• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.915-932
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• 2010

The goal of this paper is to study extension problems of several continuities in computer topology. To be specific, for a set $X\;{\subset}\;Z^n$ take a subspace (X, $T_n^X$) induced from the Khalimsky nD space ($Z^n$, $T^n$). Considering (X, $T_n^X$) with one of the k-adjacency relations of $Z^n$, we call it a computer topological space (or a space if not confused) denoted by $X_{n,k}$. In addition, we introduce several kinds of k-retracts of $X_{n,k}$, investigate their properties related to several continuities and homeomorphisms in computer topology and study extension problems of these continuities in relation with these k-retracts.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.625-642
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• 2016

The recent papers [7, 12] developed two maps named by an LA- [7] and an LMA-map [12]. The present paper studies their properties and further, develops generalized versions of an LA- and an LMA-map, which makes the LA- [7] and the LMA-map [12] improved.

• ETRI Journal
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• v.27 no.2
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• pp.235-238
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• 2005

A new mesh reconstruction scheme for approximating a surface from a set of unorganized 3D points is proposed. The proposed method, called a shrink-wrapped boundary face (SWBF) algorithm, produces the final surface by iteratively shrinking the initial mesh generated from the definition of the boundary faces. SWBF surmounts the genus-0 spherical topology restriction of previous shrink-wrapping-based mesh generation techniques and can be applied to any type of surface topology. Furthermore, SWBF is significantly faster than a related algorithm of Jeong and others, as SWBF requires only a local nearest-point-search in the shrinking process. Our experiments show that SWBF is very robust and efficient for surface reconstruction from an unorganized point cloud.

• The Bulletin of The Korean Astronomical Society
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• v.35 no.1
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• pp.82-82
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• 2010

We measure the topology of the galaxy distribution using the Seventh Data Release of the Sloan Digital Sky Survey (SDSS DR7), examining the dependence of galaxy clustering topology on galaxy properties. The observational results are used to test galaxy formation models. A volume-limited sample defined by Mr<-20.19 enables us to measure the genus curve with amplitude of G=378 at 6h-1Mpc smoothing scale, with 4.8% uncertainty including all systematics and cosmic variance. The clustering topology over the smoothing length interval from 6 to 10h-1Mpc reveals a mild scale-dependence for the shift and void abundance (A_V) parameters of the genus curve. We find strong bias in the topology of galaxy clustering with respect to the predicted topology of the matter distribution, which is also scale-dependent. The luminosity dependence of galaxy clustering topology discovered by Park et al. (2005) is confirmed: the distribution of relatively brighter galaxies shows a greater prevalence of isolated clusters and more percolated voids. We find that galaxy clustering topology depends also on morphology and color. Even though early (late)-type galaxies show topology similar to that of red (blue) galaxies, the morphology dependence of topology is not identical to the color dependence. In particular, the void abundance parameter A_V depends on morphology more strongly than on color. We test five galaxy assignment schemes applied to cosmological N-body simulations to generate mock galaxies: the Halo-Galaxy one-to-one Correspondence (HGC) model, the Halo Occupation Distribution (HOD) model, and three implementations of Semi-Analytic Models (SAMs). None of the models reproduces all aspects of the observed clustering topology; the deviations vary from one model to another but include statistically significant discrepancies in the abundance of isolated voids or isolated clusters and the amplitude and overall shift of the genus curve. SAM predictions of the topology color-dependence are usually correct in sign but incorrect in magnitude.

• Honam Mathematical Journal
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• v.29 no.1
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• pp.101-118
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• 2007

The notions of digital k-homotopy equivalence and digital ($k_0,k_1$)-homotopy equivalence were developed in [13, 16]. By the use of the digital k-homotopy equivalence, we can investigate digital k-homotopy equivalent properties of Cartesian products constructed by the minimal simple closed 4- and 8-curves in $\mathbf{Z}^2$.

• Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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• v.28 no.2
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• pp.84-91
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• 2014

This paper introduces the application of supervisory monitoring system for substation automation based on IEC 61850. The objective of proposed system is detection of such a malfunction or degradation of devices. The supervisory monitoring procedure consists of a two step - topology processor and state estimation. The topology processor using artificial intelligence is a preprocessing step of state estimation. Topology processor identifies the topology structure of switches in substation and detects an error of ON/OFF state data. The state estimation is an algorithm that minimizes an error between optimal estimation values and real values. The proposed system is applied to standard digital substation based on IEC 61850 for performance verification.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.1-16
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• 2023

In this paper, we examine the relations of two closely related concepts, the digital Lusternik-Schnirelmann category and the digital higher topological complexity, with each other in digital images. For some certain digital images, we introduce κ-topological groups in the digital topological manner for having stronger ideas about the digital higher topological complexity. Our aim is to improve the understanding of the digital higher topological complexity. We present examples and counterexamples for κ-topological groups.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.695-706
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• 2014

Various properties of digital covering spaces have been substantially used in studying digital homotopic properties of digital images. In particular, these are so related to the study of a digital fundamental group, a classification of digital images, an automorphism group of a digital covering space and so forth. The goal of the present paper, as a survey article, to speak out utility of digital covering theory. Besides, the present paper recalls that the papers [1, 4, 30] took their own approaches into the study of a digital fundamental group. For instance, they consider the digital fundamental group of the special digital image (X, 4), where X := $SC^{2,8}_4$ which is a simple closed 4-curve with eight elements in $Z^2$, as a group which is isomorphic to an infinite cyclic group such as (Z, +). In spite of this approach, they could not propose any digital topological tools to get the result. Namely, the papers [4, 30] consider a simple closed 4 or 8-curve to be a kind of simple closed curve from the viewpoint of a Hausdorff topological structure, i.e. a continuous analogue induced by an algebraic topological approach. However, in digital topology we need to develop a digital topological tool to calculate a digital fundamental group of a given digital space. Finally, the paper [9] firstly developed the notion of a digital covering space and further, the advanced and simplified version was proposed in [21]. Thus the present paper refers the history and the process of calculating a digital fundamental group by using various tools and some utilities of digital covering spaces. Furthermore, we deal with some parts of the preprint [11] which were not published in a journal (see Theorems 4.3 and 4.4). Finally, the paper suggests an efficient process of the calculation of digital fundamental groups of digital images.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.595-608
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• 2015

In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

• 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.