## 2 Materials and Methods
### Recognition of the Areas Prone to Strong Earthquakes with One Learning Class
The output value is a set \((\alpha(\beta,q))\)\(\alpha\), which is dense in each of its elements. The result of applying the DPS algorithm is as follows:  
\[\text{DPS}(q,\beta):W\rightarrow\{B_{1},\ldots,B_{n}\}, \tag{12}\]  
where parameters \(q\) and \(\beta\) determine a particular type of DPS clusters; \(B_{1},\ldots,B_{n}\), own connected subsets in a set of recognition objects. In other words, \(B=\coprod_{i=1}^{n}B_{i}\), where \(B_{i}\), \(i=1,\ldots,n\) are recognized DPS clusters, \(B,\ B_{i}\subset W\), and \(W\backslash B\) represents a significant part of the set \(W\). It is noteworthy that if the latter is false, then the recognition result is trivial. During the next phase, DPS clusters \(B_{1},\ldots,B_{n}\) are transformed by the E\({}^{2}\)XT algorithm into flat zones of the cardinality of the continuum (i.e., mapping of \(F_{\gamma}\) is performed). During this phase, mapping is constructed:  
\[\text{E}^{2}\text{XT}(\delta,C,\omega,v):\ B_{i}\to F(B_{i}), \tag{13}\]  
where \(\delta\) is the step of the geographical grid, \(\omega<0\), \(v<0\) are free parameters, \(C\) is connection type, \(B_{i}\) are determined by the formula (12), \(F(B_{i})\), \(i=1,\ldots,n\) are sought flat zones of non-zero measure. If mapping (13) meets the above conditions (a), (b), and (c), then \(F(B_{i})\) there are sought areas prone to strong earthquakes. Accordingly, the FCAZ system is a composition of two algorithms:  
\[\text{FCAZ}(\gamma)=\text{E}^{2}\text{XT}(\delta,C,\omega,v)\circ\text{DPS}(q, \beta), \tag{14}\]  
Figure 1: The flow chart of system-analytical FCAZ method. Black blocks—DPS algorithm steps; blue blocks—E\({}^{2}\)XT algorithm. and \(\gamma\) is a set of free parameters of FCAZ:  
\[\gamma=\{\delta,C,\omega,v\}\cup\{q,\beta\}=\{\delta,C,\omega,v,q,\beta\}. \tag{15}\]  
The existence of mappings (13-15) allows considering FCAZ as a systems analysis method. FCAZ(\(\gamma\)), indeed, processes input data from beginning to end in a unified system. That said, the solution presented takes shape of sought two-dimensional zones and not their palliatives, representing finite sets of points in the plane [21]. The constructions of DPS and E\({}^{2}\)XT algorithms feature artificial intelligence blocks that automatically select optimal values \(\beta\) in DPS and \(\omega\), \(v\) in E\({}^{2}\)XT. This makes the result of FCAZ recognition objective and reproducible. An optimal value \(\beta\) enables recognizing DPS clusters for which the difference between the totality of densities (in the inherent sense of the algorithm) of objects inside the clusters and the totality of densities of objects outside of the clusters will be the greatest possible. The selected \(\omega\) and \(v\) make it possible to find an optimal combination of connection and scannability of DPS clusters. A detailed description of mathematical constructions of the DPS and E\({}^{2}\)XT algorithms is provided in the papers [21; 22]. FCAZ delivers a system approach to studying strong earthquake-prone areas. Its characteristic property lies in the fact that the recognition of sought zones in the regions of the globe that differ in structure relies on universal facts and methods, which enable a uniform approach toward solving the entire class of such problems. The statement of the problem and the process of its solving represent a unified system that is sufficiently invariable relative to geological structure, the selection of threshold magnitudes of sought strong earthquakes, objects, etc. That said, the FCAZ applicability condition is the state of seismological and geological-geophysical exploration of regions, which manifests itself in the high quality of earthquake catalog. The next parts of the paper will demonstrate how FCAZ allows performing reliable recognition of the strongest (\(M\geq 7.75\)), strong (\(M\geq 6.0\)), and significant earthquake-prone areas across different mountain countries of the world.