• The Mathematical Education
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• v.43 no.4
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• pp.381-390
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• 2004

In this study, we conceptualized the ‘preformal proof’, which is a transitive level of proof from the experimental and inductive justification to the formalized mathematical proof. We investigated concrete features of the preformal proof in the historico-genetic and the didactical situations. The preformal proof can get the generality of the contents of proof, which makes a distinction from the experimental proof. And we can draw a distinction between the preformal and formal proof, in point that the preformal proof heads for the reality-oriented objects and does not use the formal language.

• Journal of Educational Research in Mathematics
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• v.8 no.1
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• pp.313-326
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• 1998

The purpose of this study is to verify the meaning of preformal proof and its didactical significance in mathematics education. A preformal proof plays a more important role in mathematics education, because nowadays in mathematics a proof is considered as an important fact from a sociological point of view. A preformal proof was classified into four categories: a) action proof, b) geometric-intuitive proof, c) reality oriented proof, d) proof by generalization from paradiam. An educational significance of a preformal proof are followings: a) A proof is not identified with a formal proof. b) A proof is not only considered from a symbolic level, but also from enactive and iconic level. c) A preformal proof generates a formal proof and convinces pupils of a formal proof d) A preformal proof is psychologically natural. e) A preformal proof changes a conception of what is a proof. Therefore a preformal proof is expected to teach in school mathematics from the elementary school to the secondary school.

• Journal for History of Mathematics
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• v.21 no.3
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• pp.127-142
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• 2008

In this paper, we introduce four different approaches of proving Cayley formula, which counts the number of trees(acyclic connected simple graphs). The first proof was done by Cayley using recursive formulas. On the other hands the core idea of the other three proofs is the bijective method-find an one to one correspondence between the set of trees and a suitable family of combinatorial objects. Each of the three bijection gives its own generalization of Cayley formula. In particular, the last proof, done by Seo and Shin, has an application to computer science(theoretical computation), which is a typical example that pure mathematics supply powerful tools to other research fields.

• The Korean Journal of Rheology
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• v.11 no.1
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• pp.9-17
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• 1999

Viscoelastic characteristics of cured phenolic resin/carbon fiber composite materials were investigated through glass transition and degradation reaction processes in the high temperature region up to $400^{\circ}C$. A typical glass transition of the cross-linked thermoset polymer was followed by irreversible degradation reactions, which were exhibited by the increasing storage modulus and loss modulus peak. A degradation master curve was constructed by using the vertical and horizontal shift factors, both of which complied well with the Arrhenius equation in light of the kinetic expression of degradation rate constants. Using an analogy to the Havriliak-Negami equation in dielectric relaxation phenomena, a viscoelastic modeling methodology was developed to characterize the frequency- and temperature-dependent complex moduli of the degrading thermoset polymer composite systems. The temperature-dependent relaxation time of the degrading composites was determined in a continuous fashion and showed a minimum relaxation time between the glass transition and degradation reaction regions. The capability of the developed modeling methodology was demonstrated by describing the complex behavior of the viscoelastic complex moduli of reacting phenolic resin composite systems.

• Korean Journal of Fisheries and Aquatic Sciences
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• v.22 no.5
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• pp.332-341
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• 1989

Stomach contents and microstructures in otolith of Chaenogobius laevis were analyzed for the determination of age, growth and food. By comparaing structural similarity in a series of growth increments from three samples collected in a two-week interval, daily deposition of growth increments in otolith can be validated. Growth in length as daily basis was well represented by Gompertz curve: $L= 5.73{\cdot}\;e^{15.06}(1-e^{-0.0015t})$ for the fish age of $46\~102$ days. Mean growth rate increased from 0.40 mm/day for the age of $50\~60$ days to 0.85 mm/day for the age of $80\~100$ days. Chasnobius laevis showed a carnivorous feeding behavior and its major food items were polychaetes, amphipods and copepods. Small individuals ($15\~30$ mm SL) preyed heavily on copepods as well as polychaetes. However, the portion of copepods in stomach contents was decreased with increasing fish siEe, and this decrease was compensated by an increased consumption of amphipods.

• Korean Chemical Engineering Research
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• v.46 no.4
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• pp.739-746
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• 2008

The aim of this study is finding the optimal design parameters and the optimal operation variables of a reactive distillation column. Different from steady state optimization, dynamic optimization makes it possible considering operation ability as well as design problems at process design step. For performing dynamic optimization, dynamic simulation should be done first. If dynamic simulation is already finished, dynamic optimization can be performed with less effort than that of dynamic simulation.Reactive distillation systems involving reaction and separation in a single unit have the potential to reduce capital and operating costs, particularly when reaction have conversion constraint or when azeotropes exist making conventional separation difficult and expensive. This study here present work on the continuous distillation process, the homogeneous catalyzed esterification of methanol and acetic acid, the synthesis of methyl acetate. Based on an equilibrium stage model of a reactive distillation column a dynamic optimization problem was formulated and solved. And the results were verified by performing dynamic simulation and showing the variation of conversion and purity as the variation of the operation variables. As the results of dynamic optimization, this study found optimal feed ratio, reflux ratio and reboiler duty of this system. And as this study applied it to dynamic simulations the dynamic characteristics of a reactive distillation column are showed under optimal operating condition.