• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.469-473
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• 2013

Let G = (V, E) be a graph and k be a positive integer. A $k$-dominating set of G is a subset $S{\subseteq}V$ such that each vertex in $V{\backslash}S$ has at least $k$ neighbors in S. A Roman $k$-dominating function on G is a function $f$ : V ${\rightarrow}$ {0, 1, 2} such that every vertex ${\upsilon}$ with $f({\upsilon})$ = 0 is adjacent to at least $k$ vertices ${\upsilon}_1$, ${\upsilon}_2$, ${\ldots}$, ${\upsilon}_k$ with $f({\upsilon}_i)$ = 2 for $i$ = 1, 2, ${\ldots}$, $k$. In the paper titled "Roman $k$-domination in graphs" (J. Korean Math. Soc. 46 (2009), no. 6, 1309-1318) K. Kammerling and L. Volkmann showed that for any graph G with $n$ vertices, ${{\gamma}_{kR}}(G)+{{\gamma}_{kR}(\bar{G})}{\geq}$ min $\{2n,4k+1\}$, and the equality holds if and only if $n{\leq}2k$ or $k{\geq}2$ and $n=2k+1$ or $k=1$ and G or $\bar{G}$ has a vertex of degree $n$ - 1 and its complement has a vertex of degree $n$ - 2. In this paper we find a counterexample of Kammerling and Volkmann's result and then give a correction to the result.

• Journal of KIISE:Software and Applications
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• v.28 no.9
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• pp.662-668
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• 2001

Since the problem of disproving a tautology is as hard as the problem of proving it, no polynomial time algorithm for falsification(or testing invalidity) is feasible. Previous algorithms are mostly based on either divide-and-conquer or graph representation. Most of them demonstrated satisfactory results on a variety of input under certain constraints. However, they have experienced difficulties dealing with big input. We propose a new falsification algorithm using a Merge Rule to produce a counterexample by constructing a minterm which is not satisfied by an input expression in DNF(Disjunctive Normal Form). We also show that the algorithm is consistent and sound. The algorithm is based on a greedy method which would seek to maximize the number or terms falsified by the assignment made at each step of the falsification process. Empirical results show practical performance on big input to falsify randomized nontautological problem instances, consuming O(nm$^2$) time, where n is the number of variables and m is number of terms.

• Journal of Korea Game Society
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• v.4 no.1
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• pp.58-66
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• 2004

This paper uses counterexamples for solving reachability games. An objective. of the game we consider here is to find out a minimal path from an initial state to the goal state. We represent initial states and game rules as finite state model and the goal state as temporal logic formula. Then, model checking is used to determine whether the model satisfies the formula. In case the model does not satisfy the formula, model checking generates a counterexample that shows how to reach the goal state from an initial state. In this way, we solve many of small-sized Push Push games. However, we cannot handle larger-sized games due to the state explosion problem. To mitigate the problem, abstraction is used to reduce the state space to be che cked. As a result, unsolved games are solved with the abstraction technique we propose inthis paper.

• Journal of Educational Research in Mathematics
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• v.21 no.4
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• pp.379-398
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• 2011

In learning environment at mathematics education, prove and refute are essential abilities to demonstrate whether and why a statement is true or false. Learning proofs and counter examples within the domain of limit of sequence is important because preservice teacher encounter limit of sequence in many mathematics courses. Recently, a number of studies have showed evidence that pre service and students have problem with mathematical proofs but many research studies have focused on abilities to produce proofs and counter examples in domain of limit of sequence. The aim of this study is to contribute to research on preservice teachers' productions of proofs and counter examples, as participants showed difficulty in writing these proposition. More importantly, the analysis provides insight and understanding into the design of curriculum and instruction that may improve preservice teachers' learning in mathematics courses.

• School Mathematics
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• v.10 no.3
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• pp.319-337
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• 2008

Intuitions in independence formed by common language help and also hinder the establishment of new conceptual system about independence as a mathematical term. Intuitions which entail such conflicts can be a driving force in explaining independence but at the same time, it is the impedimental factor causing a misconception. The goal of this paper is to help students use the intuitions properly by distinguishing helpful intuitions and impedimental intuitions. This paper suggests that we need to reveal in teaching the misconception resulting not from mathematic but from linguistic interpretation of independence. This paper points out the need for the clear distinction of independence of trials and independence of events and gives an counterexample of the case that sampling with and without replacement shouldn't be specified as a representative example of independence and dependence of events. The analysis of intuition in this parer is based on intuitions classified by Fischbein and this paper analyzed institutions applied to the concept of independence corresponding intuitions classified by Fischbein.

• Journal of Educational Research in Mathematics
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• v.18 no.2
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• pp.201-221
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• 2008

This study discusses how the humanity mathematics education can be realized in practice. The essence of mathematical concept is gradually disclosed revealing the ambiguities in the concept currently accepted and clarifying them. Historical development of mathematical concepts has progressed as such, exemplified with the group-theoretical thought and continuous function. In learning of mathematical concepts, thus, students have to recognize, reveal and clarify the ambiguities that intuitive and context-dependent definitions in school mathematics have. We present the process of improvement of definitions of a tangent and a polygon in school mathematics as examples. In the process, students may recognize the limitations of their thoughts and reform them with feelings of humility and satisfaction. Therefore this learning process would contribute to cultivating students' minds as the humanity mathematics education pursues.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.41 no.4
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• pp.220-227
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• 2018

The cooperative game theory consists of a set of players and utility function that has positive values for a subset of players, called coalition, in the game. The purpose of cost allocation method is to allocate the relevant cost among game players in a fair and rational way. Therefore, cost allocation method based on cooperative game theory has been applied in many areas for fair and reasonable cost allocation. On the other hand, the desirable characteristics of the cost allocation method are Pareto optimality, rationality, and marginality. Pareto optimality means that costs are entirely paid by participating players. Rationality means that by joining the grand coalition, players do not pay more than they would if they chose to be part of any smaller coalition of players. Marginality means that players are charged at least enough to cover their marginal costs. If these characteristics are all met, the solution of cost allocation method exists in the core. In this study, proportional method is applied to EOQ inventory game and EPQ inventory game with shortage. Proportional method is a method that allocates costs proportionally to a certain allocator. This method has been applied to a variety of problems because of its convenience and simple calculations. However, depending on what the allocator is used for, the proportional method has a weakness that its solution may not exist in the core. Three allocators such as demand, marginal cost, and cost are considered. We prove that the solution of the proportional method to demand and the proportional method to marginal cost for EOQ game and EPQ game with shortage is in the core. The counterexample also shows that the solution of the proportional method to cost does not exist in the core.