Fuzzy boundaries

Fuzzy logic gets around this difficulty by replacing hard boundaries between sets with soft divisions. Objects are allocated to fuzzy sets, which are sets with fuzzy boundaries. The membership value of an object within a fuzzy set can lie anywhere within the real number range of 0.0 to 1.0 (Figure 8.4). 

Evaluation of the concept of mental illness suggests that there are other aspects of its definition that create variability. Lilienfeld and Marino (1995) stated that there has always been great difficulty in establishing the boundary between mental illness and normalcy. These authors suggested that we need to accept this fuzzy boundary as a necessary condition of mental illness. They argued that mental illnesses should be considered open concepts (Meehl, 1977) or Roschian concepts (Rosch, 1973). A Roschian concept is essentially the same as the open concept we have already defined. It is a mental construction used to categorize natural entities that are characterized by fuzzy boundaries. Lilienfeld and Marino argued that this conception of... 

Implications of the prototypic classification method are that categories within the system will possess fuzzy boundaries. Individuals within a category will be somewhat heterogeneous and may not share any common features. In our depression example, a prototypic classification scheme may yield a diagnosis of depression in one patient who shows agitation, anhedonia, and suicidal ideation and the same diagnosis in another patient who shows sadness, fatigue, worthlessness, and decreased appetite. 

Along the same lines, there is a fuzzy boundary between hemicarceplexes and carceplexes, some hemicarceplexes having more carceplex-like traits than others. The schematic diagram illustrated in Figure 8 shows how the concepts of a pseudorotaxane and of a hemicarceplex may be explained using fuzzy sets. The sets of these species belong simulta-... 

All quantities are given in units of 10 30 Cm (see Appendix K for conversion factors). The 6-31G ab-initio results have been obtained at the SCF level, generally at the neutron crystal geometry. DI direct integration. DB discrete boundary. FB fuzzy boundary, i.e.. stockholder concept. 

Figure 6.7. Flow patterns and correlations of flow regimes of liquid-gas mixtures in pipelines, (a) Patterns in horizontal liquid-gas flow, (b) Patterns in vertical liquid-gas flow, (c) Correlations of ranges of flow patterns according to Baker [Oil Gas J. 53(72), 185 (1954)], as replotted by Bell et al. [Chem. Eng. Prog. Symp. Ser. 66, 159 (1969)] a is surface tension of the liquid, and a that of water, (d) Flow regimes of water/air at 25°C and 1 atm [Taitel and Dukler, AIChE J. 22, 47(1976)] the fuzzy boundaries are due to Mandhane et al. [Int. J. Two-Phase Flow 1, 537 (1974)].

First, a bit of history. My first paper on fuzzy sets (1965) was motivated by the realization that there is a wide gap between the precision of mathematics and the pervasive imprecision of the real world. At the center of this gap is the fact that almost all concepts in mathematics are sharply defined, whereas almost all real-world classes have unsharp, that is, fuzzy, boundaries. 

Figure 17 shows the present-day model of the atom, which takes into account both the particle and wave properties of electrons. According to this model, electrons are located in orbitals, regions around a nucleus that correspond to specific energy levels. Orbitals are regions where electrons are likely to be found. Orbitals are sometimes called electron clouds because they do not have sharp boundaries. When an orbital is drawn, it shows where electrons are most likely to be. Because electrons can be in other places, the orbital has a fuzzy boundary like a cloud. 

Figure 25.2 Illustration of multiple-type cell co-culture systems (a) Random mixed, (b) contact patterning with fuzzy boundaries, (c) co-culture without cell contact on a flat surface, (d) in chambers, (e) the trans-well system, and (/) the micro cell culture analog ( xCCA) device.

Lincoln, T.L. Privacy A real world problem with fuzzy boundaries. Methods Inf. Med. 1993, 32, 104-107. 

The distance between any two PPP types is measured and assigned into one or two distance bins. The width of each distance bin is usually 1.0 A. However, since adjacent bins are allowed to have 10% overlap with each other, the actual bin width is 1.2 A. Thus, as for geometric binding property pairs, any distance located in the overlap region is assigned to both of the bins for instance, a distance of 2.05 A, under fuzzy boundary conditions, belongs to both bin 1 and bin 2. 

The interpretation of the fundamental results of molecular informatics is rather straightforward if one considers the actual material making up the molecules. The atomic nuclei and the fuzzy, boundary-less electron density cloud are the only physical entities contained in molecules, where the electron density contains all information about the location and nature of the nuclei. Consequently, simply on information-theoretical grounds, the result of the Hohenberg-Kohn theorem [2], stating that the... 

Huisman, J., Kaiser, F. (2001). Fixed and fuzzy boundaries in higher education A comparative study of (binary) structures in nine countries. Den Haag Adviesraad voor het Wetenschaps- en Technologiebeleid. 

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