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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 meaning of verbal quantities

Milan Mareš, Radko Mesiar (2002)

Kybernetika

The aim of the paper is to summarize and interpret some ideas regarding effective processing of vague data. The main contribution of the submitted approach consists in respecting the fact that vague data can be decomposed into two parts. The numerical one, describing the quantitative value of such data, and the semantic one characterizing the qualitative structure of the vagueness included into them. This partition of vague verbal data leads to a significant simplification of their practical processing,...

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...

Dualidad en la programación lineal en subconjuntos difusos.

José Llena Sitjes (1988)

Trabajos de Investigación Operativa

La programación lineal sobre subconjuntos difusos, definida por Zimmermann, se desarrolla en estrecha relación con la definición de las funciones pertinentes funciones de pertenencia. Se estudia la dualidad difusa, ligada a la dualidad en los problemas de programación lineal con multicriterios.

Duality for Hilbert algebras with supremum: An application

Hernando Gaitan (2017)

Mathematica Bohemica

We modify slightly the definition of H -partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of H ∨ -space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.

Dynamical properties of the automorphism groups of the random poset and random distributive lattice

Alexander S. Kechris, Miodrag Sokić (2012)

Fundamenta Mathematicae

A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property....

Easton functions and supercompactness

Brent Cody, Sy-David Friedman, Radek Honzik (2014)

Fundamenta Mathematicae

Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving...

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 ...

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