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Weak Convergence and Weak Convergence

Keiko Narita, Yasunari Shidama, Noboru Endou (2015)

Formalized Mathematics

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18],...

Weakly maximal decidable structures

Alexis Bès, Patrick Cégielski (2008)

RAIRO - Theoretical Informatics and Applications

We prove that there exists a structure M whose monadic second order theory is decidable, and such that the first-order theory of every expansion of M by a constant is undecidable. 


Wide sets, deep many-valuedness and sorites arguments.

Carlos Pelta (2004)

Mathware and Soft Computing

In this article I show how to obtain a powerful and truthful explanation of the failure of sorites arguments combining an adaptation of the Wide Set Theory formulated by Formato and Gerla and the concept of deep many valuedness established by Marraud. It is shown that if the premises of a sorites argument are conceived as a succession of indexed consequence operators (where indices express the accuracy of the inferences) prefixing sentences, the argument fails because the transitive property for...

Yager’s classes of fuzzy implications: some properties and intersections

Michał Baczyński, Balasubramaniam Jayaram (2007)

Kybernetika

Recently, Yager in the article “On some new classes of implication operators and their role in approximate reasoning” [Yager2004] has introduced two new classes of fuzzy implications called the f -generated and g -generated implications. Along similar lines, one of us has proposed another class of fuzzy implications called the h -generated implications. In this article we discuss in detail some properties of the above mentioned classes of fuzzy implications and we describe their relationships amongst...

Δ -tautologies, uniform and non-uniform upper bounds in computation theory

Daniele Mundici (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Una Δ -tautologia è una tautologia del tipo H → K avente un solo interpolante di Craig J , a meno di equivalenza logica. Utilizzando misure di complessità relative al problema di trovare tale J , mostriamo come si possano ottenere limiti non uniformi di complessità mediante limiti uniformi, e viceversa.

σ-ring and σ-algebra of Sets1

Noboru Endou, Kazuhisa Nakasho, Yasunari Shidama (2015)

Formalized Mathematics

In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets [18],...

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