• Korean Journal of Mathematics
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• v.22 no.1
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• pp.123-131
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• 2014

The purpose of this paper is to study the perturbation analysis of the matrix equation $X=I-A^*X^{-1}A+B^*X^{-1}B$. Based on the matrix differentiation, we give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the shrpness of the perturbation bound.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.327-342
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• 2018

In this paper, a homotopy continuation method for obtaining the unique Hermitian positive definite solution of the nonlinear matrix equation $X-{\sum_{i=1}^{m}}A^{\ast}_iX^{-p_i}A_i=I$ with $p_i$ > 1 is proposed, which does not depend on a good initial approximation to the solution of matrix equation.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.641-653
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• 2020

To extend the well-known extremal characterization of the geometric mean of two n × n positive definite matrices A and B, we solve the following problem: $${\max}\{X:X=X^*,\;\(\array{A&V&X\\V&B&W\\X&W&C}\){\geq}0\}$$. We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.669-677
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• 1996

Let $M_n$ be the $C^*$-algebra of all $n \times n$ matrices over the complex field, and $P[M_m, M_n]$ the convex cone of all positive linear maps from $M_m$ into $M_n$ that is, the maps which send the set of positive semidefinite matrices in $M_m$ into the set of positive semi-definite matrices in $M_n$. The convex structures of $P[M_m, M_n]$ are highly complicated even in low dimensions, and several authors [CL, KK, LW, O, R, S, W]have considered the possibility of decomposition of $P[M_m, M_n] into subcones.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.439-448
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• 1994

It is known that the steepest descent(SD) method and the conjugate gradient(CG) method [1, 2, 5, 6] converge when these methods are applied to solve linear systems of the form Ax = b, where A is symmetric and positive definite. For some finite difference discretizations of elliptic problems, one gets positive definite matrices that are almost symmetric. Practically, the SD method and the CG method work for these matrices. However, the convergence of these methods is not guaranteed theoretically. The SD method is also called Orthores(1) in iterative method papers. Elman [4] states that the convergence proof for Orthores($\kappa$), with $\kappa$ a positive integer, is not heard. In this paper, we prove that the SD method and the CG method converge when the $\iota$$^2$ matrix norm of the non-symmetric part of a positive definite matrix is less than some value related to the smallest and the largest eigenvalues of the symmetric part of the given matrix.(omitted)

• Korean Journal of Materials Research
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• v.25 no.12
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• pp.678-682
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• 2015

The electrical property of polymer matrix composites with added carbon powder is studied based on the temperature dependency of the conduction mechanism. The temperature coefficient of the resistance of the polymer matrix composites below the percolation threshold (x) changed from negative to positive at 0.20 < x < 0.21; this trend decreased with increasing of the percolation threshold. The temperature dependence of the electrical property(resistivity) of the polymer matrix composites below the percolation threshold can be explained by using a tunneling conduction model that incorporates the effect of the thermal expansion of the polymer matrix composites into the tunneling gap. The temperature coefficient of the resistance of the polymer matrix composites above the percolation threshold has a positive value; its absolute value increased with increasing volume fraction of carbon powder. By assuming that the electrical conduction through the percolating paths is a thermally activated process and by incorporating the effect of thermal expansion into the volume fraction of the carbon power, the temperature dependency of the resistivity above the percolation threshold can be well explained without violating the universal law of conductivity.

• Communications for Statistical Applications and Methods
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• v.25 no.1
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• pp.61-70
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• 2018

Modeling of the random effects covariance matrix in generalized linear mixed models (GLMMs) is an issue in analysis of longitudinal categorical data because the covariance matrix can be high-dimensional and its estimate must satisfy positive-definiteness. To satisfy these constraints, we consider the autoregressive and moving average Cholesky decomposition (ARMACD) to model the covariance matrix. The ARMACD creates a more flexible decomposition of the covariance matrix that provides generalized autoregressive parameters, generalized moving average parameters, and innovation variances. In this paper, we analyze longitudinal count data with overdispersion using GLMMs. We propose negative binomial loglinear mixed models to analyze longitudinal count data and we also present modeling of the random effects covariance matrix using the ARMACD. Epilepsy data are analyzed using our proposed model.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.521-533
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• 2019

In this paper, we define a mean of two positive numbers called the k-golden mean and study some properties of it. Especially, we show that the 2-golden mean refines the harmonic and the geometric means. As an application, we define the k-golden ratio and give some properties of it as an generalization of the golden ratio. Furthermore, we define the matrix k-golden mean of two positive-definite matrices and give some properties of it. This is an improvement of Lim's results [2] for which the matrix golden mean.

• Communications for Statistical Applications and Methods
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• v.28 no.2
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• pp.161-169
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• 2021

The two-dimensional confusion matrix used in credit assessment, biostatistics, and many other fields consists of true positive, true negative, false positive, and false negative. Their rates, such as the true positive rate (TPR), true negative rate (TNR), false positive rate, and false negative rate, can be applied to measure its accuracy. In this study, we propose the TPR-TNR plot, a graphical method that can geometrically describe and explain these rates based on the confusion matrix. The proposed TPR-TNR plot consists of two right-angled triangles. We obtain that the TPR and TNR describe the acute angles of right-angled triangles in the plot. These acute angles can be used to determine optimal thresholds corresponding to lots of accuracy measures.

• Kyungpook Mathematical Journal
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• v.28 no.2
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• pp.221-226
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• 1988

Using the method of eigenfunction expantion of self adjoint operators, we establish the necessary and sufficient conditions for a reflection positive generalized matrix kernel to be represented in an integral form. This integral converges in the cosidered spaces.