• Title/Summary/Keyword: Riemann integral

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THE DENJOY EXTENSION OF THE RIEMANN INTEGRAL

  • Park, Jae Myung;Kim, Soo Jin
    • Journal of the Chungcheong Mathematical Society
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    • v.9 no.1
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    • pp.101-106
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    • 1996
  • In this paper, we will consider the Denjoy-Riemann integral of functions mapping a closed interval into a Banach space. We will show that a Riemann integrable function on [a, b] is Denjoy-Riemann integrable on [a, b] and that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b].

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The denjoy extension of the mcshane integral

  • Park, Jae-Myung;Lee, Deok-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.411-417
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    • 1996
  • Some generalizations of the Riemann integral have been studied for real-valued functions. One of these generalizations leads to an integral, often called the McShane integral, that is equivalent to the Lebesgue integral.

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NOTE ON CONVERGENCE OF EULER'S GAMMA FUNCTION

  • Choi, Junesang
    • Honam Mathematical Journal
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    • v.35 no.1
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    • pp.101-107
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    • 2013
  • The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

Some algebraic properties and a distance measure for interval-valued fuzzy numbers (쇼케이적분을 이용한 구간치 퍼지수 상의 거리측도에 관한 성질)

  • Jang, Lee-Chae;Kim, Hyun-Mee
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • pp.121-124
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    • 2005
  • Interval-valued fuzzy sets were suggested for the first time by Gorzalczang(1983) and Turken(1986). Based on this, Wang and Li extended their operations on interval-valued fuzzy numbers. Recently, Hong(2002) generalized results of Wang and Li and extended to interval-valued fuzzy sets with Riemann integral. In this paper, we define a distance measure on interval-valued fuzzy numbers using Choquet integral with respect to a classical measure and investigate their properties.

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Some properties of Choquet distance measures for interval-valued fuzzy numbers (구간치 퍼지수 상의 쇼케이 거리측도에 관한 성질)

  • Jang, Lee-Chae;Kim, Won-Joo
    • Journal of the Korean Institute of Intelligent Systems
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    • v.15 no.7
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    • pp.789-793
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    • 2005
  • Interval-valued fuzzy sets were suggested for the first time by Gorzalczang(1983) and Turken(19a6). Based on this, Wang and Li offended their operations on interval-valued fuzzy numbers. Recently, Hong(2002) generalized results of Wang and Li and extended to interval-valued fuzzy sets with Riemann integral. In this paper, using Choquet integrals with respect to a fuzzy measure instead of Riemann integrals with respect to a classical measure, we define a Choquet distance measure for interval-valued fuzzy numbers and investigate its properties.

A study on the Choquet distance measures and their applications (쇼케이 거리측도와 응용에 관한 연구)

  • Jang, Lee-Chae;Kim, Won-Joo
    • Journal of the Korean Institute of Intelligent Systems
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    • v.16 no.5
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    • pp.550-555
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    • 2006
  • Internal-valued fuzzy sets were suggested for the first time by Gorzalczang(1983). Based on this, Wang and Li extended their operations on interval-valued fuzzy numbers. Recently, Hong(2002) generalized results of Wang and Li and extended to interval-valued fuzzy numbers with Riemann integral. By using interval-valued Choquet integrals with respect to a fuzzy measure instead of Riemann integrals with respect to a classical measure, we studied some characterizations of interval-valued Choquet distance(2005). In this paper, we define Choquet distance measure for fuzzy number-valued fuzzy numbers and investigate some properties of them.

A note on the Choquet distance measures for fuzzy number-valued fuzzy numbers (퍼지수치 퍼지수 상의 쇼케이 거리측도에 관한 성질)

  • Jang Lee-Chae;Kim Won-Joo
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • pp.365-369
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    • 2006
  • Interval-valued fuzzy sets were suggested for the first time by Gorzalczang(1983) and Turken(1986). Based on this, Wang and Li extended their operations on interval-valued fuzzy numbers. Recently, Hong(2002) generalized results of Wang and Li and extended to interval-valued fuzzy sets with Riemann integral. Using interval-valued Choquet integrals with respect to a fuzzy measure instead of Riemann integrals with respect to a classical measure, we studied some characterizations of interval-valued Choquet distance(2005). In this paper, we define Choquet distance measure for fuzzy number-valued fuzzy numbers and investigate some algebraic properties of them.

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Development of the Integral Concept (from Riemann to Lebesgue) (적분개념의 발달 (리만적분에서 르베그적분으로의 이행을 중심으로))

  • Kim, Kyung-Hwa
    • Journal for History of Mathematics
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    • v.21 no.3
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    • pp.67-96
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    • 2008
  • In the 19th century Fourier and Dirichlet studied the expansion of "arbitrary" functions into the trigonometric series and this led to the development of the Riemann's definition of the integral. Riemann's integral was considered to be of the highest generality and was discussed intensively. As a result, some weak points were found but, at least at the beginning, these were not considered as the criticism of the Riemann's integral. But after Jordan introduced the theory of content and measure-theoretic approach to the concept of the integral, and after Borel developed the Jordan's theory of content to a theory of measure, Lebesgue joined these two concepts together and obtained a new theory of integral, now known as the "Lebesgue integral".

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