It is proposed the model for constructing of flexible ratings by end-users and it is analyzed applying of special weights for indicating of consumer preferences by various factors depends on the considered object. There is developed an algorithm for the building of flexible ratings. Practical testing was carried out by applying of investment indicators. 

This paper is devoted to a clustering in directed graphs using the Laplacian matrices of the digraph and their eigenprojections. The relevance of the problem due to the fact that ignoring edge directionality and considering the graph as undirected is not a meaningful way to cluster directed networks. The proposed methods for clustering in digraphs based on the coefficients of the adjacency matrix of the Laplacian matrix of the digraph.

The use of 3D tomography methods helps to the better understanding of complex structures peculiar to elements of a large-scale system, for instance, to buildings, engineering constructions, hydroelectric power plants, mines ets. Seismic tomography is important tool that the oil and gas industry applies to map the sub-surface geological structures. The LSQR algorithm by Paige and Saunders is the well-known conjugate gradient solver for the inversion tomography problem. Nevertheless, previous studies have shown that the algorithm has limitation in reconstruction of synthetic models of the large-size structure. Here two alternatives are given to resolve the complex structure: a new algorithm of the method of coordinate descent and the modification of Gaussian elimination. Both of these methods aim at constructing of the solution, which might be free from influence of determined types of error in the data. A structure with certain similar properties has been revealed for the layer from 0 to 5 km of depth by applying both methods to seismic data collected in the area around the hydrothermal gas field However, for the deeper layer the modification of Gaussian elimination produces the structure that is more realistic compared the method of coordinate descent. The reason might be in more accurate determination of the location of those seismic sources that are close to a surface. We conclude that simultaneous application of different inversion methods is useful in the presence of unclear level of errors in the data. Tomography images presented here for the area of natural concentration of gases can be applied for understanding of conditions to keep underground gases in a safe area.

A mathematical model for carrying out an express analysis of the state and forecasting the national security of states is developed. An algorithm for using the developed mathematical support in assessing the state of national security of states is presented. The method of training specialists in the field of national security in training situational centers is shown.

Mars has no active magnetic dynamo field today, but it possesses a remanent magnetic field that may reach several thousand nanoteslas locally. There is a wealth of maps and models of the magnetic field [1–4]. Recently InSight (Interior Exploration using Seismic Investigations, Geodesy, and Heat Transport) lander has provided new magnetic field strength and direction measurements. It turned out that the InSight landing site’s magnetic field is ten times stronger than predicted by satellite-based models [5]. This knowledge indicates that the magnetic field of the planet needs further study and analysis.