The Experts below are selected from a list of 159 Experts worldwide ranked by ideXlab platform
Núria Agell - One of the best experts on this subject based on the ideXlab platform.
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On fuzzy-qualitative descriptions and entropy
International Journal of Approximate Reasoning, 2016Co-Authors: Francesc Prats, Llorenç Roselló, Mónica Sánchez, Núria AgellAbstract:This paper models the assessments of a group of experts when evaluating different magnitudes, features or objects by using linguistic descriptions. A new general representation of linguistic descriptions is provided by unifying ordinal and fuzzy perspectives. Fuzzy-qualitative labels are proposed as a generalization of the concept of qualitative labels over a Well-Ordered Set. A lattice structure is established in the Set of fuzzy-qualitative labels to enable the introduction of fuzzy-qualitative descriptions as L-fuzzy Sets. A theorem is given that characterizes finite fuzzy partitions using fuzzy-qualitative labels, the cores and supports of which are qualitative labels. This theorem leads to a mathematical justification for commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy Sets. The information of a fuzzy-qualitative label is defined using a measure of specificity, in order to introduce the entropy of fuzzy-qualitative descriptions. A new general representation of linguistic descriptions by unifying ordinal and fuzzy perspectives.The construction of the extended Set of fuzzy-qualitative labels over a Well-Ordered Set.A theorem characterizing commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy Sets.The definition of the fuzzy-qualitative descriptions of a Set as L-fuzzy Sets, and the proof of its lattice structure.The entropy of a fuzzy-qualitative description providing a unified framework for the discrete and continuous cases.
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Using L-fuzzy Sets to introduce information theory into qualitative reasoning
Fuzzy Sets and Systems, 2014Co-Authors: Francesc Prats, Llorenç Roselló, Mónica Sánchez, Núria AgellAbstract:We formally construct the extended Set of qualitative labels L over a Well-Ordered Set. The qualitative descriptions of a given Set are defined as L-fuzzy Sets. In the case where the Well-Ordered Set is finite, a distance between L-fuzzy Sets is introduced based on the properties of the lattice L. The concept of the information contained in a qualitative label is introduced, leading to a formal definition of the entropy of an L-fuzzy Set as a Lebesgue integral. In the discrete case, this integral becomes a weighted average of the information of the labels, corresponding to the Shannon entropy in information theory.
Pete L. Clark - One of the best experts on this subject based on the ideXlab platform.
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A Note on Euclidean Order Types
Order, 2015Co-Authors: Pete L. ClarkAbstract:Euclidean functions with values in an arbitrary Well-Ordered Set were first considered by Motzkin in 1949 and studied in more detail by Samuel and Nagata in the 1970’s and 1980’s. Here these results are revisited, simplified, and extended. The main themes are (i) consideration of O r d -valued functions on an Artinian poSet and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. To any Euclidean ring we associate an ordinal invariant, its Euclidean order type , and we initiate a study of this invariant, especially for Euclidean rings which are not domains.
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A Note on Euclidean Order Types
arXiv: Commutative Algebra, 2012Co-Authors: Pete L. ClarkAbstract:Euclidean functions with values in an arbitrary Well-Ordered Set were first considered in a 1949 work of Motzkin and studied in more detail in work of Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are revisited, simplified, and extended. The two main themes are (i) consideration of Ord-valued functions on an Artinian poSet and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. In particular, to any Euclidean ring we associate an ordinal invariant, its Euclidean order type, and we initiate a study of this invariant. The main new result gives upper and lower bounds on the Euclidean order type of a finite product of Euclidean rings in terms of the Euclidean order types of the factor rings.
Francesc Prats - One of the best experts on this subject based on the ideXlab platform.
