• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.971-985
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• 2007

A sequential optimality condition characterizing the efficient solution without any constraint qualification for an abstract convex vector optimization problem is given in sequential forms using subdifferentials and ${\epsilon}$-subdifferentials. Another sequential condition involving only the subdifferentials, but at nearby points to the efficient solution for constraints, is also derived. Moreover, we present a proposition with a sufficient condition for an efficient solution to be properly efficient, which are a generalization of the well-known Isermann result for a linear vector optimization problem. An example is given to illustrate the significance of our main results. Also, we give an example showing that the proper efficiency may not imply certain closeness assumption.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.437-452
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• 2007

Complementarity, complementary principle and complementary approach have been often used in school mathematics but its meaning has not been obvious. Thus this paper tries to make explicit the meaning by looking around complementary characteristic of mathematical knowledge. First of all, we examines the general meaning of complementarity and Investigate complementary characteristics of mathematical concepts through incommensurability and zeno's paradox. From this, complementary approach to school mathematics is studied. To understand and uncover complementary characteristics of mathematical concepts make it possible for student to have an insight. It is the most important thing that students can have an image of mathematics as a living system rather than as a mechanical application of rules and fragmentary in formations.

• Journal of the Korean School Mathematics Society
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• v.24 no.3
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• pp.261-282
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• 2021

This study identified the meaning of mathematical justification and its characteristics for middle school math gifted students. 17 middle school math gifted students participated in questionnaires and written exams. Results show that the gifted students recognized justification in various meanings such as proof, systematization, discovery, intellectual challenge of mathematical justification, and the preference for deductive justification. As a result of justification exams, there was a difference in algebra and geometry. While there were many deductive justifications in both algebra and geometry questionnaires, the difference exists in empirical justifications: there were many empirical justifications in algebra, but there were few in geometry questions. When deductive justification was completed, the students showed satisfaction with their own justification. However, they showed dissatisfaction when they could not deductively justify the generality of the proposition using mathematical symbols. From the results of the study, it was found that justification education that can improve algebraic translation ability is necessary so that gifted students can realize the limitations and usefulness of empirical reasoning and make deductive justification.

• Communications of Mathematical Education
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• v.27 no.4
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• pp.567-580
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• 2013

As integrated essay writing is performed in university entrance examinations, teachers and students recognize the importance of integrated essay, but teachers have still difficulties of teaching methods. The purpose of this study is to derive educational implications through case of mathematics instruction for integrated essay education to pre-service mathematics teachers. The content knowledge of this class is a definition of conic section in mathematics and properties of conic section in an antenna reflector. The students have to discover them using the history of math, manipulative material, paper-folding and computer simulation. In this teaching and learning process the students can realize mathematical knowledge invented by humans through history of mathematics. The students can evaluate the validity of that as create and justify a mathematical proposition. Also, the students can explain the relation between them logically and descript cause or basis convincingly in the process of justifying. We should keep our study to instructional materials and teaching methods in integrated essay education.

• Journal for History of Mathematics
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• v.21 no.2
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• pp.55-70
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• 2008

This study is about the Data which is one of Euclid's writing. It dealt with the organization of contents, formal system and mathematical meaning. First, we investigated the organization of contents of the Data. Second, on the basis of this investigation, we analyzed the formal system of the Data. It contains the analysis of described method of definition, proposition, proof and the meaning of 'given'. Third, we explored the mathematical meaning of the Data which can be classified as algebraic point of view, geometric point of view and the opposite point of view to 'The Elements'.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.105-113
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• 1994

In the course of study of dendroids, Czuba [3] introduced a notion of $R^{i}$ -continua which is a generalization of R-arc [1]. He showed a new class of non-contractible dendroids, namely of dendroids which contain an $R^{i}$ -continuum. Subsecequently Charatonik [2] attempted to extend the notion into hyperspace C(X) of metric continuum X. In so doing, there were some oversights in extending some of the results relating $R^{i}$ -continua of dendroids for metric continua. In fact, Proposition 1 in [2] is false (see example C below) and his proof of Theorem 6 in [2] is not correct (Take Example 4 in [4] with K = [e,e'] as an $R^{1}$-continuum of X and work it out. Then one seens that K not .mem. K as he claimed otherwise.). The aims of this paper are to introduce a notion of w-regular convergence which is weaker than 0-regular convergence and to prove that the w-regular convergence of a sequence {Xn}$^{\infty}$$_{n=1}$ to $X_{0}$ of subcontinua of a metric continuum X is a necessary and sufficient for the sequence {C( $X_{n}$)}$^{\infty}$$_{n=1}$ to converge to C( $X_{0}$ ), and also to prove that if a metric continuum X contains an $R^{i}$ -continuum with w-regular convergence, then the hyperspace C(X) of X contains $R^{i}$ -continuum.inuum.uum.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.297-303
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• 2007

