By D. Drasin, C.J. Earle, F.W. Gehring, I. Kra, A. Marden

ISBN-10: 0387967664

ISBN-13: 9780387967660

ISBN-10: 3540967664

ISBN-13: 9783540967668

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**Sample text**

On the continuous extension of a four valued logic for preference modelling. In Proc. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), pages 302–309, 1998. 25. M. Roubens and P. Vincke. Preference modelling. Lecture Notes in Economics and Mathematical Systems, 250, 1985. 26. R. Sambuc. Fonctions Φ-ﬂoues. Application ` a l’aide au diagnostic en pathologie thyroidienne, 1975. Ph. D. Thesis, University of Marseille. 27. A. Tsouki` as, P. Perny, and P.

Proposition 2. [6] Every interlaced bilattice with a negation is isomorphic to a square, equipped with a negation ¬ deﬁned, for every x, y in L, by ¬(x, y) = (y, x). 2 A detailed investigation of squares and the graded versions of the logical connectives that can be deﬁned on them appears in [4, 10]. , the elements (x, y) that satisfy the condition x+ y ≤ 1. In [3, 4] it is also shown that squares are a generalization to arbitrary complete lattices (not only the unit interval) of interval-valued fuzzy sets [16, 23, 26, 30], an alternative method of extending fuzzy set theory, motivated by the need to replace crisp, [0, 1]-valued membership degrees by intervals in [0, 1] that approximate the (unknown) membership degrees.

Cp3ik end function The function estimate p(i, k) computes the ﬁnal estimated value of the missing value, cpik , as the average of all estimated values that can be calculated using all the possible intermediate alternatives xj and using the three possible expressions (3–5). Then, the iterative estimation procedure pseudo-code is as follows: 0. 1. 2. 3. 4. 5. 6. 7. ITERATIVE ESTIMATION PROCEDURE EM V0 = ∅ h=1 while EM Vh = ∅ { for every (i, k) ∈ EM Vh { estimate p(i,k) } h++ } This procedure is able to estimate all the missing values for a given incomplete fuzzy preference relation if a set of n − 1 non-leading diagonal preference values where each one of the alternatives is compared at least once is known [1].

### Holomorphic Functions and Moduli I (Proc. workshop) by D. Drasin, C.J. Earle, F.W. Gehring, I. Kra, A. Marden

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