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Divergence measure between fuzzy sets using cardinality

Vladimír Kobza (2017)

Kybernetika

In this paper we extend the concept of measuring difference between two fuzzy subsets defined on a finite universe. The first main section is devoted to the local divergence measures. We propose a divergence measure based on the scalar cardinalities of fuzzy sets with respect to the basic axioms. In the next step we introduce the divergence based on the generating function and the appropriate distances. The other approach to the divergence measure is motivated by class of the rational similarity...

Divisible ℤ-modules

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

Division schemes under uncertainty of claims

Xianghui Li, Yang Li, Wei Zheng (2021)

Kybernetika

In some economic or social division problems, we may encounter uncertainty of claims, that is, a certain amount of estate has to be divided among some claimants who have individual claims on the estate, and the corresponding claim of each claimant can vary within a closed interval or fuzzy interval. In this paper, we classify the division problems under uncertainty of claims into three subclasses and present several division schemes from the perspective of axiomatizations, which are consistent with...

Domain mu-calculus

Guo-Qiang Zhang (2003)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The basic framework of domain μ -calculus was formulated in [39] more than ten years ago. This paper provides an improved formulation of a fragment of the μ -calculus without function space or powerdomain constructions, and studies some open problems related to this μ -calculus such as decidability and expressive power. A class of language equations is introduced for encoding μ -formulas in order to derive results related to decidability and expressive power of non-trivial fragments of the domain μ -calculus....

Domain mu-calculus

Guo-Qiang Zhang (2010)

RAIRO - Theoretical Informatics and Applications

The basic framework of domain μ-calculus was formulated in [39] more than ten years ago. This paper provides an improved formulation of a fragment of the μ-calculus without function space or powerdomain constructions, and studies some open problems related to this μ-calculus such as decidability and expressive power. A class of language equations is introduced for encoding μ-formulas in order to derive results related to decidability and expressive power of non-trivial fragments of the domain...

Domain-Free λµ-Calculus

Ken-Etsu Fujita (2010)

RAIRO - Theoretical Informatics and Applications

We introduce a domain-free λµ-calculus of call-by-value as a short-hand for the second order Church-style. Our motivation comes from the observation that in Curry-style polymorphic calculi, control operators such as callcc-operators cannot, in general, handle correctly the terms placed on the control operator's left, so that the Curry-style system can fail to prove the subject reduction property. Following the continuation semantics, we also discuss the notion of values in classical system,...

Double Sequences and Iterated Limits in Regular Space

Roland Coghetto (2016)

Formalized Mathematics

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter...

Double Sequences and Limits

Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.

Double Series and Sums

Noboru Endou (2014)

Formalized Mathematics

In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In...

Dual Spaces and Hahn-Banach Theorem

Keiko Narita, Noboru Endou, Yasunari Shidama (2014)

Formalized Mathematics

In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach...

Easy lambda-terms are not always simple

Alberto Carraro, Antonino Salibra (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ-terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ-calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N, it is possible...

Easy lambda-terms are not always simple

Alberto Carraro, Antonino Salibra (2012)

RAIRO - Theoretical Informatics and Applications

A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ-terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ-calculus living in the category of complete partial orderings: given ...

Egoroff's Theorem

Noboru Endou, Yasunari Shidama, Keiko Narita (2008)

Formalized Mathematics

The goal of this article is to prove Egoroff's Theorem [13]. However, there are not enough theorems related to sequence of measurable functions in Mizar Mathematical Library. So we proved many theorems about them. At the end of this article, we showed Egoroff's theorem.MML identifier: MESFUNC8, version: 7.8.10 4.100.1011

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