Measures fuzzy

Mezey, P.G. (1997) Fuzzy measures of molecular shape and size. In Fuzzy Logic in Chemistry, Rouvray, D.H. (Ed.), Academic Press, San Diego. 

P.G. Mezey, "Fuzzy Measures of Molecular Shape and Size", in Fuzzy Logic in Chemistry, Ed. D.H. Rouvray,... 

The second broad framework for dealing with uncertainty—fuzzy measure theory—was founded by Sugeno in 1974, even though some basic ideas of fuzzy measures had already been recognized by Choquet in 1953. Fuzzy measure theory is an outgrowth of classical measure theory, which is obtained by replacing the additivity requirement of classical measures with the weaker requirements of monotonicity (with respect to set inclusion) and continuity (or semicontinuity) of fuzzy measures. 

Given a universal set X and a nonempty family W of subsets of X (usually with some algebraic structure), a fuzzy measure on (AT,g ), which... 

When A " is a finite set, requirements (g3) and (g4) are trivially satisfied. Functions g, for which only one of these requirements is satisfied, are called semicontinuousfuzzy measures. For any pair A,B W such that AnB= 0, fuzzy measures are capable of capturing any of the following situations ... 

Z. Wang and G. J. Klir, Fuzzy Measure Theory. Plenum Press, New York, 1992. 

Fuzzy Measures of Molecular Shape and Size 159 a fuzzy set A, also called the cardinality of /I, is defined as the integral... 

X. FUZZY MEASURES OF CHIRALITY AND SYMMETRY DEFICIENCY, FUZZY SYMMETRY GROUPS, AND FUZZY SYMMORPHY GROUPS BASED ON THE FUZZY SCALING-NESTING SIMILARITY MEASURE... 

The fuzzy measures of symmetry deficiency, fuzzy chirality, fuzzy symmetry, and fuzzy symmetry groups described in Section IV in terms of the fuzzy Hausdorff-type metrics provide new tools for the characterization and comparison of fuzzy objects. However, the computation of the fuzzy Hausdorff-type metric is often cumbersome and alternative formulations may have certain advantages. 

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