• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1023-1029
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• 2014

In the paper, we define a class of convex bodies in $\mathbb{R}^n$-parallel sections homothety bodies, and for some special parallel sections homothety bodies, we prove that n-cubes have the minimal Mahler volume.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.365-374
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• 1995

On a compact manifold without conjugate points, the volume entropy can be obtained as the average mean curvature of the horospheres in the universal covering space. In the case when the volume entropy is zero, we prove that the universal covering space is diffeomorphic to a product space with a line factor. This fact can be considered as a surporting evidence for the Mane's conjecture, which claims the flatness of the mainfold.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.641-693
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• 2013

We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4{\pi}^2$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.639-645
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• 2016

In [3] the main theorem was erroneously stated. We needed to assume that the irrationality measure of 1/r is finite to prove the theorem.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1087-1094
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• 2016

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.641-648
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• 2000

It has been conjectured that, on a compact 3-dimensional manifold, a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics of volume 1 is Einstein. In this paper we find a sufficient condition that a critical point is Einstein. This condition is equivalent for a critical point ot be conformally flat. Its relationship with the Fisher-Marsden conjecture is also discussed.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.129-137
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• 2014

In this paper, a new proof of the following result is given: The product of the volumes of an origin-symmetric convex bodies K in $\mathbb{R}^2$ and of its polar body is minimal if and only if K is a parallelogram.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.867-871
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• 2013

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker $s^{\prime}^*_g{\neq}0$, which generalizes the previous partial results.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.587-595
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• 2008

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.

• Journal of Korea Water Resources Association
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• v.35 no.2
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• pp.231-240
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• 2002

The wetland has very important functions in hydrologic and ecological aspects and the research of wetland functions requires the basic hydrological properties such as water quantity. However, we do not have a research work on the hydrological properties for a wetland study in Korea. Therefore, this study is to estimate the relations between the volume(V), the area(A), and the depth(h) of water in the wetland which might be the basis for the wetland research in Korea. To estimate the relations, we derive the basic equations, obtain the surveyed data and do modelling, and estimate the relations of A-h and V-h using the Surfer program. The estimated and observed volumes for 5-wetland are compared and the errors are in the range of 2 % to 11 % for 4-wetland and 34 % for the rest. The wetlands in small errors showed the similar ones with the profile of the wetted perimeter which is assumed for the derivation of the equation but the wetland of large error has much different profile with the assumed one. We re-estimate the volumes for 3-wetland(W3, W4, W5) which showed the large errors due to the bended profiles of the wetland slopes. say, after the slopes was divided into two parts of upper and lower ones, the volumes were estimated. From our re-estimation, we obtained very good results ranged from 1 % to 8 % in their errors. We conjecture that the procedure suggested in this study might be useful as a reference for the future research on the relations of V-A-h in Korea.