• Korean Journal of Mathematics
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• v.31 no.2
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• pp.145-152
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• 2023

Let $P(z):=\sum\limits^{n}_{j=0}A_jz^j$, A_j ∈ ℂ^m×m, 0 ≤ j ≤ n be a matrix polynomial of degree n, such that A_n ≥ A_n-1 ≥ . . . ≥ A₀ ≥ 0, A_n > 0. Then the eigenvalues of P(z) lie in the closed unit disk. This theorem proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007), 2151-2153] is infact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Eneström-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to Lê, Du, Nguyên [Oper. Matrices, 13(2019), 937-954] besides a matrix extention of a result proved by Mohammad [Amer. Math. Monthly, vol.74, No.3, March 1967].

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.247-260
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• 2011

Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if $f^m$ is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type $$f^n+P(f)=be^{sz}+Q(e^z)$$, where P(f) is a differential polynomial in f of degree at most n-1, and Q($e^z$) is a polynomial in $e^z$ of degree k $\leqslant$ max {n-1, s(n-1)/n} with small functions of $e^z$ as its coefficients.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.855-860
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• 2005

For the accurate measurement of acoustic properties of a surface, efforts have been made to reduce errors caused by external disturbance. If the reflection coefficient is considered as a transfer function between reflected wave and incident wave, causality is required between them and the reflection coefficient should be of minimum phase. In this thesis, the minimum phase condition is applied to measure correct reflection coefficient. The reflection coefficient is approximated as a rational function in the Z domain by minimizing the sum square error. Then the minimum phase reflection coefficient is reconstructed using the distribution of poles and zeros of the reflection coefficient model. The incident wave, the reflected wave and the impulse response function of causality are recalculated from the minimum phase reflection coefficient for further applications.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.10 s.103
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• pp.1131-1136
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• 2005

For the accurate measurement of acoustic properties of a surface, efforts have been made to reduce errors caused by external disturbance. If the reflection coefficient is considered as a transfer function between reflected wane and incident wave, the causality is required between them and the reflection coefficient should be of minimum phase. In this thesis, the minimum phase condition is applied to measure correct reflection coefficient. The reflection coefficient is approximated as a rational function In the Z domain by minimizing the sum square error. Then the minimum phase reflection coefficient is reconstructed using the distribution of poles and zeros of the reflection coefficient model.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.155-166
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• 2022

In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of f(z)ⁿL(g) - a(z) and g(z)ⁿL(f) - a(z), where L(h) takes the derivatives h^(k)(z) or the shift h(z+c) or the difference h(z+c)-h(z) or the delay-differential h^(k)(z+c), where k is a positive integer, c is a non-zero constant and a(z) is a nonzero small function with respect to f(z) and g(z). The related uniqueness problems of complex delay-differential polynomials are also considered.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.369-380
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• 2009

Let $X,X_1,X_2,\;{\cdots}$ be identically distributed and negatively associated random variables with mean zeros and positive, finite variances. We prove that, if $E{\mid}X_1{\mid}^r$ < ${\infty}$, for 1 < p < 2 and r > $1+{\frac{p}{2}}$, and $lim_{n{\rightarrow}{\infty}}n^{-1}ES^2_n={\sigma}^2$ < ${\infty}$, then $lim_{{\epsilon}{\downarrow}0}{\epsilon}^{{2(r-p}/(2-p)-1}{\sum}^{\infty}_{n=1}n^{{\frac{r}{p}}-2-{\frac{1}{p}}}E\{{{\mid}S_n{\mid}}-{\epsilon}n^{\frac{1}{p}}\}+={\frac{p(2-p)}{(r-p)(2r-p-2)}}E{\mid}Z{\mid}^{\frac{2(r-p)}{2-p}}$, where $S_n\;=\;X_1\;+\;X_2\;+\;{\cdots}\;+\;X_n$ and Z has a normal distribution with mean 0 and variance ${\sigma}^2$.

