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On fuzzy number calculus

100%
Kosiński W.
International Journal of Applied Mathematics and Computer Science
|
2006
|
tom 16
|
nr 1
51-57
EN
Some generalizations of the concept of ordered fuzzy numbers (OFN) are defined to handle fuzzy inputs in a quantitative way, exactly as real numbers are handled. Additional two structures, an algebraic one and a normed (topological) one, are introduced to allow for counting with a more general type of membership relations.
2
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The Basic Existence Theorem of Riemann-Stieltjes Integral

80%
Nakasho K., Narita K., Shidama Y.
Formalized Mathematics
|
2016
|
tom 24
|
nr 4
253-259
EN
In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].
3
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Riemann-Stieltjes Integral

80%
Narita K., Nakasho K., Shidama Y.
Formalized Mathematics
|
2016
|
tom 24
|
nr 3
199-204
EN
In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].
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