Cluster substructure

Next we discuss a fuzzy hierarchical clustering procedure proposed by Dumitrescu. The procedure starts with the computation of a binary fuzzy partition A, A2] of the data set X. To do this, the FNM algorithm or other appropriate procedure may be used. If A j and A 2 do not describe real clusters, we may conjecture there is no structure in X or that the data comprise a single compact cluster. The process then ends. If the fuzzy classes A and A2 correspond to real clusters, we set = v4j,/I2. Assume the cluster structure of each class zl, / = 1,2, is given by a binary fuzzy partition of z4,. We may compute the cluster substructure of Aj using the GFNM algorithm or one of its relatives. There are now two possibilities ... 

The existence of cluster substructure in nuclei is supported also by the observation of cluster decay. Besides a particles, the heavy nuclei can emit Ne, Mg, Si clusters, too. The partial half-lives for these decays depend on the penetrability of the Coulomb barrier in a way very similar to a decay (Mikheev and Tretyakova 1990 see Geiger-NuttaU-type relations in O Sect. 2.4.1.1). The heavy cluster decay is a very rare phenomenon. For example, in the decay of Ra there were 65 x 10 a particles observed, while only 14 cluster emissions during the same time (Rose and Jones 1984). For the detection of rare clusters solid state track detectors are very suitable. 

While we will present methods that produce a cluster substructure of the data, we need to emphasize that, sometimes, variable selection and data preprocessing may also be important. 

Another problem with such algorithms is that of determining the optimal number of classes that correspond to the cluster substructure of the data set. There are two approaches The use of validity functionals, which is a postfactum method, and the use of hierarchical algorithms, which produce not only the optimal number of classes (based on the needed granularity), but also a binary hierarchy that shows the existing relationships between the classes. 

Detection and Characterisation of Cluster Substructure I. Linear Structure Fuzzy C-lines. 

Detection and Characterization of Cluster Substructure II. Linear Structure Fuzzy C-Varieties and Convex Combinations Thereof. 

The atoms are locally arranged in a cluster substructure, in which icosahedral coordination plays a prominent role. 

In CMAs, the majority of atom coordinations are icosahedrally symmetric. Frequently, one finds a decoration of certain atomic positions with concentric shells of icosahedral polyhedra around a central position. These arrangements are referred to as cluster substructure or clusters in short. Three types of cluster substructure, based on Bergman, Mackay, and Friauf polyhedra, are found particularly often in CMAs and can be used for a classification of the latter [15]. 

CMA phases frequently exist in close relationship to other phases of similar structure, close or identical composition, which are based on the same cluster substructure but with different lattice parameters. We refer to such groups of related phases as phase families. A prominent example is the s-phase family, which is based on the phase Eg-Al-Pd-Mn. 

The Efi-phase and the described cluster substructure form the basis of the other members of the e-phase family. A sequence of related orthorhombic phases is... 

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