• The Korean Journal of Applied Statistics
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• v.30 no.4
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• pp.529-538
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• 2017

The Ghosh and Kim zero-altered distribution model is used to analyze count data that have too many or too few zeros. The dispersion type parameter ${\delta}$ in the zero-altered distribution model consists of mean, variance and zero probability and has two forms depending on the relation between ${\mu}$ and ${\sigma}^2$. We derived the influence function on ${\delta}$ when ${\sigma}^2{\geq}{\mu}$. To show the validity of the influence function, we used the Census data on the number of births of married women in Korea to compare the estimated changes in ${\delta}$ using this function with those obtained using the direct deletion method. The result proved that the obtained influence function is very accurate in estimating changes in ${\delta}$ when an observation is deleted.

• The Korean Journal of Applied Statistics
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• v.35 no.2
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• pp.311-325
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• 2022

For count responses, the situation of excess zeros often occurs in various research fields. Zero-inflated model is a common choice for modeling such count data. Bayesian inference for the zero-inflated model has long been recognized as a hard problem because the form of conditional posterior distribution is not in closed form. Recently, however, Pillow and Scott (2012) and Polson et al. (2013) proposed a Pólya-Gamma data-augmentation strategy for logistic and negative binomial models, facilitating Bayesian inference for the zero-inflated model. We apply Bayesian zero-inflated negative binomial regression model to longitudinal pharmaceutical data which have been previously analyzed by Min and Agresti (2005). To facilitate posterior sampling for longitudinal zero-inflated model, we use the Pólya-Gamma data-augmentation strategy.

• Journal of Korean Society of Disaster and Security
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• v.13 no.4
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• pp.25-36
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• 2020

An increasing frequency and intensity of natural disasters have been observed due to climate change. To better prepare for these, the MOIS (ministry of the interior and safety) announced a comprehensive plan for minimizing damages associated with natural disasters, including drought and heavy snowfall. The spatial-temporal pattern of snowfall is greatly influenced by temperature and geographical features. Heavy snowfalls are often observed in Gangwon-do, surrounded by mountains, whereas less snowfall is dominant in the southern part of the country due to relatively high temperatures. Thus, snow depth data often contains zeros that can lead to difficulties in the selection of probability distribution and estimation of the parameters. A generalized mixture distribution approach to a maximum snow depth series over the southern part of Korea (i.e., Changwon, Tongyeoung, Jinju weather stations) are located is proposed to better estimate a threshold (𝛿) classifying discrete and continuous distribution parts. The model parameters, including the threshold in the mixture model, are effectively estimated within a Bayesian modeling framework, and the uncertainty associated with the parameters is also provided. Comparing to the Daegwallyeong weather station, It was found that the proposed model is more effective for the regions in which less snow depth is observed.

• Earthquakes and Structures
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• v.14 no.4
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• pp.313-322
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• 2018

The present study is dedicated to investigate the SH body-as well as Love-waves propagation effects in porous media with uncertain porosity and permeability. A unified formulation of the governing equations for one-dimensional (1-D) wave propagation in anisotropic porous layered media is presented deterministically. The uncertainties around the above two cited parameters are taken into account by random fields with the help of Monte Carlo Simulations (MCS). Random samples of the porosity and the permeability are generated according to the normal and lognormal distribution functions, respectively, with a mean value and a coefficient of variation for each one of the two parameters. After performing several thousands of samples, the mathematical expectation (mean) of the solution of the wave propagation equations in terms of amplification functions for SH waves and in terms of dispersion equation for Love-waves are obtained. The limits of the Love wave velocity in a porous soil layer overlaying a homogeneous half-space are obtained where it is found that random variations of porosity change the zeros of the wave equation. Also, the increase of uncertainties in the porosity (high coefficient of variation) decreases the mean amplification function amplitudes and shifts the fundamental frequencies. However, no effects are observed on both Love wave dispersion and amplification function for random variations of permeability. Lastly, the present approach is applied to a case study in the Adapazari town basin so that to estimate ground motion accelerations lacked in the fast-growing during the main shock of the damaging 1999 Kocaeli earthquake.