• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2141-2147
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• 2017

Let G be a finite group and m a divisor of ${\mid}G{\mid}$. We prove that G has at least ${\tau}(m)$ cyclic subgroups whose orders divide m, where ${\tau}(m)$ is the number of divisors of m.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.21 no.1
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• pp.55-60
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• 2021

This paper relates with the performance evaluation of QE-MMA (Quantized Error-MMA) adaptive equalization algorithm based on the stepsize and quantizer bit number in order to reduce the intersymbol interference due to nonlinear distortion occurred in the time dispersive channel. The QE-MMA was proposed using the power-of-two arithmetic for the H/W implementation easiness substitutes the multiplication and addition into the shift and addition in the tap coefficient updates process that modifies the SE-MMA which use the high-order statistics of transmitted signal and sign of error signal. But it has different adaptive equalization performance by the step size and quantizer bit number for obtain the sign of error in the generation of error signal in QE-MMA, and it was confirmed by computer simulation. As a simulation, it was confirmed that the convergence speed for reaching steady state depend on stepsize and the residual quantities after steady state depend on the quantizer bit number in the QE-MMA adaptive equalization algorithm performance.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.24 no.12A
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• pp.2015-2024
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• 1999

Presented in this paper are a new complex-umber filter architecture, which is suitable for an efficient implementation of baseband signal processing of digital communication systems, and a chip-set design of adaptive decision-feedback equalizer (ADFE) employing the proposed structure. The basic concept behind the approach proposed in this paper is to apply redundant binary (RB) arithmetic instead of conventional 2’s complement arithmetic in order to achieve an efficient realization of complex-number multiplication and accumulation. With the proposed way, an N-tap complex-number filter can be realized using 2N RB multipliers and 2N-2 RB adders, and each filter tap has its critical delay of $T_{m.RB}+T_{a.RB}$ (where $T_{m.RB}, T_{a.RB}$are delays of a RB multiplier and a RB adder, respectively), making the filter structure simple, as well as resulting in enhanced speed by means of reduced arithmetic operations. To demonstrate the proposed idea, a prototype ADFE chip-set, FFEM (Feed-Forward Equalizer Module) and DFEM (Decision-Feedback Equalizer Module) that can be cascaded to implement longer filter taps, has been designed. Each module is composed of two complex-number filter taps with their LMS coefficient update circuits, and contains about 26,000 gates. The chip-set was modeled and verified using COSSAP and VHDL, and synthesized using 0.8- μm SOG (Sea-Of-Gate) cell library.

• The Journal of the Korea institute of electronic communication sciences
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• v.12 no.1
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• pp.101-108
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• 2017

As the quantum gate matrix is a $r^{n+1}{\times}r^{n+1}$ dimension when the radix is r, the number of control state vectors is n, and the number of target state vectors is one, the matrix dimension with increasing n is exponentially increasing. If the number of control state vectors is $2^n$, then the number of $2^n-1$ unit matrix operations preserves the output from the input, and only one can be performed the unitary operation to the target state vector. Therefore, this paper proposes a new method of function embedding that can replace $2^n-1$ times of unit matrix operations with deterministic contribution to matrix dimension by arithmetic power switch of the unitary gate. The proposed function embedding method uses a binary literal switch with a multivalued threshold, so that a general purpose hybrid MCU gate can be realized in a $r{\times}r$ unitary matrix.

• Education of Primary School Mathematics
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• v.4 no.1
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• pp.43-49
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• 2000

The purpose of this paper is to address the theory of counting and the development of number concepts. Leslie Steffe and his colleagues developed the theory of children's counting types using the teaching experiment. As the results of their research, they published two books： "Children's counting types" (Steffe, von Glasersfeld, Richards, '||'&'||' Cobb, 1983) and "Construction of arithmetic meanings and strategies" (Steffe, Cobb, ＆ von Glasersfeld, 1988). They classified children's counting types into five categories： Perceptual Counting Stage, Figural Counting Stages, Initial Number Sequence Stage, Tacitly Nested Number Sequence Stage, and Explicitly Nested Number Sequence Stage. The meaning of this theory is added in the last part of this paper. this paper.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.10 no.4
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• pp.81-93
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• 2000

