• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.475-486
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• 2018

We study free actions of finite nonabelian groups on 3-dimensional nilmanifolds with the first homology ${\mathbb{Z}}^2{\bigoplus}{\mathbb{Z}}_2$ which yield an orbit manifold reversing fiber orientation, up to topological conjugacy. We show that those nonabelian groups are $D_4$(the dihedral group), $Q_8$(the quaternion group), and $C_8.C_4$(the $1^{st}$ non-split extension by $C_8$ of $C_4$ acting via $C_4/C_2=C_2$).

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.609-618
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• 1994

F. Rhodes [2] introduced the fundamental group $\sigma(X, x_0, G)$ of a transformation group (X,G) as a generalization of the fundamental group $\pi_1(X, x_0)$ of a topological space X and showed a sufficient condition for $\sigma(X, x_0, G)$ to be isomorphic to $\pi_1(X, x_0) \times G$, that is, if (G,G) admits a family of preferred paths at e, $\sigma(X, x_0, G)$ is isomorphic to $\pi_1(X, x_0) \times G$. B.J.Jiang [1] introduced the Jiang subgroup $J(f, x_0)$ of the fundamental group of X which depends on f and showed a condition to be $J(f, x_0)$ = Z(f_\pi(\pi_1(X, x_0)), \pi_1(X, f(x_0)))$.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.201-215
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• 2003

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. For any connected graph $\Gamma$, by introducing the concertion of semi-arc automorphism group Aut$\_$$\frac{1}{2}$/$\Gamma$ and classifying all embedding of $\Gamma$ undo. the action of this group, the numbers r$\^$O/ ($\Gamma$) and r$\^$N/($\Gamma$) of rooted maps on orientable and non-orientable surfaces with underlying graph $\Gamma$ are found. Many closed formulas without sum ∑ for the number of rooted maps on surfaces (orientable or non-orientable) with given underlying graphs, such as, complete graph K$\_$n/, complete bipartite graph K$\_$m, n/ bouquets B$\_$n/, dipole Dp$\_$n/ and generalized dipole (equation omitted) are refound in this paper.

• Honam Mathematical Journal
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• v.37 no.1
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• pp.135-147
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• 2015

Owing to the notion of a normal adjacency for a digital product in [8], the study of product properties of digital topological properties has been substantially done. To explain a normal adjacency of a digital product more efficiently, the recent paper [22] proposed an S-compatible adjacency of a digital product. Using an S-compatible adjacency of a digital product, we also study product properties of digital topological properties, which improves the presentations of a normal adjacency of a digital product in [8]. Besides, the paper [16] studied the product property of two digital covering maps in terms of the $L_S$- and the $L_C$-property of a digital product which plays an important role in studying digital covering and digital homotopy theory. Further, by using HS- and HC-properties of digital products, the paper [18] studied multiplicative properties of a digital fundamental group. The present paper compares among several kinds of adjacency relations for digital products and proposes their own merits and further, deals with the problem: consider a Cartesian product of two simple closed $k_i$-curves with $l_i$ elements in $Z^{n_i}$, $i{\in}\{1,2\}$ denoted by $SC^{n_1,l_1}_{k_1}{\times}SC^{n_2,l_2}_{k_2}$. Since a normal adjacency for this product and the $L_C$-property are different from each other, the present paper address the problem: for the digital product does it have both a normal k-adjacency of $Z^{n_1+n_2}$ and another adjacency satisfying the $L_C$-property? This research plays an important role in studying product properties of digital topological properties.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.635-641
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• 2001

A limit point p of a Mobius group acting on$ B^m$ is called a concentration point if for every sufficiently small connected open neighborhood of p, the set of translates contains a local basis for the topology of p. For the case of two generator Schottky groups acting on $B^2$, we give characterizations for several different kinds of limit points.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.207-217
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• 2008

Digital geometry has strongly contributed to the study of a discrete topological space $X{\subset}{\mathbf{Z}}^n$ with k-adjacency of ${\mathbf{Z}}^n$. As a survey-type article, we review various utilities of digital geometry.

• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.711-719
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• 2017

Let Inn(Q) denote the inner automorphism group on a quandle Q. For a subset M of Q, let c(M) denote the orbit of M under the Inn(Q)-action on Q. Then c satisfies the axioms of the closure operator. In this paper, we study the topological space Q corresponding to the topology obtained from the closure operator c.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1139-1171
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• 1996

The purpose of our research is to understand geometric and topological aspects of real projective structures on surfaces. A real projective surface is a differentiable surface with an atlas of charts to $RP^2$ such that transition functions are restrictions of projective automorphisms of $RP^2$. Since such an atlas lifts projective geometry on $RP^2$ to the surface locally and consistently, one can study the global projective geometry of surfaces.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.587-599
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• 2000

In this paper we study some nonpositively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. Among other results it is proved that if the universal covering manifold satisfies some conditions then every nonexceptional singular orbit is a totally geodesic submanifold. When M is flat and is not toruslike, it is proved that either each orbit is isometric to $R^k\timesT^m$or there is a singular orbit. If the singular orbit is unique and nonexceptional, then it is isometric to $R^k\timesT^m$.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.445-455
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• 1996

In this paper we study the Reidemeister number $R(f_G)$ for a self-map $f_G : (X, G) \to (X, G)$ of the transformation group (X,G), as an extenstion of the Reidemeister number R(f) for a self-map $f : X \to X$ of a topological space X.