Fuzzy distance

One of the pioneers in the fuzzy distance approach that preserves uncertainty was Voxman (1998) which addresses principles of distance on the fuzzy point of view and treats the principle of convergence on the view of the Cauchy sequences. He was the first to propwse a fuzzy distance between fuzzy numbers. 

Of all works cited here that more close to our idea is the work by Chakraborty Qiakraborty (2006) which proposes a fuzzy distance for fuzzy numbers, which preserves a distance xm-certeinties. The authors put a natural question " if we do not know the numbers exactly how can the distance between them be an exact value ". At the same time, they criticise the use of the supreme, of the minimal or of any other candidate as absolute representative of the distance between two fuzzy numbers. They also consider the distance between two fuzzy numbers as a fuzzy number, saying that the distance between two numbers with xmcertainties must be a number with uncertainty. In addition, for fuzzy sets, Grzegorzewski (2004) uses a Hausdorff metric in the construction of a fuzzy metric, that unfortunately it does not preserve uncertainty. In their works they use metric spaces and topological spaces. 

Chakraborty, C. Chakraborty, D. (2006). A theoretical development on a fuzzy distance measure for fuzzy numbers. Mathematical and Computer Modelling (43) 254-261. 

We now detail a second approach to the evaluation of the degree of symmetry deficiency, from a different point of view, namely the fuzzy sets approach. Fuzzy set methods are especially suitable to reflect the Heisenberg uncertainty relation and other aspects of quantum chemistry " and computational chemistry. Symmetry in molecules relies on the concepts of distance and metric. For the approximate symmetries of fuzzy electron densities, fuzzy set methods, fuzzy distance or fuzzy metric, and fuzzy symmetry are of importance. In Section 3.3 below, we compare and generalize the two approaches. 

On the other hand, if we want to characteri2e objects which are described by the rather fuzzy statement "numbers dose to three", we then need a membership function which describes the doseness to three. An adequate membership function could be the one plotted in Figure 9-25 m x) has its maximum value of m x) = 1 for value x = 3. The greater the distance from x to 3 gets, the smaller is the value of m x). until it reaches its minimum m x) = 0 if the distance from x to 3 is greater than say 2, thus for x > 5 or x < 1. 

Data collected by modern analytical instalments are usually presented by the multidimensional arrays. To perform the detection/identification of the supposed component or to verify the authenticity of a product, it is necessary to estimate the similarity of the analyte to the reference. The similarity is commonly estimated with the use of the distance between the multidimensional arrays corresponding to the compared objects. To exclude within the limits of the possible the influence of the random errors and the nonreproductivity of the experimental conditions and to make the comparison of samples more robust, it is possible to handle the arrays with the use of the fuzzy set theory apparatus. 

Distances in these spaces should be based upon an Zj or city-block metric (see Eq. 2.18) and not the Z2 or Euclidean metric typically used in many applications. The reasons for this are the same as those discussed in Subheading 2.2.1. for binary vectors. Set-based similarity measures can be adapted from those based on bit vectors using an ansatz borrowed from fuzzy set theory (41,42). For example, the Tanimoto similarity coefficient becomes... 

THINK considers fuzzy two-, three- or four-center pharmacophores. If a given molecule contains more than three or four centers, then all possible groups of two to four centers are taken. The distances (including a tolerance) between the pharmacophore centers are measured exactly and then allocated to distance bins, each distance being represented by the bin into whose range it falls. The distance bins are used to transform the distances within each pharmacophore into a set of integers that give a more compact representation of the pharmacophores. 

Core-sphere radius Rs, fuzzy-layer thickness ARf, center-to-center distance z Small steps allowed in s s at Rs and Rs + ARf ... 

Electron density decreases exponentially with distance that suggests that an Additive Fuzzy Density Fragmentation (AFDF) approach can be used for both a fuzzy decomposition and construction of molecular electron densities. The simplest AFDF technique is the Mulliken-Mezey density matrix fragmentation [12,13], that is the basis of both the Molecular Electron Density Loge Assembler (MEDLA) [14-17] and the Adjustable Density Matrix Assembler (ADMA) [18-21] macromolecular quantum chemistry methods. 

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