• Communications of the Korean Mathematical Society
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• v.6 no.1
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• pp.145-158
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• 1991

• Bulletin of the Korean Mathematical Society
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• v.24 no.1
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• pp.41-41
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• 1987

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.15-24
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• 1992

The purpose of this paper is to parameterize range-equivalence classes of all eigenmaps of flat 3-tori $T^{3}$= $R^{3}$.LAMBDA., .LAMBDA.= $c_{1}$ $e_{1}$ + $c_{2}$ $e_{2}$ + $c_{3}$ $e_{3}$, into the standard unit spheres. In this paper, we classify $A_{ 0 \lambda}$( $T^{3}$)(cf..cint.1) which is contained in $A_{\lambda}$( $T^{3}$), are belonging to $A_{ 0 \lambda}$( $T^{3}$). Moreover, as an application, we show that the only minimally imbedded flat torus into ( $S^{5}$ , can) which is contained in $A_{ 0 \lambda}$( $T^{3}$) is the generalized Clifford torus.rd torus.

• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.67-71
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• 2005

We call K a (1, 1)-knot in M if M is a union of two solid tori $V_1\;and\;V_2$ glued along their boundary tori ${\partial}V_1\;and\;{\partial}V_2$ and if K intersects each solid torus $V_i$ in a trivial arc $t_i$ for i = 1 and 2. Note that every (1, 1)-knot is a tunnel number one knot. In this article, we determine when a tunnel number one knot is a (1, 1)-knot. In other words, we show that any tunnel number one knot with bridge number 3 is a (1, 1)-knot.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1527-1575
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• 2017

In a spirit of Givental's constructions Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested Landau-Ginzburg models for smooth Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections in complex tori equipped with special functions called superpotentials. We provide a particular algorithm for constructing birational isomorphisms of these models for complete intersections in Grassmannians of planes with complex tori. In this case the superpotentials are given by Laurent polynomials. We study Givental's integrals for Landau-Ginzburg models suggested by Batyrev, Ciocan-Fontanine, Kim, and van Straten and show that they are periods for pencils of fibers of maps provided by Laurent polynomials we obtain. The algorithm we provide after minor modifications can be applied in a more general context.

• Proceedings of the Korean Information Science Society Conference
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• 2008.06a
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• pp.353-354
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• 2008

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.409-424
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• 2010

We express Gauss sums for symplectic and orthogonal groups over finite fields as averages of exponential sums over certain maximal tori. Together with our previous results, we obtain some interesting identities involving various classical Gauss and Kloosterman sums.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.897-910
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• 2019

In this paper we introduce the notion of "robust special Anosov endomorphisms", and show that Anosov endomorphisms of tori which are not neither an Anosov diffeomorphism nor an expanding map, are not robust special.

• The Bulletin of The Korean Astronomical Society
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• v.40 no.2
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• pp.37.2-38
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• 2015

According to the unified model of AGNs, type 1 and 2 AGNs are intrinsically the same objects but seem different due to an obscuring matter which can block lights from the central engine of the AGN depending on the viewing angle. The obscuring object is thought to be shaped in a toroidal form and thus the geometry of tori of AGNs is an important factor to determine the fraction of type 1 (or type 2) AGNs. Oh et al. (2015) provides a new catalog of type 1 AGNs from SDSS DR7 in the nearby universe (z < 0.2) and it contains nearly 50% more type 1 AGNs than previously known. Using this new catalog, we test the fraction of type 1 AGNs along the black hole mass (MBH) and the bolometric luminosity of AGNs (Lbol), which are regarded as key parameters of the AGNs. First of all, because the methods to derive the black hole mass and the bolometric luminosity bear uncertainties, we test how the different methods lead to different values of type 1 fraction. We found that the fraction of type 1 AGNs varies with both MBH and Lbol. The extensively-studied, "receding torus model" can only explain the trend along Lbol and hence fails to explain the trend. To understand the new trend, we test the geometry of the torus based on the "clumpy torus model". We present our results on the basic properties of the torus such as a column density or opening angle and compare with those from previous studies based on other wavelengths (e.g. Infrared or X-ray).

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.335-340
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• 2007

Let M be a hyperbolic 3-manifold such that ${\partial}M$ has at least two boundary tori ${\partial}_oM$ and ${\partial}_1M$. Suppose that M contains an essential orientable surface P of genus $g$ with one outer boundary component ${\partial}_oP$, lying in ${\partial}_oM$ and having slope ${\lambda}$ in ${\partial}_oM$, and $p$ inner boundary components ${\partial}_iP$, $i=1$, ${\cdots}$, $p$, each having slope ${\alpha}$ in ${\partial}_1M$. Let ${\beta}$ be a slope in ${\partial}_1M$ and suppose that $M({\beta})$ is toroidal. Let $\hat{T}$ be a minimal essential torus in $M({\beta})$, which means that $\hat{T}$ is pierced a minimal number of times by the core of the ${\beta}$-Dehn filling, among all essential tori in $M({\beta})$. Let $T=\hat{T}{\cap}M$ and denote by $t$ the number of components of ${\partial}T$. In this paper we prove: (i) if $t{\geq}3$, then ${\Delta}({\alpha},{\beta}){\leq}6+\frac{10g-5}{p}$, (ii) If $t=2$, then ${\Delta}({\alpha},{\beta}){\leq}13+\frac{24g-12}{p}$, (iii) If $t=1$, then ${\Delta}({\alpha},{\beta}){\leq}1$.