By Gleb Beliakov

A large creation to the subject of aggregation capabilities is to be present in this publication. It additionally presents a concise account of the homes and the most periods of such capabilities. a few cutting-edge options are offered, in addition to many graphical illustrations and new interpolatory aggregation capabilities. specific awareness is paid to id and development of aggregation features from software particular specifications and empirical facts.

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They have been introduced by Yager [263] and have become very popular in the fuzzy sets community. Let x be the vector obtained from x by arranging its components in non-increasing order x(1) ≥ x(2) ≥ . . ≥ x(n) . 72 (OWA). Given a weighting vector w, the OWA function is n OW Aw (x) = wi x(i) =< w, x >. i=1 Note that calculation of the value of the OWA function can be done by using a sort() operation. If all weights are equal, OWA becomes the arithmetic mean. The vector of weights w = (1, 0, .

However, general non-monotone functions can be continuous in each variable without being continuous). , a polynomial). Thus based on their values, or graphs, we cannot distinguish between continuous and discontinuous integrable functions, as the values of both functions coincide up to a tiny difference (which we can make as small as we want). A computer will not see any difference between the two types of functions. Mathematically speaking, the subset of continuous functions C(Ω) is dense in the set of integrable functions L1 (Ω) on a compact set.

Xi−1 , a, xi+1 , . . , xn ) = 0, can hold for some x > 0 with a at any position. 35. Because of monotonicity of f , if a is a zero divisor, then all values b ∈]0, a] are also zero divisors. The interpretation of zero divisors is straightforward: if one of the inputs takes the value a, or a smaller value, then the aggregated value could be zero, for some x. So it is possible to have the aggregated value zero, even if all the inputs are positive. The largest value a (or rather an upper bound on a) plays the role of a threshold, the lower bound on all the inputs which guarantees a non-zero output.

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