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Topology from Neighbourhoods

Roland Coghetto (2015)

Formalized Mathematics

Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this...

Torsion Part of ℤ-module

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

Torsion Z-module and Torsion-free Z-module

Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2014)

Formalized Mathematics

In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

Totally coherent set-valued probability assessments

Angelo Gilio, Salvatore Ingrassia (1998)

Kybernetika

We introduce the concept of total coherence of a set-valued probability assessment on a family of conditional events. In particular we give sufficient and necessary conditions of total coherence in the case of interval-valued probability assessments. Some relevant cases in which the set-valued probability assessment is represented by the unitary hypercube are also considered.

Totally proper forcing and the Moore-Mrówka problem

Todd Eisworth (2003)

Fundamenta Mathematicae

We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.

Toward a mathematical analysis for a model of suspension flowing down an inclined plane

Matsue, Kaname, Tomoeda, Kyoko (2017)

Proceedings of Equadiff 14

We consider the Riemann problem of the dilute approximation equations with spatiotemporally dependent volume fractions from the full model of suspension, in which the particles settle to the solid substrate and the clear liquid film flows over the sediment [Murisic et al., J. Fluid. Mech. 717, 203–231 (2013)]. We present a method to find shock waves, rarefaction waves for the Riemann problem of this system. Our method is mainly based on [Smoller, Springer-Verlag, New York, second edition, (1994)]....

Towards an extension of the 2-tuple linguistic model to deal with unbalanced linguistic term sets

Mohammed-Amine Abchir, Isis Truck (2013)

Kybernetika

In the domain of Computing with words (CW), fuzzy linguistic approaches are known to be relevant in many decision-making problems. Indeed, they allow us to model the human reasoning in replacing words, assessments, preferences, choices, wishes... by ad hoc variables, such as fuzzy sets or more sophisticated variables. This paper focuses on a particular model: Herrera and Martínez' 2-tuple linguistic model and their approach to deal with unbalanced linguistic term sets. It is interesting since the...

Towards the properties of fuzzy multiplication for fuzzy numbers

Alexandru Mihai Bica, Dorina Fechete, Ioan Fechete (2019)

Kybernetika

In this paper, by using a new representation of fuzzy numbers, namely the ecart-representation, we investigate the possibility to consider such multiplication between fuzzy numbers that is fully distributive. The algebraic and topological properties of the obtained semiring are studied making a comparison with the properties of the existing fuzzy multiplication operations. The properties of the generated fuzzy power are investigated.

Towers of measurable functions

James Hirschorn (2000)

Fundamenta Mathematicae

We formulate variants of the cardinals f, p and t in terms of families of measurable functions, in order to examine the effect upon these cardinals of adding one random real.

Traced premonoidal categories

Nick Benton, Martin Hyland (2003)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in cartesian categories.

Traced Premonoidal Categories

Nick Benton, Martin Hyland (2010)

RAIRO - Theoretical Informatics and Applications

Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in Cartesian categories.

Trames, classifications, définitions

Daniel Parrochia (1991)

Mathématiques et Sciences Humaines

L'article part d'une analogie entre trames et partitions, définitions conceptuelles et optiques. On montre que les divisions d'un espace de concepts ressemblent souvent à celles de l'espace réel. On étudie alors quelques exemples de pavage d'un espace conceptuel (Aristote) et on compare les processus dichotomiques platoniciens (générateurs de définitions) aux filtres d'une algèbre booléenne. Par la suite, on généralise ces modèles, considérant des structures floues et des «ensembles approximatifs»...

Transfinite inductions producing coanalytic sets

Zoltán Vidnyánszky (2014)

Fundamenta Mathematicae

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that consistently there exists an uncountable coanalytic subset of the plane that intersects every C¹ curve in a countable set.

Transition of Consistency and Satisfiability under Language Extensions

Julian J. Schlöder, Peter Koepke (2012)

Formalized Mathematics

This article is the first in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [17] for uncountably large languages. We follow the proof given in [18]. The present article contains the techniques required to expand formal languages. We prove that consistent or satisfiable theories retain these properties under changes to the language they are formulated in.

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