speed upto at 500 rpm and increased with the increase of air flow rate upto at 1.0 vvm. The values calculated from the above equation confirmed that the experimental maximum production of glucoamylase was achieved when the $K_La$ and the apparent viscosity of the broth were $260\;hr^{-1}$ and 1800 cps, respectively.