• 대한수학회보
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• 제42권4호
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• pp.807-815
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• 2005

Let R be a ring with identity, X the set of all nonzero, nonunits of Rand G the group of all units of R. We will consider two group actions on X by G, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if G is a finitely generated abelian group, then the orbit O(x) under the regular action on X by G is finite for all nilpotents x $\in$ X. Secondly, if F is a field in which 2 is a unit and F $\backslash\;\{0\}$ is a finitley generated abelian group, then F is finite. Finally, if G in a unit-regular ring R is a torsion group and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.

• 대한수학회논문집
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• 제17권2호
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• pp.253-260
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• 2002

For an outer action $\alpha$ of a finite group G on a factor M, it was proved that H is a, normal subgroup of G if and only if there exists a finite group F and an outer action $\beta$ of F on the crossed product algebra M $\times$$_{\alpha}$ G = (M $\times$$_{\alpha}$ F. We generalize this to infinite group actions. For an outer action $\alpha$ of a discrete group, we obtain a Galois correspondence for crossed product algebras related to normal subgroups. When $\alpha$ satisfies a certain condition, we also obtain a Galois correspondence for fixed point algebras. Furthermore, for a minimal action $\alpha$ of a compact group G and a closed normal subgroup H, we prove $M^{G}$ = ( $M^{H}$)$^{{beta}(G/H)}$for a minimal action $\beta$ of G/H on $M^{H}$.f G/H on $M^{H}$.TEX> H/.

• Korean Journal of Mathematics
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• 제7권1호
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• pp.71-77
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• 1999

We show that if the stable rank of $B^{\alpha}$ is one, then the stable rank of B is less than or equal to the order of G for any action of a finite group G. Also we give a short proof to the known fact that if the action of a finite group on a $C^*$-algebra B is saturated then the canonical conditional expectation from B to $B^{\alpha}$ is of index-finite type and the crossed product $C^*$-algebra is isomorphic to the algebra of compact operators on the Hilbert $B^{\alpha}$-module B.

• 대한수학회지
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• 제47권5호
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• pp.1097-1106
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• 2010

Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering $\Gamma$(R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of $\Gamma$(R) which is adjacent to every other vertex in $\Gamma$(R) if and only if R is a local ring or $R\;{\simeq}\;\mathbb{Z}_2\;{\times}\;F$ where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, $J^2$, $\ldots$, $J^n$, R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.

• 한국게임학회 논문지
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• 제10권3호
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• pp.93-102
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• 2010

본 논문은 비둘기들의 다양한 상태에 대하여 집단행동을 자연스럽게 시뮬레이션하는 시스템을 설계하고 구현하는 것이다. 이것을 하기 위하여 비둘기들의 집단행동은 '날아가기', '내려앉기', '먹이먹기' 및 '날아오르기'와 같이 4개의 액션모델로 나뉘었다. 각 액션모델을 구성하는 조종힘들이 찾아졌으며, 유한상태기계 기법을 사용하여 설계되었다. 설계된 시스템은 오우거 엔진과 집적하여 구현되었다. 구현된 시스템의 시뮬레이션으로부터 비둘기들의 자연스러운 집단행동을 표현하는 조종힘에 대한 다양한 파라미터 값들을 찾을 수 있었다.

• East Asian mathematical journal
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• 제25권4호
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• pp.433-440
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• 2009

Let R be a ring with unity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will consider some group actions on X by G, the left (resp. right) regular action and the conjugate action. In this paper, by investigating these group actions we can have some results as follows: First, if E(R), the set of all nonzero nonunit idempotents of a unit-regular ring R, is commuting, then $o_{\ell}(x)\;=\;o_r(x)$, $o_c(x)\;=\;\{x\}$ for all $x\;{\in}\;X$ where $o_{\ell}(x)$ (resp. $o_r(x)$, $o_c(x)$) is the orbit of x under the left regular (resp. right regular, conjugate) action on X by G and R is abelian regular. Secondly, if R is a unit-regular ring with unity 1 such that G is a cyclic group and $2\;=\;1\;+\;1\;{\in}\;G$, then G is a finite group. Finally, if R is an abelian regular ring such that G is an abelian group, then R is a commutative ring.

• 대한수학회보
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• 제25권2호
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• pp.179-183
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• 1988

In this paper we study a $C^{*}$-dynamical system (A, G, .alpha.) where A is a UHF-algebra, G is a finite abelian group and .alpha. is a *-automorphic action of product type of G on A. In [2], A. Kishimoto considered the case G= $Z_{n}$, the cyclic group of order n and investigated a condition in order that the fixed point algebra $A^{\alpha}$ of A under the action .alpha. is UHF. In later N.J. Munch studied extremal tracial states on $A^{\alpha}$ by employing the method of A. Kishimoto [3], where G is a finite abelian group. Generally speaking, when G is compact (not necessarily discrete and abelian), $A^{\alpha}$ is an AF-algebra and its ideal structure was well analysed by N. Riedel [4]. Here we obtain some conditions for $A^{\alpha}$ to be UHF, where G is a finite abelian group, which is an extension of the result of A. Kishimoto.oto.

• 대한수학회보
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• 제38권1호
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• pp.143-148
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• 2001

First, we show the finiteness property of the homotopy fixed point set of p-discrete toral group. Let $G_\infty$ be a p-discrete toral group and X be a finite complex with an action of $G_\infty such that X^K$ is nilpotent for each finit p-subgroup K of $G_\infty$. Assume X is $F_\rho-complete$. Then X(sup)hG$\infty$ is F(sub)p-finite. Using this result, we give the condition so that X$^{hG}$ is $F_\rho-finite for \rho-compact$ toral group G.

• 대한수학회보
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• 제33권2호
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• pp.253-258
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• 1996

Let X be a finite connected CW-complex, $\Gamma = \pi_1(X)$ its fundamental group, $\tilde{X}$ its universal covering space. Then $\Gamma$ acts on $\tilde{X}$ by covering transformations and on the homology group $H_*(\tilde{X})$. In this note we establish the following vanishing result for the Euler characteristic $x(X)$ of X.

• Korean Journal of Mathematics
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• 제19권2호
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• pp.181-189
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• 2011

Let G be a group and X be a nonempty set and H be a subgroup of G. For a given ${\phi}_G\;:\;G{\times}X{\rightarrow}X$, a group action of G on X, we define ${\phi}_H\;:\;H{\times}X{\rightarrow}X$, a subgroup action of H on X, by ${\phi}_H(h,x)={\phi}_G(h,x)$ for all $(h,x){\in}H{\times}X$. In this paper, by considering a subgroup action of H on X, we have some results as follows: (1) If H,K are two normal subgroups of G such that $H{\subseteq}K{\subseteq}G$, then for any $x{\in}X$ ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_H}(x)$) = ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_K}(x)$) = ($orb_{{\phi}_K}(x)\;:\;orb_{{\phi}_H}(x)$); additionally, in case of $K{\cap}stab_{{\phi}_G}(x)$ = {1}, if ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}H}(x)$) and ($orb_{{\phi}_K}(x)\;:\;orb_{{\phi}_H}(x)$) are both finite, then ($orb_{{\phi}_G}(x)\;:\;orb_{{\phi}_H}(x)$) is finite; (2) If H is a cyclic subgroup of G and $stab_{{\phi}_H}(x){\neq}$ {1} for some $x{\in}X$, then $orb_{{\phi}_H}(x)$ is finite.