• The Mathematical Education
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• v.60 no.1
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• pp.21-39
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• 2021

The aim of this study was to deduce implications of the growth of mathematics teachers' specialty for effective instruction about the formulae of exponentiation with rational exponents by analyzing teachers' understanding of the definition of rational exponent. In order to accomplish the aim, this study ascertained the nature of the definition of rational exponent through examining previous literature and established specific research questions with reference to the results of the examination. A questionnaire regarding the nature of the definition was developed in order to solve the questions and was taken to 50 in-service high school teachers. By analysing the data from the written responses by the teachers, this study delineated four characteristics of the teachers' understanding with regard to the definition of rational exponent. Firstly, the teachers did not explicitly use the condition of the bases with rational exponents while proving f'(x)=rxr-1. Secondly, few teachers explained the reason why the bases with rational exponents must be positive. Thirdly, there were some teachers who misunderstood the formulae of exponentiation with rational exponents. Lastly, the majority of teachers thought that $(-8)^{\frac{1}{3}}$ equals to -2. Additionally, several issues were discussed related to teacher education for enhancing teachers' knowledge about the definition, features of effective instruction on the formulae of exponentiation and improvement points to explanation methods about the definition and formulae on the current high school textbooks.

• The Mathematical Education
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• v.50 no.1
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• pp.61-67
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• 2011

There may be two methods defining the rational exponent $a^{\frac{m}{n}}$ for any positive real number a. The one which is used in all korean highschool mathematics textbooks is to define it as $\sqrt[n]{a^m}$, that is $(a^m)^{\frac{1}{n}}$. The other is to define it as $(\sqrt[n]a)^m}$, that is $(a^{\frac{1}{n}})^m$. In this paper, we insist that the latter is more appropriate and universal, and that the contents of current textbooks on the definition of the rational exponent should be corrected.

• Communications of Mathematical Education
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• v.25 no.2
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• pp.323-339
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• 2011

The expansion of exponential law as the law of calculation of integer numbers can be a good material for the students to experience an extended configuration which is based on an algebraic principle of the performance of equivalent forms. While current textbooks described that exponential law can be expanded from natural number to integer, rational number and real number, most teachers force students to accept intuitively that the exponential law is valid although exponent is expanded into real number. However most teachers overlook explaining the value of exponent of rational number or exponent of irrational number so most students have a lot of questions whether this value is a rational number or a irrational number. Related to students' questions, most teacher said that it is out of the current curriculum and students will learn it after going to college instead of detailed answers. In this paper, we will present several examples and the values about irrational exponents of a positive rational and irrational exponents of a positive irrational number, and study the recognition and fallacy of would-be teachers about the cases of irrational exponents of a positive rational and irrational exponents of a positive irrational number at the expansion of exponential law.

• The Journal of the Korea institute of electronic communication sciences
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• v.14 no.1
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• pp.185-190
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• 2019

This study aims to apply the mathematical method of fractional order derivatives to the controller that controls the system response. In general, the Laplace transform of the PID controller has an exponent of the integer order of s. The derivative of the fractional order has a fractional exponent of s when it is transformed by Laplace transform. Therefore, this controller proposes a design method with the result of discrete time conversion. Because controllers with fractional exponents of s are not easy to design. This controller is applied to a standard secondary system and its performance is examined. Then, it applies to solenoid valve which is widely used in industrial field. A Luenberger's observer was designed to estimate the disturbance state and the observed state was applied to the fractional order controller. As a result, uniform and precise control performance was obtained. It was confirmed that the position error of the steady state is within 0.1 [%] and the rising time is within about 0.03 [s].

• The Journal of the Korea institute of electronic communication sciences
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• v.16 no.3
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• pp.503-510
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• 2021

This study intends to apply the mathematical method of fractional order differentiation to a controller that controls the response of the system. Therefore, we design integrator for the fractional index by converting it into discrete time to construct a controller. The IP controller composes an integral controller for errors and the proportional controller applies only the system output. The controller is designed by using the fractional order integrator to the integral controller of the IP controller. First, the performance of the PI controller and the IP controller is compared, and the designed controller is applied to the speed control of the motor. As a result, the motor output speed was uniformed and precise control performance could be obtained. It was confirmed that the speed error in the steady state is within 0.1 [%], and it has precise and uniform speed control performance without overshoot.

• Journal of Nuclear Fuel Cycle and Waste Technology(JNFCWT)
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• v.16 no.2
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• pp.183-193
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• 2018

Currently, the Korea Atomic Energy Research Institute (KAERI) is planning to build the Ki-Jang Research Reactor (KJRR) in Ki-Jang, Busan. It is important to safely dispose of low-level radioactive waste from the operation of the reactor. The most efficient way to treat radioactive waste is cement solidification. For a radioactive waste disposal facility, cement solidification is performed based on specific waste acceptance criteria such as compressive strength, free-standing water, immersion and leaching tests. Above all, the leaching test is important to final disposal. The leakage of radioactive waste such as $^{137}Cs$ causes not only regional problems but also serious global ones. The cement solidification method is simple, and cheaper than other solidification methods, but has a lower leaching resistance. Thus, this study was focused on the development of cement solidification for an enhancement of cesium leaching resistance. We used Zeolite and Loess to improve the cesium leaching resistance of KJRR cement solidification containing simulated KJRR liquid waste. Based on an SEM-EDS spectrum analysis, we confirmed that Zeolite and Loess successfully isolated KJRR cement solidification. A leaching test was carried out according to the ANS 16.1 test method. The ANS 16.1 test is performed to analyze cesium ion concentration in leachate of KJRR cement for 90 days. Thus, a leaching test was carried out using simulated KJRR liquid waste containing $3000mg{\cdot}L^{-1}$ of cesium for 90 days. KJRR cement solidification with Zeolite and Loess led to cesium leaching resistance values that were 27.90% and 21.08% higher than the control values. In addition, in several tests such as free-standing water, compressive strength, immersion, and leaching tests, all KJRR cement solidification met the waste acceptance or satisfied the waste acceptance criteria for final disposal.