• School Mathematics
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• v.9 no.4
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• pp.523-533
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• 2007

This article focused the meaning of Pythagoras' and Euclid's proof about the Pythagorean theorem in a historical and mathematical perspective. Pythagoras' proof using similarity is based on the arithmetic assumption about commensurability. However, Euclid proved the Pythagorean theorem again only using the concept of dissection-rearrangement that is purely geometric so that it does not need commensurability. Pythagoras' and Euclid's different approaches to geometry have to do with Birkhoff's axiom system and Hilbert's axiom system in the school geometry Birkhoff proposed the new axioms for plane geometry accepting real number that is strictly defined. Thus Birkhoff's metrical approach can be defined as a Pythagorean approach that developed geometry based on number. On the other hand, Hilbert succeeded Euclid who had pursued pure geometry that did not depend on number. The difference between the proof using similarity and dissection-rearrangement is related to the unsolved problem in the geometry curriculum that is conflict of Euclid's conventional synthetical approach and modern mathematical approach to geometry.

• Communications of Mathematical Education
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• v.32 no.2
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• pp.207-223
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• 2018

In this study, we studied Bumwoo KIM Chi Young's cogitation on mathematics, and analyzed his typical 3 essays on mathematics by KoNLP. Approximately 80% of Bumwoo's sentences consist of less than 30. His writing became clearer over the years. It is verified from the mean and standard deviation of the number of words in a sentence are decreasing. Bumwoo emphasized the structure in mathematics, and he was a strong advocate of importancy on axiom, topolized and category as the characteristics of modern mathematics. In particular, it can be seen that the relations between 'mathematics', 'axiom', 'structure', 'Euclid', 'axiomatic system' and 'set' were his main topic.

• Journal of KIISE:Software and Applications
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• v.35 no.12
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• pp.764-773
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• 2008

The web ontology language(OWL) has become a W3C recommendation to publish and share ontologies on the semantic web. In order to check the satisfiablity of concepts in OWL ontology, OWL reasoners have been introduced. But most reasoners simply report check results without providing a justification for any arbitrary entailment of unsatisfiable concept in OWL ontologies. In this paper, we propose dependency label based causing inconsistency axiom (CIA) detection for debugging unsatisfiable concepts in ontology. CIA is a set of axioms to occur unsatisfiable concepts. In order to detect CIA, we need to find axiom to cause inconsistency in ontology. If precise CIA is gave to ontology building tools, these ontology tools display CIA to debug unsatisfiable concepts as suitable presentation format. Our work focuses on two key aspects. First, when a inconsistency ontology is given, it detect axioms to occur unsatisfiable and identify the root of them. Second, when particular unsatisfiable concepts in an ontology are detected, it extracts them and presents to ontology designers. Therefore we introduce a tableau-based decision procedure and propose an improved method which is dependency label based causing inconsistency axiom detection. Our results are applicable to the very expressive logic SHOIN that is the basis of the Web Ontology Language.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1131-1142
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• 2018

A notion of measure expansivity for homeomorphisms was introduced by Morales recently as a generalization of expansivity, and he obtained many interesting dynamic results of measure expansive homeomorphisms in [8]. In this paper, we introduce a concept of weak measure expansivity for homeomorphisms which is really weaker than that of measure expansivity, and show that a diffeomorphism f on a compact smooth manifold is $C^1$-stably weak measure expansive if and only if it is ${\Omega}$-stable. Moreover we show that $C^1$-generically, if f is weak measure expansive, then f satisfies both Axiom A and the no cycle condition.

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.127-144
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• 2009

Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.10 s.181
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• pp.2438-2450
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• 2000

The Design Axioms provide a general framework for design methodologies. The axiomatic design framework has been successfully applied to various design tasks. However, the axiomatic design has been rarely utilized in the detailed design process of structures where the optimization technology is generally carried out. The relationship between the axiomatic design and the optimization is investigated and Logical Decomposition method is developed for a systematic structural optimization. The entire optimization process is decomposed to satisfy the Independence Axiom. In the decomposition process, design variables are grouped according to sensitivities. The sensitivities are evaluated by the Analysis of Variance(ANOVA) to avoid considering only local values. The developed method is verified through examples such as the twenty -five members transmission tower and the two -bay-six-story frame.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2002.05a
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• pp.555-560
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• 2002

This paper addresses the methodologies for manufacturing systems design. While a number of design methods are used in product or part design, methods for manufacturing systems design are rarely known. Two approaches, simulation and axiomatic design theory, are discussed with respective case examples. The usual purpose of using simulation is to identify the bottleneck of a manufacturing system or to evaluate its performance with the aim of configuring the manufacturing system. The simulation typically proceeds in steps such as problem definition, model building, numerical experimentation, analysis and evaluation. The axiomatic design method transforms customer attributes into functional requirements and repeats mapping processes between functional domain and physical one until a satisfactory level of refinement of the functional requirements and the design parameters is reached. Possible design alternatives are evaluated by applying the independence axiom as well as the information axiom.

• Honam Mathematical Journal
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• v.35 no.1
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• pp.101-107
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• 2013

The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.4
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• pp.1333-1346
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• 1996

As the automation system in manufacturing field works more efficientely, the automation scheme is applied to many areas. In order to reduce the entire manufacturing, cost the design process must be automated. However, design process is so complicated, it is very difficult to construct the design automation system. The axiomatic approach to design provides a general theoretical framework for all design fields, including mechanical design. The key concepts of axiomatic design are : the existence of domains, the characteristic vectors within the domains that can be decomposed into hierarchies through zigzagging between the domains, and the design axioms. Using this approach, the glass bulb design process was analyzed and the design automation software was developed. Through menu display, a user can select or furnish the design input and generate the drawing with ease.

• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.363-447
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• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.