Possibility theory

Iand67] Landauer, R., Wanted a physically possible theory of physics , IEEE Spectrum 4 (1967) 105-109. 

The construction of schemes with the indicated properties and a desired quality is one of the outlines of the possible theory. 

Such locally one-dimensional schemes were investigated before, all the triks and turns remain here unchanged. The following issues are yet to be answered in the possible theory ... 

We begin our exposition with a discussion of examples that make it possible to draw fairly accurate outlines of the possible theory regarding these questions and with a listing of the basic results together with the development desired for them. Common practice involves the Laplace operator as the operator R in the case of difference elliptic operators A. The present section is devoted to rather complicated difference problems of the elliptic type. Here and below it is supposed that the domain of interest is a p-dimensional parallelepiped G = 0 < < / , a = 1,2,..., p) with... 

S. Claudel, C. Fonteix, J.P. Leclerc, and H.G. Lintz. Application of the possibility theory to the compartment modelling of flow pattern in industrial processes. Chemical Engineering Science, 58 4005-4016, 2003. 

The various possible theories provide different functions 11(1) and through the above tests may in principle be distinguished by observations. However,... 

A possible theory concerning adsorption was developed by Langmuir... 

The vast array of experiments in the cuprates narrows the possible theories for the phenomena. One can be certain that the theories should have a QCP with a phase with a pseudogap ending at it. One can also be certain that in the long wavelength limit, the fluctuations about the QCP must have the form of Eq. (2) because that is what is observed [29],... 

The problem of the second virial coeffident of flexible chain polymers has three phases. In the first place, there is the problem of interchain interactions in the second, the problem of the inteochain interactions and finally, the problem of the coupling of the inter- and intra-chain interactions. Different approximations are possible for each phase of the problem, various combinations of such approximations are also possible, and consequently we are confronted by a wide variety of possible theories. For example, the present development of a new excluded-volume equation for the intrachain problem almost doubles the variety of Az-theories, since most existing theories have been developed on the basis of the Flory expression for the excluded volume effect and the new theory can be combined, in prindple, with all of these theories. On the other hand, we have as yet no computational data for A2 and furthermore, the experimental accuracy of A2 measurements is still rather poor, as illustrated in Fig. 2. It is therefore unlikely that one particular combination of several approximations can be properly chosen from among the... 

Cirdte pearls are nucleated cultured pearls from marine pearl oysters. The round to oval pearls display indented rings around the middle of the pearl. It is not known eatacdy why this patterning occurs. Two possible theories are that the mantle tissue was twisted when inserted into the oyster, or that the pearl turned firequendy while it was growit. ... 

Zadeh 1975-1978 Development of possibility theory fuzzy sets act as elastic constraints on system parameters... 

D. Dubois and H. Prade, Possibility Theory. Plenum, New York, 1988. 

G. de Cooman, D. Ruan, and E. E. Kerre, Foundations and Applications of Possibility Theory. World Scientific, Singapore, 1995. 

Propositions may also contain uncertainties of both types. To deal with information contained in such propositions, a measure-theoretic counterpart of fuzzy set theory was introduced by Zadeh under the name possibility theory.The following are basic notions of the theory. 

From these basic properties of possibility measures, the full calculus of possibility theory, analogous to the calculus of probability theory, has been developed. Its primary role is to deal with incomplete information expressed in terms of fuzzy propositions. Due to limited space, it is not possible to cover here details of this calculus. 

We can see in these debates once again that most opponents of fuzzy set theory from the area of probability theory attempt to compare probabilities with degrees of truth (or degrees of membership). However, these are not comparable. As explained in Section III in the context of possibility theory, degrees of truth result from linguistic uncertainty, while probabilities result from information deficiency. These two types of uncertainty may be combined, but their comparison is meaningless. 

If the suspension behaviour can not be predicted satisfactorily, at least not qualitatively, we suggest that the effect of solvation forces should be investigated. The origin of these forces and their effects are still being studied and possible theories are being discussed (see for example Ref. [27]). 

Other possible theories of stream humic substance formation are the browning reactions (the black substances produced by the action of acids on sugars) (Haworth, 1971) and the melanoidin reaction (the dark pigments produced when glucose, alanine or ammonia, and phenolic substances react with one another) (Hoering, 1973 Filip et al., 1974 Ertel and Hedges, 1983). These theories are discussed in detail by Stevenson in Chapter 2 of this book. 

Oden GC, Lopes L (1982) In Yager RR (ed) Recent developments in fuzzy set and possibility theory. Pergamon, Blmsford, New York, p 75... 

In order to introduce notation and to emphasize the range of possible theories that can be termed statistical, we first summarize some essential aspects of statistical theories. An extension31 of the bimolecular collision formulation of Wagner and Parks3 provides a convenient starting point. 

When all the preparation work has been completed and sample is introduced in the column, analysis can begin. The analysis result consists of a chromatogram with many peaks that correspond to the compounds present in the sample mixture. At the beginning of this chapter, it was mentioned that the main idea of chromatography is the differential adsorption or absorption of the components in a sample mixture followed by desorption. However, what is observed in the chromatogram are wide peaks instead of thin lines that should represent the compounds of the sample. Why do we have wide peaks One of the possible theories used to explain this occurrence is the theory of resistances. The main idea of this theory is presented in Figure 2.13. 

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