• The Pure and Applied Mathematics
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• v.28 no.2
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• pp.91-102
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• 2021

In this paper, we introduce Lie bracket Jordan derivations in Banach Jordan algebras. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of Lie bracket Jordan derivations in complex Banach Jordan algebras.

• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.353-357
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• 2007

Let $(g,\;\delta)$ be a Lie bialgebra. Here we give an explicit formula for the Poisson bracket on a subalgebra of $U(g)^{\circ}$ induced by the given cobracket $\delta$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.57-60
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• 2008

Let $U(sp_4)$ be the universal enveloping algebra of the symplectic Lie algebra $sp_4$. Then the restricted dual $U(sp_4)^{\circ}$ becomes a Poisson Hopf algebra with the Sklyanin Poisson bracket determined by the standard classical r-matrix. Here we illustrate a method to obtain the Lie bialgebra from a Poisson bialgebra $U(sp_4)^{\circ}$.

• Proceedings of the Korean Statistical Society Conference
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• 2003.05a
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• pp.25-30
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• 2003

We prove that the logarithm of the flow of stochastic differential equations is an element of the free Lie algebra generated by a finite set consisting of vector fields being coefficients of equations. As an application, we directly obtain a formula of the solution of stochastic differential equations given by Castell(1993) without appealing to an expansion for ordinary differential equations given by Strichartz (1987).

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.17-21
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• 2011

Let A be a co-Poisson Hopf algebra with Poisson co-bracket $\delta$. Here it is shown that the Hopf dual $A^{\circ}$ is a Poisson Hopf algebra with Poisson bracket {f, g}(x) = < $\delta(x)$, $f\;{\otimes}\;g$ > for any f, g $\in$ $A^{\circ}$ and x $\in$ A if A is an almost normalizing extension over the ground field. Moreover we get, as a corollary, the fact that the Hopf dual of the universal enveloping algebra U(g) for a finite dimensional Lie bialgebra g is a Poisson Hopf algebra.

• Proceedings of the KIEE Conference
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• 1997.07b
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• pp.422-424
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• 1997

In this paper, we characterize the whole class of vector fields that can be linearized by a given nominal state transformation and a feedback linearizing controller. The necessary and sufficient condition for a given uncertain vector field to be so-called "completely linearizable by the nominal coordinate transformation" is given in terms of Lie Bracket of uncertain vector fields and some suitable vector fields of the nominal system.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.1
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• pp.142-149
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• 1995

This paper presents a new kinematic control strategy for serial redundant manipulators which gives repeatability in the joint space when the end-effector undergoes some general cyclic motions. Theoretical development has been accomplished by deriving a new inverse kinematic equation that is based on springs being conceptually located in the joints of the manipulator. Although some inverse kinematic equations for serial redundant manipulators have been derived by many researchers, the new strategy is the first to include the free angles of torsional springs and the free lengths of the translational springs. This is important because it ensures repeatability in the joint space of a serial redundant manipulator whose end-effector undergoes a cyclic type motion. Numerical verification for repeatability is done in terms of Lie Bracket Condition. Choices for the free angle and torsional stiffness of a joint (or the free length and translational stiffness) are made based upon the mechanical limits of the joints.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.43-46
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• 1989

In this paper, we are concerned with the pointwise-hypoellipticity (see Definition 2.1) of an m-dimensional Frobenious Lie algebra L of analytic complex vector fields in somel open subset .ohm. of $R^{m+1}$. That is, L is a set of complex vector fields in .ohm. with (real-) analytic coefficients satisfying: (A) each point of .ohm. has an open neighborhood in which L is generated by m linearly independent elements of L; (B) L is closed under the commutation bracket [A, B]. The pointwise-analytic hypoellipticity of L is completely characterized by M.S. Baouendi and F. Treves in [1]. Here, we shall prove that if L is hypoelliptic at a point then it must be analytic hypoelliptic in a full neighborhood of the same point. When the coefficients are $C^{\infty}$, hypoellipticity of L was discussed in [2].2].