In a methodological approach to improve the processing performance of modulo exponentiation which is the primary arithmetic in RSA crypto algorithm, we present a new RSA hardware architecture based on high-radix modulo multiplication and CRT(Chinese Remainder Theorem). By implementing the modulo multiplier using radix-16 arithmetic, we reduced the number of PE(Processing Element)s by quarter comparing to the binary arithmetic scheme. This leads to having the number of clock cycles and the delay of pipelining flip-flops be reduced by quarter respectively. Because the receiver knows p and q, factors of N, it is possible to apply the CRT to the decryption process. To use CRT, we made two s/2-bit multipliers operating in parallel at decryption, which accomplished 4 times faster performance than when not using the CRT. In encryption phase, the two s/2-bit multipliers can be connected to make a s-bit linear multiplier for the s-bit arithmetic operation. We limited the encryption exponent size up to 17-bit to maintain high speed, We implemented a linear array modulo multiplier by projecting horizontally the DG of Montgomery algorithm. The H/W proposed here performs encryption with 15Mbps bit-rate and decryption with 1.22Mbps, when estimated with reference to Samsung 0.5um CMOS Standard Cell Library, which is the fastest among the publications at present.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1345-1356
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• 2017

For a given number field K, we show that the ranks of elliptic curves over K are uniformly finitely bounded if and only if the weak Mordell-Weil property holds in all (some) ultrapowers $^*K$ of K. We introduce the nonstandard weak Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*{\mathbb{Z}}$-module, where $^*{\mathbb{Z}}$ is an ultrapower of ${\mathbb{Z}}$, and we show that the nonstandard weak Mordell-Weil property is equivalent to the weak Mordell-Weil property in $^*K$. In a saturated nonstandard number field, there is a nonstandard ring of integers $^*{\mathbb{Z}}$, which is definable. We can consider definable abelian groups as $^*{\mathbb{Z}}$-modules so that the nonstandard weak Mordell-Weil property is well-defined, and we conclude that the nonstandard weak Mordell-Weil property and the weak Mordell-Weil property are equivalent. We have valuations induced from prime numbers in nonstandard rational number fields, and using these valuations, we identify two nonstandard rational numbers.

• School Mathematics
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• v.4 no.4
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• pp.643-655
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• 2002

In this paper we emphasize the introduction of ‘incommensurability’ on the teaching and learning the irrational number because we think of the origin of number as ‘ratio’. According to Greek classification of continuity as a ‘never ending’ divisibility, discrete number and continuous magnitude belong to another classes. That is, those components were dealt with respectively in category of arithmetic and that of geometry. But the comparison between magnitudes in terms of their ratios took the opportunity to relate ratios of magnitudes with numerical ratios. And at last Stevin coped with discrete and continuous quantity at the same time, using his instrumental decimal notation. We pay attention to the fact that Stevin constructed his number conception in reflecting the practice of measurement : He substituted ‘subdivision of units’ for ‘divisibility of quantities’. Number was the result of such a reflective abstraction. In other words, number was invented by regulation of measurement. Therefore, we suggest decimal representation from the point of measurement, considering the foregoing historical development of number. From the perspective that the conception of real number originated from measurement of ‘continuum’ and infinite decimals played a significant role in the ‘representation’ of measurement, decimal expression of real number should be introduced through contexts of measurement instead of being introduced as a result of algorithm.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.1
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• pp.129-142
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• 2013

The purpose of this study was to investigate the typical characteristics of mathematics education of Japan. In order to achieve this goal, we focused the journal 'Arithmetic Education' from 2007 to 2011. This journal has published by the Japan Society of Mathematical Education and 6 issues each year. A total of 133 articles related with mathematics education were analyzed. The typical characteristics of Japan's research for mathematics education were as follows: The number of single author of article were 98 cases (74%), and those of two co-authors were 21 cases (16%). There are some unusual research topic for mathematics education such as 'combined class'. 'cultural pluralism' and 'mathematics learning disabled children'. The articles 'statistical methods related to educational evaluation', 'statistical analysis for educational evaluation' and 'the relationship between number and quantity on the quaternion number' are very interesting results to the readers who know the basics of statistics and algebra. We may find many researcher who majored pure mathematics in the University of Educations in Korea. So we hope that they may write the paper which combine the pure mathematics and mathematics education. The education survey conducted by the policy is actually very meaningful. If the researcher can connect these surveys to the field of education, then the readers can see a nice paper in the journal of elementary mathematics education in Korea. Finally, it is very difficult to find that counterstatement paper for the results of the other's.

• Journal of Educational Research in Mathematics
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• v.21 no.3
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• pp.239-259
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• 2011

This study investigated the elementary school students' ability on the algebraic reasoning as generalized arithmetic. It analyzed the written responses from 648 second graders, 688 fourth graders, and 751 sixth graders using tests probing their understanding of the properties of whole number operations. The result of this study showed that many students did not recognize the properties of operations in the problem situations, and had difficulties in applying such properties to solve the problems. Even lower graders were quite successful in using the commutative law both in addition and subtraction. However they had difficulties in using the associative and the distributive law. These difficulties remained even for upper graders. As for the associative and the distributive law, students had more difficulties in solving the problems dealing with specific numbers than those of arbitrary numbers. Given these results, this paper includes issues and implications on how to foster early algebraic reasoning ability in the elementary school.