• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1155-1162
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• 2010

In this paper, we find the maximum exponent of D ${\times}$ E, the cartesian product of two digraphs D and E on n, n + 2 vertices, respectively for an even integer $n\geq4$. We also characterize the extrema cases.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.279-287
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• 2015

A subset S of V (G), where G is a graph without isolated vertices, is a double dominating set of G if for each $x{{\in}}V(G)$, ${\mid}N_G[x]{\cap}S{\mid}{\geq}2$. This paper, shows that any positive integers a, b and n with $2{\leq}a<b$, $b{\geq}2a$ and $n{\geq}b+2a-2$, can be realized as domination number, double domination number and order, respectively. It also characterize the double dominating sets in the Cartesian and tensor products of two graphs and determine sharp bounds for the double domination numbers of these graphs. In particular, it show that if G and H are any connected non-trivial graphs of orders n and m respectively, then ${\gamma}_{{\times}2}(G{\square}H){\leq}min\{m{\gamma}_2(G),n{\gamma}_2(H)\}$, where ${\gamma}_2$, is the 2-domination parameter.

• Korean Journal of Mathematics
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• v.2
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• pp.73-77
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• 1994

• Korean Journal of Mathematics
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• v.25 no.2
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• pp.279-301
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• 2017

An L(3, 2, 1)-labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that ${\mid}f(u)-f({\upsilon}){\mid}{\geq}4-k$ when $dist(u,{\upsilon})=k{\leq}3$. The L(3, 2, 1)-labeling number, denoted by ${\lambda}_{3,2,1}(G)$, for G is the smallest number N such that there is an L(3, 2, 1)-labeling for G with span N. In this paper, we compute the L(3, 2, 1)-labeling number ${\lambda}_{3,2,1}(G)$ when G is a cylindrical grid, which is the cartesian product $P_m{\Box}C_n$ of the path and the cycle, when $m{\geq}4$ and $n{\geq}138$. Especially when n is a multiple of 4, or m = 4 and n is a multiple of 6, then we have ${\lambda}_{3,2,1}(G)=11$. Otherwise ${\lambda}_{3,2,1}(G)=12$.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.213-226
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• 2017

The concept of fuzzy transitive filters is introduced in BE-algebras. Some sufficient conditions are established for every fuzzy filter of a BE-algebra to become a fuzzy transitive filter. Some properties of fuzzy transitive filters are studied with respect to fuzzy relations and cartesian products.

• Proceedings of the Korea Multimedia Society Conference
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• 1998.10a
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• pp.202-207
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• 1998

The network reliability is to be computed in terms of the terminal reliability. The computation of a terminal reliability is started with a Boolean sum of products expression corresponding to simple paths of the pair of nodes. This expression is then transformed into another equivalent expression to be a Disjoint Sum of Products form. But this computation of the terminal reliability obviously does not consider the communication between any other nodes but for the source and the sink. In this paper, we derive the overall network reliability which all other remaining nodes. For this, we propose a method to make the SOP disjoint for deriving the network reliability expression from the system success expression using the modified Sheinman's method. Our method includes the concept of spanning trees to find the system success function by the Cartesian products of vertex cutsets.

• Journal of Korea Foundry Society
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• v.43 no.6
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• pp.302-309
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• 2023

In general, castings often have complex shapes and significant variations in thickness within a single product, making grid generation for simulations challenging. Casting flows involve multiphase flows, requiring the tracking of the boundary between air and molten metal. Additionally, considerable time is spent calculating pressure fields due to density differences in a numerical analysis. For these reasons, the Cartesian grid system has traditionally been used in mold filling simulations. However, orthogonal grids fail to represent shapes accurately, leading to a momentum loss caused by the stair-like grid patterns on curved and sloped surfaces. This can alter the flow of molten metals and result in incorrect casting process designs. To address this issue, simulations in the Cartesian grid system involve creating a large number of grids to represent shapes more accurately. Alternatively, the Cut Cell method can be applied to address the problems arising from the Cartesian grid system. In this study, analysis results based on the number of grid in the Cartesian grid system for a casting flow analysis were compared with results obtained using the Cut Cell method. Casting flow simulations of actual products during various casting processes were also conducted, and these results were analyzed with and without applying the Cut Cell method.

• 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 $\mathbf{Z}^2$.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.175-190
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• 2019

In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of the product of arbitrary graphs G and H based on their respective clique numbers.

• 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.