Fuzzy formula

In order to keep the input as flexible as possible, we allow the input of several fuzzy formulas, each of them accompanied by its own restrictions. 

Where fine structure is present, this information can be used to define a more exact fuzzy formula for molecular formula candidate generation, significantly limiting the number of candidates. 

Variations in the local properties can be monitored by calculating the variations of the fuzzy membership function i-Fi,x(r)- If two functional groups Fj and Fj of a molecule X have the same chemical formula, then their differences must be due to their different local surroundings within the molecule X. These differences are necessarily manifested in differences of their fuzzy membership functions iFi,x(r) and PFj,x(r) >n molecule X. 

Considering a collection of functional groups of the same chemical formula but located in a series of different molecules, local similarity measures among these functional groups can be based on their fuzzy membership functions (for a discussion of local similarity measures see sections 5 and 6). These similarity measures provide clues for the similarities and differences in their reactivities, caused by the similarities and differences in their molecular environments. 

In this chapter, it is sufficient to introduce only the most common fuzzy operations, which are the only fuzzy operations that are cutworthy (and also strongly cutworthy). These operations, usually called the standard fuzzy operations, are defined for all xElX by the following formulas ... 

Given two fuzzy binary relationships R and Q defined on Z X y and YxZ, respectively, their standard composition R°Q is defined by the formula... 

According to this birmirig scheme, the two distance bins, whose centers are the closest ones to the distance Vy of a given combination of two features, are incremented proportionally leading to fuzzy counts of the binding property pairs. A simple formula for the calculation of the increments A for the main bin m, that is, the bin where the distance actually falls, and the adjacent bin a is... 

The problems encountered in mathematical modeling of tumble/growth agglomeration do not relate to the theories, formulas, and possibilities to solve the ever more complicated equations. With modem computing possibilities, a whole series of assumptions can be introduced into the model equations and responses to certain imaginary process conditions can be predicted. However, the real system often produces unexpected results intermittently or even consistently without offering a clear indication of why such deviations occur. Introduction of new mathematical methods, such as, for example, fuzzy logic or chaos theory, produce more complicated model equations and closer to life results but still are not able to serve as unequivocal bases for control schemes. 

The fuzzy comprehensive assessment method is adopted to integrate the index weighing formula (2) and single-factor grey assessment matrix formula. (6). It is assumed that the whitened fuzzy comprehensive assessment matrix is... 

According to the formula (7), the comprehensive fuzzy assessment matrix for the whitening can be obtained. 

This method is based on fuzzy preferential relations. Degree of preference variant a,- before Uj is (Formula 17) ... 

WiUett, Barnard, and Downs provide an extensive Usting of many types of similarity functions [61]. Five similarity functions will be considered here to illustrate how the current formulation provides a unified description Tanimoto ( Tan ), Cosine ( Cos ), Dice, and the related pair Max and Min. Table 15.3 summarizes the various forms. Importantly, the formulas apply to all three types of sets—classical, fuzzy, and multisets—providing a unity to the set-based approach. 

As the generation is formula-based, the molecular formula is required. A generalization also allows input of a fuzzy molecular formula, i.e. with intervals for multiplicities ... 

Definition (Fuzzy molecular formula) Let denote a set of chemical elements and... 

It should be pointed out that in order to avoid the case of oo — oo, at least one of the two integrals on the right side of formula (6.1) needs to be limited. Furthermore, the expression E on both sides of formula (6.1) have different meanings. The E on the left side stands for the expression of the fuzzy random variable while the E on the right side stands for the expression of the fuzzy variable, which is already reloaded. The reloading enables the expression E to have different definitions, which are indicated by the variables. 

Note that the formulas depend on the specifications given in the fuzzy table shown in Fig. 19.33(a). 

As a result of fuzzy rule evaluation, each analog output variable is represented by several fuzzy variables. The purpose of defuzzification is to obtain analog outputs. This can be done by using a membership function similar to that shown in Fig. 19.32. In the first step, fuzzy variables obtained from rule evaluations are used to modify the membership function employing the formula... 

A fuzzy reasoning system implemented in the SciLab package and applied in Section 5 uses the above inference reasoning method specified in formulas (8-10). 

The structure of a real Mathcad worksheet is usually more complex and has the fuzzy borders between these three areas the results may be displayed just after the formula, the souse data may be duplicated in a summary table etc. If the area containing calculations itself occupies several pages it is usually collapsed during checkout to see the source data and the results at the same time. We may divide the worksheet into several areas, place the results between them and expand them in turn. Figure 6.97 shows that a user changes the values of variables a and (or) b and... 

According to fuzzy comprehensive evaluation formula, this research uses Af (, )operation. Given the judgment set = i = 1,2,- -,h, with... 

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