• 대한수학회논문집
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• 제9권3호
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• pp.565-570
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• 1994

In this article we prove a version of the Perron-Frobenius Theorem in linear algebra using the Brouwer's Fixed Point Theorem in topology. We will mostly concentrate on he qualitative aspect of the Perron-Frobenius Theorem rather than quantitative formulas, which would be enough for theoretical investigations in ergodic theory. By the nature of the method of the proof, we do not expect to obtain a numerical estimate. But we may regard it worthwhile to see why a certain type of result should be true from a topological and geometrical viewpoint. However, a geometric argument alone would give us a sharp numerical bounds on the size of the eigenvalue as shown in Section 2. Eigenvectors of a matrix A will be fixed points of a certain mapping defined in terms of A. We shall modify an existing proof of Frobenius Theorem and that will do the trick for Perron-Frobenius Theorem.

• 대한수학회지
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• 제46권5호
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• pp.1087-1103
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• 2009

Given a system of s independent 1-forms on a smooth manifold M of dimension m, we study the existence of integral manifolds by means of various generalized versions of the Frobenius theorem. In particular, we present necessary and sufficient conditions for there to exist s'-parameter (s' < s) family of integral manifolds of dimension p := m-s, and a necessary and sufficient condition for there to exist integral manifolds of dimension p', p' $\leq$ p. We also present examples and applications to complex analysis in several variables.

• 한국전자통신학회논문지
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• 제7권5호
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• pp.1133-1138
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• 2012

본 논문은 최적제어에 기초하여 비선형 슬라이딩 평면을 설계하는 것이다. 최적제어입력에 의한 상태 궤적을 Frobenius 정리와 matrix decomposition 방법에 의해 구하였고, 이 궤적을 시스템의 슬라이딩 평면으로 설정하였다. 상태는 초기부터 슬라이딩 평면을 유지하며, 그 결과 초기상태 단계로부터 전 영역까지 시스템의 강인성은 보장 받을 수 있으며, 도달시간 동안 발생 될 수 있는 불확실성과 외란의 영향을 제거되고, 큰 제어 입력의 문제도 해결할 수 있었다. 그리고 최적경로를 슬라이딩 평면으로 설정함으로 추적시간을 줄일 수 있었다. 역진자 시스템을 사용하여 그 타당성을 보인다.

• 제어로봇시스템학회논문지
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• 제6권7호
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• pp.523-528
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• 2000

In this short note the characteristics of a nonlinear system of which the state trajectories are oscillating in the phase plane are overviewed. The physical concept of stuck and sliding patch phenomena are also introduced by adding some switching functions and their stability on the sliding patches are analyzed by using the Lyapunov stability theory and Frobenius theorem.

• 대한수학회지
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• 제47권5호
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

• 대한수학회지
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• 제39권2호
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• pp.237-252
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• 2002

We study the compatibility conditions and the existence of solutions or overdetermined PDE systems that admit complete prolongation. For a complete system of order k there exists a submanifold of the ($\kappa$-1)st jet space of unknown functions that is the largest possible set on which the initial conditions of ($\kappa$-1)st order may take values. There exists a unique solution for any initial condition that belongs to this set if and only if the complete system satisfies the compatibility conditions on the initial data set. We prove by applying the Frobenius theorem to a Pfaffian differential system associated with the complete prolongation.

• 대한수학회보
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• 제46권5호
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• pp.931-940
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• 2009

Given a system of smooth 1-forms $\theta$ = ($\theta^1$,...,$\theta^s$) on a smooth manifold $M^m$, we give a necessary and sufficient condition for M to be foliated by integral manifolds of dimension n, n $\leq$ p := m - s, and construct an integrable supersystem ($\theta,\eta$) by finding additional 1-forms $\eta$ = ($\eta^1$,...,$\eta^{p-n}$). We also give a necessary and sufficient condition for M to be foliated by reduced submanifolds of dimension n, n $\geq$ p, and construct an integrable subsystem ($d\rho^1$,...,$d\rho^{m-n}$) by finding a system of first integrals $\rho=(\rho^1$,...,$\rho^{m-n})$. The special case n = p is the Frobenius theorem on involutivity.

• 대한수학회지
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• 제37권4호
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• pp.503-519
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• 2000

It is well-known that an analytic generic CR submainfold M of codimension m in Cn+m is locally transformed by a biholomorphic mapping to a plane Cn$\times$Rm ⊂ Cn$\times$Cm whenever the Levi form L on M vanishes identically. We obtain such a normalizing biholomorphic mapping of M in terms of the defining function of M. Then it is verified without Frobenius theorem that M is locally foliated into complex manifolds of dimension n.