• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.541-545
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• 2013

In this paper, we explain how to get defining equations of the modular curves $X_1(2,2N)$ which show the moduli problems and present defining equations of $X_1(2,2N)$ for $N=2,3,{\ldots},8$.

• Korean Journal of Mathematics
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• v.21 no.3
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• pp.305-309
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• 2013

In this paper, we give a new method to get defining equations of modular curves $X_1(2N)$ which show the moduli problems.

• East Asian mathematical journal
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• v.35 no.1
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• pp.51-58
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• 2019

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations. In this paper we precisely determine the defining equations of some rational curves of maximal regularity in ${\mathbb{P}}^4$ according to their rational parameterizations.

• East Asian mathematical journal
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• v.34 no.1
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• pp.21-28
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• 2018

For a nondegenerate irreducible projective variety, it is a classical problem to study the defining equations of a variety with respect to the given embedding. In this paper we precisely determine the defining equations of certain types of rational curves in ${\mathbb{P}}^3$.

• East Asian mathematical journal
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• v.38 no.3
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• pp.347-355
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• 2022

For a nondegenerate projective variety, it is a classical problem to study its defining equations with respect to a given embedding. In this paper, we precisely determine minimal sets of generators of the defining ideals of some curves of maximal regularity in ℙ⁵.

• East Asian mathematical journal
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• v.40 no.3
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• pp.287-293
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• 2024

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ³.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.399-409
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• 2007

In this paper we use iterative dynamic programming in the discrete case to solve a wide range of the nonlinear equations systems. First, by defining an error function, we transform the problem to an optimal control problem in discrete case. In using iterative dynamic programming to solve optimal control problems up to now, we have broken up the problem into a number of stages and assumed that the performance index could always be expressed explicitly in terms of the state variables at the last stage. This provided a scheme where we could proceed backwards in a systematic way, carrying out optimization at each stage. Suppose that the performance index can not be expressed in terms of the variables at the last stage only. In other words, suppose the performance index is also a function of controls and variables at the other stages. Then we have a nonseparable optimal control problem. Furthermore, we obtain the path from the initial point up to the approximate solution.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.683-692
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• 2024

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in ℙ⁹. In this paper, we study Keum's fake projective plane (a = 7, p = 2, {7}, D₃2₇) and use the equations of [1] to construct an embedding of fake projective plane in ℙ⁵. We also simplify the 84 cubic equations defining the fake projective plane in ℙ⁹.

• International Union of Geodesy and Geophysics Korean Journal of Geophysical Research
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• v.26 no.1
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• pp.1-14
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• 1998

Various numerical methods for the two dimensional shallow water equations have been applied to the problems of flood routing, tidal circulation, storm surges, and atmospheric circulation. These methods are often based on the Alternating Direction Implicity(ADI) method. However, the ADI method results in inaccuracies for large time steps when dealing with a complex geometry or bathymetry. Since this method reduces the performance considerably, a fully implicit method developed by Wilders et al. (1998) is used to improve the accuracy for a large time step. Finite Difference Methods are defined on a rectangular grid. Two drawbacks of this type of grid are that grid refinement is not possibile locally and that the physical boundary is sometimes poorly represented by the numerical model boundary. Because of the second deficiency several purely numerical boundary effects can be involved. A boundary fitted curvilinear coordinate transformation is used to reduce these difficulties. It the curvilinear coordinate transformation is used to reduce these difficulties. If the coordinate transformation is orthogonal then the transformed shallow water equations are similar to the original equations. Therefore, an orthogonal coorinate transformation is used for defining coordinate system. A multigrid (MG) method is widely used to accelerate the convergence in the numerical methods. In this study, a technique using a MG method is proposed to reduce the computing time and to improve the accuracy for the orthogonal to reduce the computing time and to improve the accuracy for the orthogonal grid generation and the solutions of the shallow water equations.

• Journal of the Korean Geotechnical Society
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• v.23 no.11
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• pp.39-48
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• 2007

The soil-water characteristic curve(SWCC) is a useful tool in the prediction of the engineering behavior of unsaturated soils. Several equations are available in the literature to mathematically represent the experimental behavior of the SWCC. Some equations are based on the assumption that the shape of curve is dependent upon pore-size distribution. Other equations assume that SWCC can be estimated from the grain size distribution and the physical properties of soils. This study evaluated the suitability of using two different SWCC equations for defining the relationship between water content and matric suction. Various parameters that influence the SWCC behavior are also briefly discussed.