• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.425-443
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• 2010

We investigate the Schr$\ddot{o}$dinger type operator $H_2\;=\;(-\Delta_{\mathbb{H}^n})^2+V^2$ on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sublaplacian and the nonnegative potential V belongs to the reverse H$\ddot{o}$lder class $B_q$ for $q\geq\frac{Q}{2}$, where Q is the homogeneous dimension of $\mathbb{H}^n$. We shall establish the estimates of the fundamental solution for the operator $H_2$ and obtain the $L^p$ estimates for the operator $\nabla^4_{\mathbb{H}^n}H^{-1}_2$, where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n$.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.551-560
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• 2007

We prove the existence of solutions for the system of the nonlinear biharmonic equations with Dirichlet boundary condition $$\{^{-{\Delta}^2u-c{\Delta}u+{\gamma}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega},\;}_{-{\Delta}^2u-c{\Delta}u+{\delta}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega}}$$, where $u^+$ = max{u, 0}, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition.

• Journal of the Korean Mathematical Society
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• v.15 no.1
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• pp.39-57
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• 1978

It is given in the Lecture Note [1] of Berger, Gauduchon and Mazet that, if ${\pi}$n: (${\tilde{M}}$, ${\tilde{g}}$)${\rightarrow}$(${\tilde{M}}$, ${\tilde{g}}$) is a Riemannian submersion with totally geodesic fibers, ${\Delta}$ and ${\tilde{\Delta}}$ are Laplace operators on (${\tilde{M}}$, ${\tilde{g}}$) and (M, g) respectively and f is an eigenfunction of ${\Delta}$, then its lift $f^L$ is also an eigenfunction of ${\tilde{\Delta}}$ with the common eigenvalue. But such a simple relation does not hold for an eigenform of the Laplace-Beltrami operator ${\Delta}=d{\delta}+{\delta}d$. If ${\omega}$ is an eigenform of ${\Delta}$ and ${\omega}^L$ is the horizontal lift of ${\omega}$, ${\omega}^L$ is not in genera an eigenform of the Laplace-Beltrami operator ${\tilde{\Delta}}$ of (${\tilde{M}}$, ${\tilde{g}}$). The present author has obtained a set of formulas which gives the relation between ${\tilde{\Delta}}{\omega}^L$ and ${\Delta}{\omega}$ in another paper [7]. In the present paper a Sasakian submersion is treated. A Sasakian manifold (${\tilde{M}}$, ${\tilde{g}}$, ${\tilde{\xi}}$) considered in this paper is such a one which admits a Riemannian submersion where the base manifold is a Kaehler manifold (M, g, J) and the fibers are geodesics generated by the unit Killing vector field ${\tilde{\xi}}$. Then the submersion is called a Sasakian submersion. If ${\omega}$ is a eigenform of ${\Delta}$ on (M, g, J) and its lift ${\omega}^L$ is an eigenform of ${\tilde{\Delta}}$ on (${\tilde{M}}$, ${\tilde{g}}$, ${\tilde{\xi}}$), then ${\omega}$ is called an eigenform of the first kind. We define a relative eigenform of ${\tilde{\Delta}}$. If the lift ${\omega}^L$ of an eigenform ${\omega}$ of ${\Delta}$ is a relative eigenform of ${\tilde{\Delta}}$ we call ${\omega}$ an eigenform of the second kind. Such objects are studied.

• Journal of Institute of Control, Robotics and Systems
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• v.4 no.2
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• pp.170-177
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• 1998

In this paper, a unified solution of Hankel norm approximation problem is proposed by $\delta$-operator. To derive the main result, all-pass property is derived from the inner and co-inner property in the $\delta$-domain. The solution of all-pass becomes an optimal Hankel norm approximation problem in .delta.-domain through LLFT(Low Linear Fractional Transformation) inserting feedback term $\phi(\gamma)$, which is a free design parameter, to hold the error bound desired against the variance between the original model and the solution of Hankel norm approximation problem. The proposed solution does not only cover continuous and discrete ones depending on sampling interval but also plays a key role in robust control and model reduction problem. The verification of the proposed solution is exemplified via simulation for the zero-order Hankel norm approximation problem and the model reduction problem applied to a 16th order MIMO system.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1113-1122
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• 2016

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator $-{\Delta}_{\phi}+{\frac{R}{2}}$ under the Yamabe flow, where ${\Delta}_{\phi}$ is the Witten-Laplacian operator, ${\phi}{\in}C^2(M)$, and R is the scalar curvature with respect to the metric g(t). As a consequence, we construct some monotonic quantities under the Yamabe flow.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.689-695
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• 2011

The main object of this paper is to prove several inclusion relations associated with (j, ${\delta}$)-neighborhoods of various subclasses defined by Salagean operator by making use of the familiar concept of neighborhoods of analytic functions. Special cases of some of these inclusion relations are shown to yield known results.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.501-509
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• 2013

In this paper we give a definition of the Hodge type Laplacian ${\Delta}$ on a non-commutative manifold which is the smooth dense subalgebra of a $C^*$-algebra. We prove that the Laplacian on a quantum Heisenberg manifold is an elliptic operator in the sense that $({\Delta}+1)^{-1}$ is compact.

• 제어로봇시스템학회:학술대회논문집
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• 1998.10a
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• pp.386-390
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• 1998

A master-slave system is proposed as a teaching device for a dual arm robot. The slave robots are remotely controlled by two delta-type master arms. In order to help the operator to observe the target object from the desired position and desired direction, cameras are mounted on a specialized manipulator, Movements of two slave arms are coordinated with that of the cameras. Due to this coordinated movements, the operator needs not to care the geometrical relation between the cameras and the slave robots.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.135-143
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• 2008

We investigate the existence of nontrivial solutions of the nonlinear biharmonic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}{\Delta}^2{\xi}+c{\Delta}{\xi}={\mu}h({\xi}+{\eta})\;in{\Omega},\\{\Delta}^2{\eta}+c{\Delta}{\eta}={\nu}h({\xi}+{\eta})\;in{\Omega},\end{array}$$ where $c{\in}R$ and ${\Delta}^2$ denote the biharmonic operator.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.529-541
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• 2005

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.