In this paper, we study on C-closed spaces, SC-closed spaces and related spaces. We show that a sequentially compact SC-closed space is sequential and as corollaries obtain that a sequentially compact space with unique sequential limits is sequential if and only if it is C-closed [7, 1.19 Proposition] and every sequentially compact SC-closed space is C-closed. We also show that a countably compact WAP and C-closed space is sequential and obtain that a countably compact (or compact or sequentially compact) WAP-space with unique sequential limits is sequential if and only if it is C-closed as a corollary. Finally we prove that a weakly discretely generated AP-space is C-closed. We then obtain that every countably compact (or compact or sequentially compact) weakly discretely generated AP-space is $Fr\acute{e}chet$-Urysohn with unique sequential limits, for weakly discretely generated AP-spaces, unique sequential limits ${\equiv}KC{\equiv}C-closed{\equiv}SC-closed$, and every continuous surjective function from a countably compact (or compact or sequentially compact) space onto a weakly discretely generated AP-space is closed as corollaries.

• Honam Mathematical Journal
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• v.8 no.1
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• pp.51-74
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• 1986

The thery of measure is significant in that we extend from it to the theory of integration. AS specific metric outer measures we can take Hausdorff outer measure and Lebesgue-Stieltjes outer measure connecting measure with monotone functions.([12]) The purpose of this paper is to find some properties of Lebesgue-Stieltjes measure by extending it from $R^1$ to $R^n(n{\geq}1)$ $({\S}3)$ and differentiation of the integral defined by Borel measure $({\S}4)$. If in detail, as follows. We proved that if $_n{\lambda}_{f}^{\ast}$ is Lebesgue-Stieltjes outer measure defined on a finite monotone increasing function $f:R{\rightarrow}R$ with the right continuity, then $$_n{\lambda}_{f}^{\ast}(I)=\prod_{j=1}^{n}(f(b_j)-f(a_j))$$, where $I={(x_1,...,x_n){\mid}a_j$<$x_j{\leq}b_j,\;j=1,...,n}$. (Theorem 3.6). We've reached the conclusion of an extension of Lebesgue Differentiation Theorem in the course of proving that the class of continuous function on $R^n$ with compact support is dense in $L^p(d{\mu})$ ($1{\leq$}p<$\infty$) (Proposition 2.4). That is, if f is locally $\mu$-integrable on $R^n$, then $\lim_{h\to\0}\left(\frac{1}{{\mu}(Q_x(h))}\right)\int_{Qx(h)}f\;d{\mu}=f(x)\;a.e.(\mu)$.

• Communications of Mathematical Education
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• v.26 no.1
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• pp.51-69
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• 2012

The cognition of an intrinsic attribute play an important role in the process of theoretical generalization. It is the aim of this paper to study how the theoretical generalization is made. First of all, we suggest the What-if-only-strategy(WIOS) which is the strategy helping the cognition of an intrinsic attribute. And we propose the process of the theoretical generalization that go on the cognitive stage, WIOS stage, conjecture stage, justification stage and insight into an intrinsic attribute in order. We propose the process of generalization adding the concrete process cognizing an intrinsic attribute to the existing process of generalization. And we applied the proposed process of generalization to two mathematical theorem which is being managed in middle school. We got a conclusion that the what-if-only strategy is an useful method of generalization for the proposition. We hope that the what-if-only strategy is helpful for both teaching and learning the mathematical generalization.

• Journal of Mechanical Science and Technology
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• v.14 no.7
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• pp.796-806
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• 2000

To understand the transient behavior of droplets after impingement in a diesel engine, a numerical model for diesel sprays impinging on a flat wall is newly developed by the proposition of several mathematical formulae to determine the post-impingement characteristics of droplets. The new model consists of three representative regimes such as rebound, deposition and splash. The gas phase is modeled in terms of the Eulerian conservation equations, and the dispersed phase is calculated using a discrete droplet model. To validate the new model, the calculated results are compared with several experimental data. The results show that the new model is generally in good agreement with the experimental data. Therefore, it is thought that the new model is acceptable for the prediction of transient behavior of wall sprays.