• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.251-264
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• 1995

Applications of the classical Knaster-Kuratowski-Mazurkiewicz (si-mply, KKM) theorem and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde [L1]. In this direction, the first author [P5] found that certain coincidence theorems on a large class of composites of upper semicontinuous multifunctions imply many fundamental results in the KKM theory.

• Journal of Korean Society of Rural Planning
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• v.11 no.1 s.26
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• pp.9-15
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• 2005

Preservation of geometrical context of terrains in a digitized format is useful in handling and making modification to the data. Digitization of three-dimensional terrain still proves a great challenge due to heavy load of context required to retain details of topological and geometrical information. Methods of simplification, restoration and multi-level terrain generation are often employed to transform the original data into a compressed digital format. However, reduction of the stored data size comes at an expense of loss of details in the original data set. This article reports on an alternative scheme for simplification and restoration of terrain data. The algorithm utilizes the fact that the terrain formation and patterns can be predicted and modeled through the fractal algorithm. This method was used to generate multi-level terrain model based on NGIS digital maps with preserving geometrical context of terrains.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1371-1385
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• 2019

Given a topological space X and a non-negative integer k, ${\varepsilon}^{\sharp}_k(X)$ is the set of all self-homotopy equivalences of X that do not change maps from X to an t-sphere $S^t$ homotopically by the composition for all $t{\geq}k$. This set is a subgroup of the self-homotopy equivalence group ${\varepsilon}(X)$. We find certain homotopic tools for computations of ${\varepsilon}^{\sharp}_k(X)$. Using these results, we determine ${\varepsilon}^{\sharp}_k(M(G,n))$ for $k{\geq}n$, where M(G, n) is a Moore space type of (G, n) for a finitely generated abelian group G.

• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.105-127
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• 2022

It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the first minimal image set. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or topologically almost continuous endomorphisms. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite length, which we call the maximal invariance order, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number ξ, there exists a topologically almost continuous endomorphism f on a compact Hausdorff space X with the maximal invariance order ξ. We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.

• Journal of The Korean Astronomical Society
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• v.29 no.spc1
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• pp.19-21
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• 1996

We apply topological measures of clustering such as percolation and genus curves (PC & GC) and shape statistics to a set of scale free N-body simulations of large scale structure. Both genus and percolation curves evolve with time reflecting growth of non-Gaussianity in the N-body density field. The amplitude of the genus curve decreases with epoch due to non-linear mode coupling, the decrease being more noticeable for spectra with small scale power. Plotted against the filling factor GC shows very little evolution - a surprising result, since the percolation curve shows significant evolution for the same data. Our results indicate that both PC and GC could be used to discriminate between rival models of structure formation and the analysis of CMB maps. Using shape sensitive statistics we find that there is a strong tendency for objects in our simulations to be filament-like, the degree of filamentarity increasing with epoch.

• Journal of the Korea Society of Computer and Information
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• v.18 no.4
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• pp.27-34
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• 2013

Mobile robot has the exploration function in order to perform its own task. Robot exploration can be used in many applications such as surveillance, rescue and resource detection. The workspace that robots performed in was complicated or quite wide, the multi search using the several mobile robots was mainly used. In this paper, we proposed a scheme that all areas are searched for by using one robot. The method to be proposed extract a area that can be explored in the workspace then the robot investigates the area and updates the map at the same time. The explored area is saved as a hybrid map that combines the nice attributes of the grid and topological maps. The robot can produce the safe navigation route without the obstacles by using hybrid map. The proposed hybrid map uses less memory than a grid map, but it can be used for complete coverage with the same efficiency of a topological map. Experimental results show that the proposed scheme can generate a map of an environment with only 6% of the memory that a grid map requires.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.139-149
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• 1989

pp.Hartman and G. Stampacchia [6] proved the following theorem in 1966: If f:X.rarw. $R^{n}$ is a continuous map on a compact convex subset X of $R^{n}$ , then there exists $x_{0}$ ..mem.X such that $x_{0}$ , $x_{0}$ -x>.geq.0 for all x.mem.X. This remarkable result has been investigated and generalized by F.E. Browder [1], [2], W. Takahashi [9], S. Park [8] and others. For example, Browder extended this theorem to a map f defined on a compact convex subser X of a topological vector space E into the dual space $E^{*}$; see [2, Theorem 2]. And Takahashi extended Browder's theorem to closed convex sets in topological vector space; see [9, Theorem 3]. In Section 2, we obtain some variational inequalities, especially, generalizations of Browder's and Takahashi's theorems. The generalization of Browder's is an earlier result of the first author [8]. In Section 3, using Theorem 1, we improve and extend some known fixed pint theorems. Theorems 4 and 8 improve Takahashi's results [9, Theorems 5 and 9], respectively. Theorem 4 extends the first author's fixed point theorem [8, Theorem 8] (Theorem 5 in this paper) which is a generalization of Browder [1, Theroem 1]. Theorem 8 extends Theorem 9 which is a generalization of Browder [2, Theorem 3]. Finally, in Section 4, we obtain variational inequalities for multivalued maps by using Theorem 1. We improve Takahashi's results [9, Theorems 21 and 22] which are generalization of Browder [2, Theorem 6] and the Kakutani fixed point theorem [7], respectively.ani fixed point theorem [7], respectively.

