By Michał Baczyński (auth.), Humberto Bustince, Javier Fernandez, Radko Mesiar, Tomasas Calvo (eds.)

ISBN-10: 3642391648

ISBN-13: 9783642391644

ISBN-10: 3642391656

ISBN-13: 9783642391651

This quantity collects the prolonged abstracts of forty five contributions of members to the 7th overseas summer time institution on Aggregation Operators (AGOP 2013), held at Pamplona in July, 16-20, 2013. those contributions conceal a truly wide variety, from the only theoretical ones to these with a extra utilized concentration. in addition, the summaries of the plenary talks and tutorials given on the similar workshop are included.

Together they supply an outstanding evaluate of modern developments in learn in aggregation capabilities which might be of curiosity to either researchers in Physics or arithmetic engaged on the theoretical foundation of aggregation capabilities, and to engineers who require them for applications.

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J. : Copulas, marginals, and joint distributions. In: Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993). IMS Lecture Notes Monogr. , vol. 28, pp. 213–222. Inst. Math. : Quantitative risk management. Concepts, Techniques and Tools. Princeton Series in Finance. : On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econom. : Clustering heteroskedastic time series by model–based procedures. Comput. Statist. Data Anal. : On the return period and design in a multivariate framework.

On the other hand, for each c ∈]0, 12 ], Cp is a copula. 44 R. Mesiar, J. Komorník, and M. Komorníková Open problems: i) For each convex strong negation N, putting p = (Π , Π , λ , N, N), the function Cp is a copula for λ ∈ {−1, 0}. Is this claim valid for each λ ∈ [−1, 0]? Are there some other constant λ so that Cp is a copula? ii) For two convex strong negations N1 , N2 , and some λ ∈ ℜ, does p = (Π , Π , λ , N1 , N2 ) generate a copula Cp applying (9)? Example 3. Consider the standard negation Ns .

It is well known that any fuzzy implication I provides a fuzzy negation NI , the so-called natural negation of I, defined as NI (x) = I(x, 0) for any x ∈ [0, 1], and that the former can be generalized as follows (see [4]): Proposition 3. Let I be a fuzzy implication and let α ∈ [0, 1[ verify I(1, α ) = 0. Then the function NIα : [0, 1] → [0, 1], given by NIα (x) = I(x, α ) for any x ∈ [0, 1], is a fuzzy negation called the natural negation of I with respect to α . Therefore, any implication has at least one natural negation associated to it (NI , which is equal to NI0 ), but can have a family of them as long as there exists α > 0 such that I(1, α ) = 0: indeed, in such cases, thanks to the non-decreasing monoβ tonicity of I in its second variable, any NI with β ≤ α is also a natural negation of β I, which, in addition, verifies NI ≤ NIα .

### Aggregation Functions in Theory and in Practise: Proceedings of the 7th International Summer School on Aggregation Operators at the Public University of Navarra, Pamplona, Spain, July 16-20, 2013 by Michał Baczyński (auth.), Humberto Bustince, Javier Fernandez, Radko Mesiar, Tomasas Calvo (eds.)

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