• 호남수학학술지
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• 제43권4호
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• pp.679-689
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• 2021

The first author suggested an exact volume formula of the hypercubes [0, 1]ⁿ clipped by several hyperplanes expressed directly in terms of linear coefficients of the hyperplanes. However, it requires awkward assumptions to apply the formula to various situations. We suggest a concrete method to overcome those restrictions for two or three hyperplanes using 𝜖-perturbation, which gives an exact value applicable for any kind of arrangement of hyperplanes with no consideration.

• 한국수학사학회지
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• 제15권1호
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• pp.147-168
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• 2002

In this paper, we introduce the arrangement of hyperplanes and the graph theory. In particular, we explain how to study the 4-color problem by using characteristic polynomials of the arrangement of hyperplanes. The 4-color problem was appeared in 1852 at first and Appel and Haken proved it by using computer in 1976. The arrangement of hyperplanes induced from a graph is called a graphic arrangement. Graphic arrangement is a subarrangement of Braid arrangement. Thus the chromatic function of a graph is equal to the characteristic polynomial of a graphic arrangement. If we use this result, we can apply the theory of the arrangement of hyperplanes to the study for the chromatic functions.

• 대한수학회보
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• 제57권1호
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• pp.179-194
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• 2020

In this article, we establish some new second main theorems for meromorphic mappings of ℂ^m into ℙⁿ(ℂ), in which the truncated counting functions have different weights. As for application, we deal with the algebraic dependence problem of meromorphic mappings sharing moving hyperplanes in general position.

• 호남수학학술지
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• 제31권1호
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• pp.25-29
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• 2009

In this paper, we classify essential arrangements $\mathcal{A}$ of hyperplanes in $\mathbb{R}^3$ with ${\mid}\mathcal{A}{\mid}$ = 5 and give a full description of each case in terms of freeness.

• 대한수학회보
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• 제59권5호
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• pp.1105-1117
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• 2022

In this paper, we obtain a second main theorem for holomorphic curves and moving hyperplanes of Pⁿ(C) where the counting functions are truncated multiplicity and have different weights. As its application, we prove a uniqueness theorem for holomorphic curves of finite growth index sharing moving hyperplanes with different multiple values.

• 대한기계학회논문집
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• 제18권8호
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• pp.1985-1996
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• 1994

A new design method of sliding hyperplanes is proposed in the synthesis of a variable structure controller for robust tracking of general nonlinear multi-input-output(MIMO) uncertain systems of relative degree higher than two. Input/ output(I/O) linearzation is firstly undertaken by employing the concept of relative degree and minimum phase followed by the construction of sliding mode controllers. Sliding hyperplanes are then derived from the inherent properties of companion matrix and ideal sliding mode characterized in I/O linearized system. Subsequently, the gradient magnitudes of the sling hyperplanes are determined in an optimal manner by considering a quadratic performance index to be evaluated at two phases; a reaching phase and a sliding phase. The proposed design methodology is relatively straightforward and systematic compared with conventional strategies such as geometric approach or pole assignment technique. A nonlinear governor and exciter control problem for a power system is adopted herein in order to demonstrate the design efficiency and also favorable and robust control performances.

• 대한수학회보
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• 제55권1호
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• pp.205-226
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• 2018

In this paper, we show the Second Main Theorems for zero-order meromorphic mapping of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n({\mathbb{C}})$ intersecting hyperplanes in subgeneral position without truncated multiplicity by considering the p-Casorati determinant with $p{\in}{\mathbb{C}}^m$ instead of its Wronskian determinant. As an application, we give some unicity theorems for meromorphic mapping under the growth condition "order=0". The results obtained include p-shift analogues of the Second Main Theorem of Nevanlinna theory and Picard's theorem.

• 대한전기학회:학술대회논문집
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• 대한전기학회 2005년도 심포지엄 논문집 정보 및 제어부문
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• pp.122-124
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• 2005

In this paper, we propose a switching control method for 2nd order nonlinear systems. The main idea behind the method is changing the control law before the trajectory of the solution arrives at the singularities imposed on the denominator of the control law. We show that the control system is asymptotically stable from the fact that the sector consisting of the singular hyperplanes is an invariant set. Illustrative examples are given.

• International Journal of Control, Automation, and Systems
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• 제4권1호
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• pp.124-135
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• 2006

In this paper, we propose a switching control method for a class of 2nd order nonlinear systems with single input. The main idea is to switch the control law before the trajectory of the solution arrives at singular hyperplanes which are defined by the denominator of the control law. The proposed method can handle a class of nonlinear systems which is difficult to be stabilized by the existing methods such as feedback linearization, backstepping, control Lyapunov function, and sliding mode control.

• 대한수학회지
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• 제58권2호
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• pp.451-471
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• 2021

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (ℂ∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.