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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

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.

The Basic Existence Theorem of Riemann-Stieltjes Integral

80%

Nakasho K. , Narita K. , Shidama Y.

Formalized Mathematics

2016

tom 24

nr 4

253-259

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].

Riemann-Stieltjes Integral

80%

Narita K. , Nakasho K. , Shidama Y.

Formalized Mathematics

2016

tom 24

nr 3

199-204

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].

A Representation Theorem for '-Bounded Variation of Functions in the Sense of Riesz

70%

Aziz W. , Leiva H. , Merentes N. , Rzepka B.

2010

tom Vol. 50, [Z] 2

109-120

In this paper we extend the well known Riesz lemma to the class of bounded φ-variation functions in the sense of Riesz defined on a rectangle [...].This concept was introduced in [2], where the authors proved that the space [...] of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].

Generalized Köthe-Toeplitz dual of some double sequence spaces

51%

Tripathy B. C. , Sarma B.

2008

tom Nr 40

119-125

In this article we introduce the notion of η-dual of double sequence spaces. We find the η-dual of some double sequence spaces. We verify the perfectness of different double sequence spaces relative to η-dual.

On the Kantorovich variant of generalized Bernstein type rational functions

51%

Gupta V. , Ispir N.

2006

tom Vol. 39, nr 1

117-130

In the present paper we define Kantorovich variant of generalized Bernstein type rational functions. We establish the order of approximation for continuous functions in different normed spaces and also estimate the rate of convergence for functions of bounded variation.