• 한국수학사학회지
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• 제26권5_6호
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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 applied mathematics & informatics
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• 제42권4호
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• pp.969-983
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• 2024

For an integer ℓ ≥ 1, let ${\bar{B}}_{\ell}(n)$ denotes the number of ℓ-regular over partition pairs of n. For certain conditions of ℓ, we study the divisibility of ${\bar{B}}_{\ell}(n)$ and arithmetic properties for ${\bar{B}}_{\ell}(n)$. We further obtain infinite family of congruences modulo 2^t satisfied by ${\bar{B}}_3(n)$ employing a result of Ono and Taguchi (2005) on nilpotency of Hecke operators.

• 대한수학회보
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• 제44권1호
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• pp.131-150
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• 2007

In this paper, we will define direct producted $W^*-porobability$ spaces over their diagonal subalgebras and observe the amalgamated free-ness on them. Also, we will consider the amalgamated free stochastic calculus on such free probabilistic structure. Let ($A_{j},\;{\varphi}_{j}$) be a tracial $W^*-porobability$ spaces, for j = 1,..., N. Then we can define the corresponding direct producted $W^*-porobability$ space (A, E) over its N-th diagonal subalgebra $D_{N}\;{\equiv}\;\mathbb{C}^{{\bigoplus}N}$, where $A={\bigoplus}^{N}_{j=1}\;A_{j}\;and\;E={\bigoplus}^{N}_{j=1}\;{\varphi}_{j}$. In Chapter 1, we show that $D_{N}-valued$ cumulants are direct sum of scalar-valued cumulants. This says that, roughly speaking, the $D_{N}-freeness$ is characterized by the direct sum of scalar-valued freeness. As application, the $D_{N}-semicircularityrity$ and the $D_{N}-valued$ infinitely divisibility are characterized by the direct sum of semicircularity and the direct sum of infinitely divisibility, respectively. In Chapter 2, we will define the $D_{N}-valued$ stochastic integral of $D_{N}-valued$ simple adapted biprocesses with respect to a fixed $D_{N}-valued$ infinitely divisible element which is a $D_{N}-free$ stochastic process. We can see that the free stochastic Ito's formula is naturally extended to the $D_{N}-valued$ case.

• Korean Journal of Mathematics
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• 제8권2호
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• pp.111-119
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• 2000

For a linear operator Q from $R^d$ into $R^d$. ${\alpha}$ > 0 and 0 < $b$ < 1, the Gaussian (Q, $b$, ${\alpha}$)-semi-stability of probability measures on $R^d$ is investigated.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제6권1호
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• pp.9-12
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• 1999

The purpose of this paper is to show that by the divisibility of a direct injective module, we obtain some results with respect to a direct injective module.

• 대한수학회논문집
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• 제10권4호
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• pp.811-813
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• 1995

The purpose of this paper is to prove the divisibility of a direct injective module and every closed submodule of a $\pi$-injective module M is a direct summand of M.

• 대한수학교육학회지:학교수학
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• 제4권4호
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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.

• 한국멀티미디어학회:학술대회논문집
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• 한국멀티미디어학회 1998년도 추계학술발표논문집
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• pp.49-54
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• 1998

초기의 전자 지불 시스템은 안전성과 사용자의 익명성이 주요 관심사였다. 하지만 최근에는 그러한 안정성과 익명성 뿐만 아니라 전자화폐의 분할성과 불법적인 사용자에 대한 부분적인 추적성이 커다란 관심사가 되고 있다. 본 논문에서는 전자화폐의 분할성을 기존의 이진 트리 구조를 이용한 방식과 다른 방식으로 해결함으로써 보다 간단하고 효율적인 방법을 제시하고 부분적인 추적성을 가질 수 있는 방식을 제안하도록 한다.

• 대한수학회지
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• 제38권1호
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• pp.101-123
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• 2001

For a linear operator Q from R(sup)d into R(sup)d and 0$\alpha$ and parameter b on the other. characterization of strictly (Q,b)-semi-stable distributions among (Q,b)-semi-stable distributions is made. Existence of (Q,b)-semi-stable distributions which are not translation of strictly (Q,b)-semi-stable distribution is discussed.

• 대한수학회지
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• 제47권1호
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• pp.101-112
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• 2010

In [3] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. In this paper, we consider the polynomials with coefficients in a field and divisibility of a polynomial by a polynomial with a certain degree is equivalent to the existence of common solution to a system of Diophantine equations. As an application we construct a family of irreducible quartics over $\mathbb{Q}$ which are not of Eisenstein type.