• Journal for History of Mathematics
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• v.26 no.5_6
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• pp.323-328
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• 2013

The Division Algorithm is known to be the fundamental foundation for Number Theory and it leads to the Euclidean Algorithm and hence the whole theory of divisibility properties. In JiuZhang SuanShu(九章算術), greatest common divisiors are obtained by the exactly same method as the Euclidean Algorithm in Elements but the other theory on divisibility was not pursued any more in Chinese mathematics. Unlike the other authors of the traditional Chinese mathematics, Zhu ShiJie(朱世傑) noticed in his SuanXue QiMeng(算學啓蒙, 1299) that the Division Algorithm is a really important concept. In [4], we claimed that Zhu wrote the book with a far more deeper insight on mathematical structures. Investigating the Division Algorithm in SuanXue QiMeng in more detail, we show that his theory of Division Algorithm substantiates his structural apporaches to mathematics.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.17-35
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• 2013

The purpose of this study was to analyze the extendibility of division algorithm and Euclid algorithm for integers to algorithms for rational numbers based on word problems of fraction division. This study serviced to upgrade professional development of elementary and secondary mathematics teachers. In this paper, fractions were used as expressions of rational numbers, and they also represent rational numbers. According to discrete context and continuous context, and measurement division and partition division etc, divisibility was classified into two types; one is an abstract algebraic point of view and the other is a generalizing view which preserves division algorithms for integers. In the second view, we raised some contextual problems that can be used in school mathematics and then we discussed division algorithm, the greatest common divisor and the least common multiple, and Euclid algorithm for fractions.

• The Mathematical Education
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• v.50 no.3
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• pp.309-327
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• 2011

In this paper, we extended the division algorithm for the integers to the finite decimals. Though the remainder for the finite decimals is able to be defined as various ways, the remainder could be defined as 'the remained amount' which is the result of the division and as "the remainder" only if 'the remained amount' is decided uniquely by certain conditions. From the definition of "the remainder" for the finite decimal, it could be inferred that 'the division by equal part' and 'the division into equal parts' are proper for the division of the finite decimal concerned with the definition of "the remainder". The finite decimal, based on the unit of measure, seemed to make it possible for us to think "the remainder" both ways: 1" in the division by equal part when the quotient is the discrete amount, and 2" in the division into equal parts when the quotient is not only the discrete amount but also the continuous amount. In this division context, it could be said that the remainder for finite decimal must have the meaning of the justice and the completeness as well. The theorem of the division algorithm for the finite decimal could be accomplished, based on both the unit of measure of "the remainder", and those of the divisor and the dividend. In this paper, the meaning of the division algorithm for the finite decimal was investigated, it is concluded that this theory make it easy to find the remainder in the usual unit as well as in the unusual unit of measure.

• East Asian mathematical journal
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• v.19 no.1
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• pp.81-89
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• 2003

There are many algorithms for factoring integers. The trial division algorithm is one of the most efficient algorithms for factoring small integers(say less than 10,000,000,000). For a number n to be factored, the runtime of the trial division algorithm depends mainly on the size of a nontrivial factor of n. In this paper, we create a function named factors that can implement the trial division algorithm in ActionScript and using the factors function we construct an interactive Prime Factorization Movie and an interactive GCD Movie.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.3
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• pp.395-411
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• 2013

There are two concepts of division: measurement division and partitive or fair-sharing division. Students are expected to understand comprehensively division algorithm in both contexts. Contents of textbooks and teachers' guidebooks should be suitable for helping students develop comprehensive understanding of algorithm for whole-number division in both contexts. The results of the analysis of textbooks and teachers' guidebooks shows that they fail to connect two division contexts with division algorithm comprehensively. Their expedient and improper use of two division contexts would keep students from developing comprehensive understanding of algorithm for whole-number division. Based on the results of analysis, some ways of improving textbooks and teachers' guidebooks are suggested.

• Communications of Mathematical Education
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• v.19 no.4 s.24
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• pp.715-731
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• 2005

There are five contexts of division algorithm of fractions such as measurement division, determination of a unit rate, reduction of the quantities in the same measure, division as the inverse of multiplication and analogy with multiplication algorithm of fractions. The division algorithm, however, should be taught by 'dividing by using reciprocals' via 'measurement division' because dividing a fraction by a fraction results in 'multiplying the dividend by the reciprocal of the divisor'. If a fraction is divided by a large fraction, then we can teach the division algorithm of fractions by analogy with 'dividing by using reciprocals'. To achieve the teaching-learning methods above in elementary school, it is essential for children to use the maniplatives. As Piaget has suggested, Cuisenaire color rods is the most efficient maniplative for teaching fractions. The instruction, therefore, of division algorithm of fractions should be focused on 'dividing by using reciprocals' via 'measurement division' using Cuisenaire color rods through analogy if necessary.

• Journal of Communications and Networks
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• v.14 no.2
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• pp.195-198
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• 2012

An efficient sliding window algorithm employing an adaptive-length guard window for turbo decoders is proposed. The proposed algorithm results in significant complexity reductions, especially for small sliding window lengths where the additional computational complexity required for the guard window is critical.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.521-539
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• 2016

The structures of partitive and quotitive division of fractions are dealt with differently, and this led to using partitive division context for helping develop invert-multiply algorithm and quotitive division for common denominator algorithm. This approach is unlikely to provide children with an opportunity to develop an understanding of common structure involved in solving different types of division. In this study, I propose two approaches, measurement approach and isomorphism approach, to develop a unifying understanding of fraction division. From each of two approaches of solving quotitive division based on proportional reasoning, I discuss an idea of constructing a measure space, unit of which is a quantity of divisor, and another idea of constructing an isomorphic relationship between the measure spaces of dividend and divisor. These ideas support invert-multiply algorithm for quotitive as well as partitive division and bring proportional reasoning into the context of fraction division. I also discuss some curriculum issues regarding fraction division and proportion in order to promote the proposed unifying understanding of partitive and quotitive division of fractions.

• Journal of the Korean Society of Manufacturing Technology Engineers
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• v.25 no.6
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• pp.486-491
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• 2016

Disturbance/vibration reduction is critical in many applications using machine vision. The off-focusing or blurring error caused by vibration degrades the machine performance. In line with this, real-time disturbance estimation and avoidance are proposed in this study instead of going with a more familiar approach, such as the vibration absorber. The instantaneous motion caused by the disturbance is sensed by an attitude heading reference system module. A periodic vibration modeling is conducted to provide a better performance. The algorithm for vibration avoidance is described according to the vibration modeling. The vibration occurrence function is also proposed, and its parameters are determined using the genetic algorithm. The proposed algorithm is experimentally tested for its effectiveness in the vision inspection system.

• Journal of the Korea Society for Simulation
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• v.12 no.2
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• pp.13-24
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• 2003

In this study, we propose a scheduling analysis method of the Flexible management system using the transitive matrix. The Scheduling problem is a combination-optimization problem basically, and a complexity is increased exponentially for a size of the problem. To reduce an increase of a complexity, we define that the basic unit of concurrency (short BUC) is a set of control flows based on behavioral properties in the net. And we propose an algorithm to divide original system into some BUC. To sum up, we divide a petri net model of the Flexible management system Into the basic unit of concurrency through the division algorithm using the transitive matrix. Then we apply it to the division-scheduling algorithm to find an efficient scheduling. Finally, we verify its efficiency with an example.