Riemann-Stieltjes Integral

Pole DC	Wartość	Język
dc.contributor.author	Narita, Keiko	-
dc.contributor.author	Nakasho, Kazuhisa	-
dc.contributor.author	Shidama, Yasunari	-
dc.date.accessioned	2017-06-02T11:53:00Z	-
dc.date.available	2017-06-02T11:53:00Z	-
dc.date.issued	2016	-
dc.identifier.citation	Formalized Mathematics, Volume 24, Issue 3, pp. 199-204	pl
dc.identifier.issn	1426-2630	pl
dc.identifier.issn	1898-9934	pl
dc.identifier.uri	http://hdl.handle.net/11320/5554	-
dc.description.abstract	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].	-
dc.language.iso	en	-
dc.publisher	De Gruyter Open	-
dc.subject	Riemann-Stieltjes integral	-
dc.subject	bounded variation	-
dc.subject	linearity	-
dc.title	Riemann-Stieltjes Integral	-
dc.type	Article	-
dc.identifier.doi	10.1515/forma-2016-0016	-
dc.description.Affiliation	Narita Keiko - Hirosaki-city Aomori, Japan	-
dc.description.Affiliation	Nakasho Kazuhisa - Akita Prefectural University Akita, Japan	-
dc.description.Affiliation	Shidama Yasunari - Shinshu University Nagano, Japan	-
dc.description.references	Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.	-
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dc.description.references	H. Kestelman. Modern theories of integration. Dover Publications, 2nd edition, 1960.	-
dc.description.references	Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.	-
dc.description.references	Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from ℝ into real normed space. Formalized Mathematics, 19(1):17-22, 2011.	-
dc.description.references	Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Springer Science & Business Media, 1999.	-
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