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On fuzzy-qualitative descriptions and entropy
International Journal of Approximate Reasoning, 2016Co-Authors: Francesc Prats, Llorenç Roselló, Mónica Sánchez, Núria AgellAbstract:This paper models the assessments of a group of experts when evaluating different magnitudes, features or objects by using linguistic descriptions. A new general representation of linguistic descriptions is provided by unifying ordinal and fuzzy perspectives. Fuzzy-qualitative labels are proposed as a generalization of the concept of qualitative labels over a Well-Ordered Set. A lattice structure is established in the Set of fuzzy-qualitative labels to enable the introduction of fuzzy-qualitative descriptions as L-fuzzy Sets. A theorem is given that characterizes finite fuzzy partitions using fuzzy-qualitative labels, the cores and supports of which are qualitative labels. This theorem leads to a mathematical justification for commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy Sets. The information of a fuzzy-qualitative label is defined using a measure of specificity, in order to introduce the entropy of fuzzy-qualitative descriptions. A new general representation of linguistic descriptions by unifying ordinal and fuzzy perspectives.The construction of the extended Set of fuzzy-qualitative labels over a Well-Ordered Set.A theorem characterizing commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy Sets.The definition of the fuzzy-qualitative descriptions of a Set as L-fuzzy Sets, and the proof of its lattice structure.The entropy of a fuzzy-qualitative description providing a unified framework for the discrete and continuous cases.
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Using L-fuzzy Sets to introduce information theory into qualitative reasoning
Fuzzy Sets and Systems, 2014Co-Authors: Francesc Prats, Llorenç Roselló, Mónica Sánchez, Núria AgellAbstract:We formally construct the extended Set of qualitative labels L over a Well-Ordered Set. The qualitative descriptions of a given Set are defined as L-fuzzy Sets. In the case where the Well-Ordered Set is finite, a distance between L-fuzzy Sets is introduced based on the properties of the lattice L. The concept of the information contained in a qualitative label is introduced, leading to a formal definition of the entropy of an L-fuzzy Set as a Lebesgue integral. In the discrete case, this integral becomes a weighted average of the information of the labels, corresponding to the Shannon entropy in information theory.
Overtoun M. G. Jenda - One of the best experts on this subject based on the ideXlab platform.
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Closure under transfinite extensions
Illinois Journal of Mathematics, 2007Co-Authors: Edgar E. Enochs, Alina Iacob, Overtoun M. G. JendaAbstract:The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen’s theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions. 1. Definitions and basic results Throughout this paper A will be a Grothendieck category with a fixed projective generator U . We will be concerned with direct and inverse limits of systems of objects of A indexed by the Well Ordered Set of ordinals α, where α ≤ λ (or α < λ) for some ordinal λ. To simplify notation, we will denote such a system (direct or inverse) by (Xα | α ≤ λ) with the associated morphisms understood. Definition 1.1. A direct (inverse) system (Xα | α ≤ λ) is said to be continuous if X0 = 0 and if for each limit ordinal β ≤ λ we have Xβ = lim −→α (or Xβ = lim ←− Xα) with the limit over the α < β. The direct (inverse) system (Xα | α ≤ λ) is said to be a system of monomorphisms (epimorphisms) if all the morphisms in the system are monomorphisms (epimorphisms). In order for a continuous direct system (Xα | α ≤ λ) to be a system of monomorphisms it suffices that Xα → Xα+1 be monomorphism whenever α+1 ≤ λ. This follows from what is called the AB5 axiom of a Grothendieck category. If (Xα | α ≤ λ) is a continuous inverse system such that each Xα+1 → Xα (when α + 1 ≤ λ) is an epimorphism, then (Xα | α ≤ λ) is a system of epimorphisms. This is a consequence of the existence of a projective generator U and the fact that (Hom(U,Xα) | α ≤ λ) is a continuous Received May 13, 2005; received in final form July 10, 2006. 2000 Mathematics Subject Classification. 18E15, 18A30. c ©2007 University of Illinois
Peter Schust - One of the best experts on this subject based on the ideXlab platform.
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Well-order as a construction principle for physical theories
The European Physical Journal C, 2019Co-Authors: Peter SchustAbstract:Physics has up to now missed to express in mathematical terms the fundamental idea of events of a path in time and space uniquely succeeding one another. An appropriate mathematical concept that reflects this idea is a Well-Ordered Set. In such a Set every subSet has a least element. Thus every element of a Well-Ordered Set has as its definite successor the least element of the subSet of all elements larger than itself. This is apparently contradictory to the densely Ordered real number lines which conventionally constitute the coordinate axes in any representation of time and space and in which between any two numbers exists always another number. In this article it is shown how decomposing this disaccord in favour of Well-Ordered Sets causes spacetime to be discontinuous.