• 대한수학회보
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• 제55권5호
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• pp.1577-1586
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• 2018

In this paper, we prove that if every bounded ${\mathcal{A}}$-harmonic function on a complete Riemannian manifold M is asymptotically constant at infinity of p-nonparabolic ends of M, then each bounded ${\mathcal{A}}$-harmonic function is uniquely determined by the values at infinity of p-nonparabolic ends of M, where ${\mathcal{A}}$ is a nonlinear elliptic operator of type p on M. Furthermore, in this case, every bounded ${\mathcal{A}}$-harmonic function on M has finite energy.

• 대한수학회보
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• 제37권2호
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• pp.317-336
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• 2000

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future (or past) complete Lorentzian metrics on $M=(-{\infty},{\;}\infty){\;}{\times}f^N$ with specific scalar curvatures.

• 대한수학회논문집
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• 제33권1호
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• pp.317-323
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• 2018

We study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact manifolds with zero boundary condition. In particular, we obtain some geometric estimates for the first eigenvalue.

• Smart Structures and Systems
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• 제3권4호
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• pp.423-437
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• 2007

The empirical mode decomposition (EMD) method is well-known for its ability to decompose a multi-component signal into a set of intrinsic mode functions (IMFs). The method uses a sifting process in which local extrema of a signal are identified and followed by a spline fitting approximation for decomposition. This method provides an effective and robust approach for decomposing nonlinear and non-stationary signals. On the other hand, the IMF components do not automatically guarantee a well-defined physical meaning hence it is necessary to validate the IMF components carefully prior to any further processing and interpretation. In this paper, an attempt to use the EMD method to identify properties of nonlinear elastic multi-degree-of-freedom structures is explored. It is first shown that the IMF components of the displacement and velocity responses of a nonlinear elastic structure are numerically close to the nonlinear normal mode (NNM) responses obtained from two-dimensional invariant manifolds. The IMF components can then be used in the context of the NNM method to estimate the properties of the nonlinear elastic structure. A two-degree-of-freedom shear-beam building model is used as an example to illustrate the proposed technique. Numerical results show that combining the EMD and the NNM method provides a possible means for obtaining nonlinear properties in a structure.

• 대한기계학회논문집
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• 제19권2호
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• pp.537-547
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• 1995

Numerical study on the chaotic stirring of viscous flow in an alternately driven cavity has been performed. Even under the Stokes-flow assumption, the inherent singularity at the corners made the problem not so easily accessible. With some special treatments to the region near the corners, the biharmonic equation was solved numerically by using the fully implicit method. The velocity field was then used in obtaining the trajectories of passive particles for studying the stirring effect. The three tools developed in the field of the nonlinear dynamics and chaos, that are the Poincare sections, the unstable manifolds, and the Lyapunov exponents, were used in analysing the stirring effect. It was shown that the unstable manifolds obtained in this study well fit the experimental results given by the previous investigators. It is predicted that the best stirring can be obtained when the aspect ratio a is near 0.8 and the dimensionless period T is in the range 4.3 - 4.7.

• 대한수학회보
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• 제58권3호
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• 한국HCI학회:학술대회논문집
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• 한국HCI학회 2008년도 학술대회 1부
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• pp.342-347
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• 2008

사람의 얼굴은 강체(rigid object)가 아니기 때문에 얼굴을 추적하거나 인식하기는 쉽지 않다. 또한 시스템에 미리 학습되어 있지 않은 임의의 얼굴의 경우 지속적으로 얼굴의 변화를 추적하고 인식하기는 어렵다. 본 논문에서는 시스템에 저장되어 있는 얼굴들에 대해 비선형적 매니폴드 모델을 구축하고 각 모델을 선형적으로 결합함으로써 비디오 기반의 영상으로부터 시스템이 알지 못하는 임의의 얼굴에 대해 추적하고 인식하는 방법을 제안한다. 입력된 임의의 얼굴은 얼굴 포즈나 표정 혹은 주위 환경 등에 따라 시스템에 저장되어 있는 서로 다른 얼굴들과 서로 다른 유사성을 갖는다. 따라서 입력 얼굴과 시스템에 저장되어 있는 얼굴들과의 확률적인 접근을 통해 유사성을 추정할 수 있고 추정된 유사성을 이용하여 입력 얼굴에 대한 새로운 비선형적 매니폴드 모델을 구축한다. 또한 추정된 모델은 매 프레임마다 입력 얼굴에 따라 실시간으로 갱신된다. 본 논문에서 제안하는 방법은 실험 결과를 통하여 효율적으로 임의의 얼굴에 대해 추적하고 인식할 수 있음을 보인다.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.631-644
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• 2021

In the present paper, we obtain gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds $$\frac{{\partial}u}{{\partial}t}={\Delta}u+a(x,t)u^p+b(x,t)u^q$$ where, 0 < p, q < 1 are real constants and a(x, t) and b(x, t) are functions which are C² in the x-variable and C¹ in the t-variable. We shall get an interesting Harnack inequality as an application.

• 대한수학회지
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• 제37권6호
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• pp.1031-1042
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• 2000

The equations prescribed in Ω⊂R(sup)n are called loaded, if they contain some operations of the traces of desired solution on manifolds (of dimension which is strongly less than n) from closure Ω. These equations result from approximations of nonlinear equations by linear ones, in the problems of optimal control when the control when the control actions depends on a part of independent variables, in investigations of the inverse problems and so on. In present work we study the nonlocal boundary value problems for first-order loaded differential operator equations. Criterion of unique solvability is established. We illustrate the obtained results by examples.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권2호
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• pp.81-87
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• 2017

In this note, we show that the cone property is satisfied for a class of dissipative equations of the form $u_t={\Delta}u+f(x,u,{\nabla}u)$ in a domain ${\Omega}{\subset}{\mathbb{R}}^2$ under the so called exactness condition for the nonlinear term. From this, we see that the global attractor is represented as a Lipshitz graph over a finite dimensional eigenspace.