• Journal of the Korean association of regional geographers
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• v.6 no.3
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• pp.83-99
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• 2000

This paper purposes to provide a new perspective for better development of geography texts. For this purpose, we have applied spatial cognition development theory to children's drawings. We have suggested that children's spatial operation ability has three development stages according to their age: topological space, projective space, euclidean space. This study turns out that Piaget and Inhelder's spatial concept development theory is on the right track. However, we make clear that their division according to the age is not always accurate due to children's individual differences. These findings have educational implications as the following: First, it is dubious that most children can understand pictures, pictorial maps and illustrations in the third grader's textbook. Second, current textbooks require pictorial map understanding and drawing to third grade students and map drawing to fourth grade students. However, according to this study, the placement of these tasks are not fit for children's developmental stage because both tasks correspond to euclidean space operation. Therefore, we should remove them from the textbook for children at the age.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.189-196
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• 1990

In this paper, we try to generalize ultraproducts in the category of locally convex spaces. To do so, we introduce D-ultracolimits. It is known [7] that the topology on a non-trivial ultraproduct in the category T $V^{ec}$ of topological vector spaces and continuous linear maps is trivial. To generalize the category Ba $n_{1}$ of Banach spaces and linear contractions, we introduce the category L $C_{1}$ of vector spaces endowed with families of semi-norms closed underfinite joints and linear contractions (see Definition 1.1) and its subcategory, L $C_{2}$ determined by Hausdorff objects of L $C_{1}$. It is shown that L $C_{1}$ contains the category LC of locally convex spaces and continuous linear maps as a coreflective subcategory and that L $C_{2}$ contains the category Nor $m_{1}$ of normed linear spaces and linear contractions as a coreflective subcategory. Thus L $C_{1}$ is a suitable category for the study of locally convex spaces. In L $C_{2}$, we introduce $l_{\infty}$(I. $E_{i}$ ) for a family ( $E_{i}$ )$_{i.mem.I}$ of objects in L $C_{2}$ and then for an ultrafilter u on I. we have a closed subspace $N_{u}$ . Using this, we construct ultraproducts in L $C_{2}$. Using the relationship between Nor $m_{1}$ and L $C_{2}$ and that between Nor $m_{1}$ and Ba $n_{1}$, we show thatour ultraproducts in Nor $m_{1}$ and Ba $n_{1}$ are exactly those in the literatures. For the terminology, we refer to [6] for the category theory and to [8] for ultraproducts in Ba $n_{1}$..

• Journal of The Korean Society of Agricultural Engineers
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• v.49 no.6
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• pp.105-114
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• 2007

This study is to predict the spatial expansion of urban areas by applying CA(Cellular Automata)-Markov technique considering MCE(multi-criteria evaluation) and MOLA(multi-objective land allocation) of factor analysis. For the 10 administration districts$(3677.3km^2)$ including the whole Anseong-cheon watershed, the past six temporal land use data(1973, 1981, 1985, 1990, 1994, 2000) from Landsat satellite images were prepared. During this period, the urban area increased $233.71km^2$. Using the 36 indices composed of topological characteristics, population and land use change, the final factor map of MOLA was produced through 5 maps of MCE. Using 1990 and 1994 land use data, the 2000 predicted urban area of CA-Markov with factor map showed 0.06% improvement of absolute error comparing with that of CA-Markov without factor map. By the CA-Markov technique considering factor map, the 2030 and 2060 urban area increased $58.94km^2(0.78%)\;and\;60.14km^2(0.81%)$ respectively comparing with 2000 urban area$(313.19km^2)$. The 2030 and 2060 paddy area decreased $93.28km^2(2.54%)\;and\;93.65km^2(2.55%)$ respectively comparing with 2000 paddy area$(1383.23km^2)$.