• 한국수학교육학회지시리즈E:수학교육논문집
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• 제29권3호
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• pp.353-372
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• 2015

본 연구에서는 초등학교 수학 교과서에서 수 개념 학습과 관련하여 수직선이 어떻게 활용되고 있는지 살펴보고, 현재 교과서의 수직선 활용과 관련된 문제점을 자연수와 유리수(분수, 소수)의 개념 학습을 중심으로 분석하였다. 초등학교 수학 교과서에서 수직선의 도입 시기, 도입 내용, 활용 방법에 대한 분석을 바탕으로 수직선 활용 방안과 관련한 시사점을 도출하고자 하였다.

• 대한수학회논문집
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• 제38권4호
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• pp.1309-1320
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• 2023

Let f : X → Y be a map between simply connected CW-complexes of finite type with X finite. In this paper, we prove that the rational cohomology of mapping spaces map(X, Y ; f) contains a polynomial algebra over a generator of degree N, where N = max{i, π_i(Y)⊗ℚ ≠ 0} is an even number. Moreover, we are interested in determining the rational homotopy type of map(𝕊ⁿ, ℂP^m; f) and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

• 대한수학교육학회지:수학교육학연구
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• 제14권1호
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• pp.1-17
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• 2004

본 연구는 중학교 과정에서 순환소수의 취급에 관하여 알아보는 것으로 교육과정에 제시된 관련 내용을 분석하고 그에 따른 현행 교과서를 살펴보아 문제점을 알아보았고, 다음에 관련된 분야의 일부 외국교과서를 비교 분석하여 보았다. 현행 교육과정과 교과서 보다 바람직한 지도방안은 우선 체계적인 학습을 할 수 있도록 교육과정에서보다 적합한 학습내용과 그 취급을 제시해야만하고, 이에 따라 교과서도 보다 적합하게 순환소수를 취급하고 그에 따른 무리수를 도입하는 것이 바람직 할 것이다. 특히 순환소수는 무한소수가 아닌 그냥 소수로 도입하여 숫자 0을 순환마디로 사용할 것을 제시하고, 교육의 다양성을 위해서 직관적이기는 하지만 현행교과서에서의 취급보다는 일반적인 방법으로 순환소수와 유리수의 관계를 명확히 규명하여 무리수의 도입을 무한소수로서 잘 도입하도록 제시하였다. 그리고 무리수라는 용어의 도입만은 현행 교육과정과는 달리 순환소수의 취급 과정에서 함께 다루는 것이 바람직함을 제시하였다.

• Journal of Mechanical Science and Technology
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• 제20권5호
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• pp.658-667
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• 2006

Fractional derivative models, which are used to describe the viscoelastic behavior of material, have received considerable attention. Thus it is necessary to put forward the analysis solutions of dynamic systems containing a fractional derivative. Although previously reported such kind of fractional calculus-based constitutive models, it only handles the particularity of rational number in part, has great limitation by reason of only handling with particular rational number field. Simultaneously, the former study has great unreliability by reason of using the complementary error function which can't ensure uniform real number. In this paper, a new approach is proposed for an analytical scheme for dynamic system of a spring-mass-damper system of single-degree of freedom under general forcing conditions, whose damping is described by a fractional derivative of the order $0<{\alpha}<1$ which can be both irrational number and rational number. The new approach combines the fractional Green's function and Laplace transform of fractional derivative. Analytical examples of dynamic system under general forcing conditions obtained by means of this approach verify the feasibility very well with much higher reliability and universality.

• 산업기술연구
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• 제18권
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• pp.181-186
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• 1998

The flatness is the most important nature for the surface table. For finding such a flatness, the surface is surveyed along a number of straight lines parallel to the edges of table, which form a grid. Next, the variations in height of the grid points are measured relative to a datum point. If the number of such points is increased. It is not necessarily to use many grid points for finding the original flatness of a measured surface table. So, it is necessary to find the rational quantity of such grid points. It is found that about 220 points per $1m^2$ of surface table for measurement is the rational quantity with less than about 15% error of the original flatness.

• 한국수학사학회지
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• 제13권2호
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• pp.49-64
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• 2000

Every real number can be expressed as a simple continued fraction. In particular, a number is rational if and only if its simple continued fraction has a finite number of terms. Owing to this property, continued fractions have been a powerful tool which determines a real number to be rational or not. Continued fractions provide not only a series of best estimate for a real number, but also a useful method for finding near commensurabilities between events with different periods. In this paper, we investigate the history and some properties of continued fractions, and then consider their applications in several examples. Also we explain why the Fibonacci numbers and the Golden section appear in nature in terms of continued fractions, with some examples such as the arrangements of petals round a flower, leaves round branches and seeds on seed head.

• 대한수학회지
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• 제54권4호
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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.

• 대한수학회지
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• 제46권1호
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• pp.151-170
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• 2009

Let m be a positive integer. We show that for any given real number ${\alpha}\;{\in}\;[0,\;1]$ and complex number $\mu$ with $|\mu|{\leq}1$ which satisfy $e^{2{\pi}i{\alpha}}{\mu}^m\;{\neq}\;1$, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number $\alpha$. Moreover if the given real number $\alpha$ is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier ${\mu}^m$ at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제8권1호
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• pp.1-17
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• 2014

Cooperative spectrum sensing (CSS) is considered as a powerful approach to improve the utilization of scarce radio spectrum resources. However, most of CSS schemes assume all secondary users (SU) are honest, and thus offering opportunities for malicious SUs to launch the spectrum sensing data falsification attack (SSDF attack). To combat such misbehaved behaviors, recent efforts have been made to trust schemes. In this paper, we argue that powering CSS with traditional trust schemes is not enough. The rational SSDF attack is found in this paper. Unlike the simple SSDF attack, rational SSDF attackers send out false sensing data on a small number of interested primary users (PUs) rather than all PUs. In this case, rational SSDF attackers can keep up high trustworthiness, resulting in difficultly detecting malicious SUs in the traditional trust schemes. Meanwhile, a defense scheme using a novel trust approach is proposed to counter rational SSDF attack. Simulation results show that this scheme can successfully reduce the power of rational SSDF, and thus ensure the performance of CSS.

• 대한수학회지
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• 제56권2호
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• pp.559-578
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• 2019

We list all subfields of cyclotomic function fields over rational function fields with class number three. We also determine all the imaginary abelian extensions with relative class number three, explicitly.