• Education of Primary School Mathematics
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• v.8 no.1
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• pp.33-50
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• 2004

This study is intended to investigate the effect on the development of multiplicative thinking and multiplicative ability by teaching repeated addition, rate, comparison, area-array, and combination problems. Two research questions are established: first, is there any difference of multiplicative thinking between the experimental group(the modeling of problem situation learning group) and the control group(the traditional learning group)\ulcorner Second, is there any difference of multiplicative ability between the experimental group and the control group\ulcorner The treatment process for the experimental group is based on modeling problem situations for nine lesson periods. In order to answer the research questions the chi-square analysis was used for the first research question and the t-test was used for the second one. The findings are summarized as follows: there is no significant difference of multiplicative thinking be1ween the experimental and the control group but there is significant difference of multiplicative ability.

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.205-221
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• 2016

We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.

• Journal of Mechanical Science and Technology
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• v.17 no.4
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• pp.571-580
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• 2003

Tomography has been investigated to observe bubble behaviors in two-phase flows. A bubbly flow and an annular flow have been reconstructed by tomography methods such as an algebraic reconstruction technique (ART) and a multiplicative algebraic reconstruction technique (MART) . Computer synthesized phantom fields have been used to calculate asymmetric density distributions for limited cases of 3, 5, and 7 projection angles. As a result of comparison of two tomography methods, the MART method has shown a significant improvement in the reconstruction accuracy for analysis of the two-phase flows.

• The Mathematical Education
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• v.50 no.3
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• pp.337-354
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• 2011

The purpose of the study was to investigate how students understand multiplication and division of fractions and how their understanding influences the solutions of fractional word problems. Thirteen students from 5th to 6th grades were involved in the study. Students' understanding of operations with fractions was categorized into "a part of the parts", "multiplicative comparison", "equal groups", "area of a rectangular", and "computational procedures of fractional multiplication (e.g., multiply the numerators and denominators separately)" for multiplications, and "sharing", "measuring", "multiplicative inverse", and "computational procedures of fractional division (e.g., multiply by the reciprocal)" for divisions. Most students understood multiplications as a situation of multiplicative comparison, and divisions as a situation of measuring. In addition, some students understood operations of fractions as computational procedures without associating these operations with the particular situations (e.g., equal groups, sharing). Most students tended to solve the word problems based on their semantic structure of these operations. Students with the same understanding of multiplication and division of fractions showed some commonalities during solving word problems. Particularly, some students who understood operations on fractions as computational procedures without assigning meanings could not solve word problems with fractions successfully compared to other students.

• Journal of Educational Research in Mathematics
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• v.25 no.1
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• pp.21-58
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• 2015

The aim of this study is to look into the didactical background for teaching proportional reasoning in elementary school mathematics and offer suggestions to improve teaching proportional reasoning in the future. In order to attain these purposes, this study extracted and examined key ideas with respect to the didactical background on teaching proportional reasoning through a theoretical consideration regarding various studies on proportional reasoning. Based on such examination, this study compared and analyzed textbooks used in the United States, the United Kingdom, and South Korea. In the light of such theoretical consideration and analytical results, this study provided suggestions for improving teaching proportional reasoning in elementary schools in Korea as follows: giving much weight on proportional reasoning, emphasizing multiplicative comparison and discerning between additive comparison and multiplicative comparison, underlining the ratio concept as an equivalent relation, balancing between comparisons tasks and missing value tasks inclusive of quantitative and qualitative, algebraic and geometrical aspects, emphasizing informal strategies of students before teaching cross-product method, and utilizing informal and pre-formal models actively.

• Education of Primary School Mathematics
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• v.21 no.2
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• pp.237-260
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• 2018

This study investigated the models of four countries' elementary mathematics textbooks in Ratio and Rate and identified how multiplicative perspectives on proportional relationships and the structure of proportion situations are reflected in the textbooks. For this, textbooks of 5th and 6th grade textbooks in Korea Japan, Singapore and U.S. are compared and analyzed. As a result, we can find multiplicative perspectives on proportional relationships and the structure of proportion situations on pictorial models, ratio tables, double number lines and double tape diagrams. Also, the development of Japanese textbooks from multiple batches perspectives to variable parts perspectives and the examples of the use with two models together implied the connection and union of two multiplicative perspectives. Based on these results, careful verification and discussion for the next textbook is needed to develop students' proportional reasoning and teach some effective reasoning strategies. And this study will provide the implication for what kinds of and how visual models are presented in the next textbook.

• Wind and Structures
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• v.27 no.3
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• pp.149-161
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• 2018

A reliability analysis method is proposed in this paper based on the maximum entropy (MaxEnt) principle in which constraints are specified in terms of the fractional moments instead of integer moments. Then a multiplicative dimensional reduction method (M-DRM) is introduced to compute the fractional moments. The method is applicable for both explicit and implicit limit state functions of complex structures. After two examples illustrate the accuracy and efficiency of this method in comparison to the Monte Carlo simulation (MCS), the method is used to analyze the flutter reliability of suspension bridge. The results show that the empirical formula method in which the limit state function is explicitly represented as a function of variables is only a too conservative estimate for flutter reliability analysis but is not accurate adequately. So it is not suitable for reliability analysis of bridge flutter. The actual flutter reliability analysis should be conducted based on a finite element method in which limit state function is implicitly represented as a function of variables. The proposed M-DRM provide an alternate and efficient way to analyze a much more complicated flutter reliability of long span suspension bridge.

• Journal of the Korean Society for Nondestructive Testing
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• v.22 no.5
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• pp.545-556
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• 2002

Bubble behaviors in two-phase flows have been analyzed by tomography methods such as an algebraic reconstruction technique (ART) and a multiplicative algebraic reconstruction technique (MART). Initially, a bubbly flow and an annular flow have been investigated by cross-sectional view using computer synthesized phantoms. Two tomography methods have been compared to obtain more accurate results of the two-phase flows. Then, reconstruction of three-dimensional density distributions of phantoms with two and three bubbles have been accomplished by the MART method which provided the better results for the two-dimensional reconstructions accurately to analyze the bubble behaviors in the two-phase flow.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.2A
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• pp.102-109
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• 2003

In this paper, the new multiplicative algorithm using standard basis over GF(2m) is proposed. The multiplicative process is simplified by data select method in proposed algorithm. After multiplicative operation, the terms of degree greater than m can be expressed as a polynomial of standard basis with degree less than m by irreducible polynomial. For circuit implementation of proposed algorithm, we design the circuit using multiplexer and show the example over GF(24). The proposed architectures are regular and simple extension for m. Also, the comparison result show that the proposed architecture is more simple than privious multipliers. Therefore, it well suited for VLSI realization and application other operation circuits.

• Transactions of The Korea Fluid Power Systems Society
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• v.3 no.2
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• pp.14-20
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• 2006

A controller design procedure for an electro-hydraulic positioning systems has been developed using $H{\infty}$ control. The generalized plant models and weighting function for multiplicative uncertainty modelling error was presented along with $H{\infty}$ controller designs in order to investigate the robust stability and performance. Both disturbance rejection and command tracking performances were improved with the $H{\infty}$ controller, and the better uniformity of time response is achieved across wide range of operating conditions than the PID, LQR and LQG control scheme. The multiplicative uncertainty case was specifically suited for the design of an electro-hydraulic positioning control systems using $H{\infty}$ control.