• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.763-770
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• 2006

Let X be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor K with every very general curve is positive (K is almost numerically positive) then every very general proper subvariety of X is of general type in ';he viewpoint of geometric Kodaira dimension. We note that the converse does not hold for simple abelian varieties.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1347-1370
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• 2016

On a closed eta-Einstein Sasakian spin manifold of dimension $2m+1{\geq}5$, $m{\equiv}0$ mod 2, we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair (${\varphi}_{{\frac{m}{2}}-1}$, ${\varphi}_{\frac{m}{2}}$) of spinor fields (called Sasakian duo, see Definition 2.1) that solves a special system of two differential equations.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1539-1561
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• 2021

We prove Perelman type 𝒲-entropy formulae and differential Harnack estimates for positive solutions to weighed doubly nonlinear diffusion equation on weighted Riemannian manifolds with CD(-K, m) condition for some K ≥ 0 and m ≥ n, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1129-1135
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• 2011

In this note, we prove that every subscalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent subscala operator is nilpotent. We also prove that every subscalar operator with property (${\delta}$) on a Banach space of dimension greater than 1 has a nontrivial invariant closed linear subspace.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.911-920
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• 2019

In this paper we characterize semiconformally flat spacetimes and a spacetime with harmonic semiconformal curvature tensor. At first in a semiconformally flat perfect fluid spacetime we obtain a state equation and prove that in particular for dimension n = 4, the spacetime represents a model for incoherent radiation. Next we prove that perfect fluid spacetime with harmonic semiconformal curvature tensor is of Petrov type I, D or O and the spacetime is a GRW spacetime. As a consequence we obtain several corollaries.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.611-622
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• 2023

In this paper, we prove that every weakly Einstein slope metric, which is conformally flat on a manifold M of dimension n ≥ 3, is either a locally Minkowski metric or a Riemannian metric. We also prove the same result for conformally flat weakly Einstein Kropina metrics.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.157-171
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• 2014

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let $E_{{\alpha},{\beta}}$ be the set of points with lower and upper recurrence rates ${\alpha},{\beta}$, ($0{\leq}{\alpha}{\leq}{\beta}{\leq}{\infty}$), we prove that all the sets $E_{{\alpha},{\beta}}$, are of full Hausdorff dimension. Then the recurrence sets $E_{{\alpha},{\beta}}$ have constant multifractal spectra.

• Proceedings of the Korean Society of Machine Tool Engineers Conference
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• 2003.04a
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• pp.255-259
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• 2003

Moire topography method isa well-known non-contacting 3-D measurement method as afast non-contact test for three-dimension shape measuring method. Recently, it's important to study the automatic three-dimension measurement by moire topography because it is frequently applied to the reverse engineering , the medical , the entertainment fields. Three-dimension measurement using projection of moire topography is very attractive because of its high measuring speed and high sensitivity. In this paper, the classical moire method is computerized-so called digital moire when a virtual grating pattern is projected on a surface, the captured image by the CCD camera has three-dimension information of the objects. The moire image can be obtained through a simple image processing and a reference grating pattern. and it provides similar results without physical grating pattern. digital projection moire topography turn out to be very effective for the three-dimension measurement of objects. Using different N-bucket algorithm method of digital projection moire topography is tested to measuring object with the 2-ambiguity problem. Experimental results prove that the proposed scheme is capable of finding measurement errors that decreased more by using the four-three step algorithm method instead of the same step in the phase shifting of different pitch.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.961-976
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• 2019

Let R be any commutative ring and S be any multiplicative closed set. We introduce an S-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as R is noetherian if and only if $R_S$ is noetherian and every $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension of R, and prove that $R_S$ is noetherian if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is zero; $R_S$ is coherent if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is at most one.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.339-352
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• 1994

A sumbanifold M of a quaternionic Kaehlerian manifold $\tilde{M}^m$ of real dimension 4m is called a generic submanifold if the normal space N(M) of M is always mapped into the tangent space T(M) under the action of the quaternionic Kaehlerian structure tensors of the ambient manifold at the same time.The purpose of the present paper is to study generic submanifold of quaternionic Kaehlerian manifold of constant Q-sectional curvature with nonvanishing parallel mean curvature vector. In section 1, we state general formulas on generic submanifolds of a quaternionic Kaehlerian manifold of constant Q-sectional curvature. Section 2 is devoted to the study generic submanifolds with nonvanishing parallel mean curvature vector and compute the restricted Laplacian for the second fundamental form in the direction of the mean curvature vector. As applications of those results, in section 3, we prove our main theorems. In this paper, the dimension of a manifold will always indicate its real dimension.