## Mathematical Structures and Modeling N4(48)

 Mathematical Structures and Modeling. - Omsk : OmSU, 2018. N4(48), 165 p. ISSN (print): 2222-8772 ISSN (online): 2222-8799 For researchers, post-graduate students and senior students. Journal issue in one file

Fundamental Mathematics and Physics

The necessary and sufficient conditions for convergence of the distributions of symmetric calibration functions from dependent random variables to max-stable distributions are obtained in this article. These conditions include the so-called minimal conditions of the weak dependence.
Keywords: calibration functions of random variables, max-stable distributions, minimal conditions of the weak dependence.

The article investigated the critical behavior of three-dimensional semi-infinite antiferromagnetic Ising model in a magnetic field by computer simulation. The values of the Neel temperature for surface and bulk phase transitions at different magnetic field values are received. The phase diagram of the system is built. The position of tricritical point of a special phase transition depending on the strength of the magnetic fields is determined.
Keywords: antiferromagnetic Ising model, surface phase transition, special phase transition.

The author proved all curvatures of a regular curve in $$n$$-dimensional pseudo-Euclidean space of an arbitrary index, $$n\geq 2$$, are constant if and only if the curve is an orbit of a one-parameter isometry group of the space.
Keywords: curvature, Frenet frame, orbit of a one-parameter isometry group, pseudo-Euclidean space.

R.Yu. Simanchev, P.V. Solov'eva.
$$b\mathcal{H}$$-bases for a Some Class of Facet of a Clique Partitioning Polytope.

Let $$K_n=(V,E)$$ be a complete undirected $$n$$-vertex graph without loops and multiple edges. A spanning subgraph $$H\subset K_n$$ is called an $$M$$-graph if each of its connected components (possibly one-vertex) is a clique. We denote the family of all $$M$$-graphs in $$K_n$$ by $$\mathcal{H}$$. This family is the set of feasible solutions of the clique partitioning problem, consisting in finding in the complete edge-weighted graph of an $$M$$-graph of minimal weight [GroWak1990,GroWak1989,SU1]. In the mentioned papers, the polyhedral properties [GroPad1985] of this problem are considered. The classes of inequalities that generate the faces of the problem polytope are constructed, on the basis of which algorithms for branches and cuts are developed. In this paper, using the technique proposed in [Sim2017], we prove the facetness of inequalities of a special class with respect to the clique partitioning polytope.
The work was supported by the Russian Foundation for Basic Research (project 18-07-00599).
Keywords: polytope, facet, the clique partitioning problem.

In work (see [1]) a class of inequalities supporting to the polyhedron of the given problem was proposed. Conditions are found under which the constructed inequalities are faceted. When these inequalities are used in cutting algorithms, the Separation problem becomes an actual problem. In the paper (see [2]) it was shown that the problem of identifying the proposed inequalities is $NP$-hard. In this paper, a local search procedure has been developed to identify clipping. To analysis the effectiveness of the proposed methods, a computer experiment was carried out.
This work was supported by the Russian Foundation for Basic Research (project 18-07-00599).
Keywords: separation problem, plyhedra, facet, approximation problem.

O. Kosheleva, V. Kreinovich.
In the Discrete Case, Averaging Cannot Be Consistent.

When we have two estimates of the same quantity, it is desirable to combine them into a single more accurate estimate. In the usual case of continuous quantities, a natural idea is to take the arithmetic average of the two estimates. If we have four estimates, then we can divide them into two pairs, average each pair, and then average the resulting averages. Arithmetic average is {\em consistent} in the sense that the result does not depend on how we divide the original four estimates into two pairs. For discrete quantities - e.g., quantities described by integers - the arithmetic average of two integers is not always an integer. In this case, we need to select one of the two integers closest to the average. In this paper, we show that no matter how we select - even if we allow probabilistic selection - the resulting averaging cannot be always consistent.
Keywords: averaging, processing estimates, consistency, discrete case

Ladner's 1975 result says that any NP-complete problem - i.e., in effect, any maximally complex problem - can be reduced to solving two easier problems. This result sounds counterintuitive: if a problem is maximally complex, how can it be reduced to simpler ones? In this paper, we provide an intuitive explanation for this result. Our main argument is that since complexity and easiness-to-divide are not perfectly correlated, it is natural to expect that a maximally complex problem is not maximally difficult to divide. Our related argument is that - as this result shows - NP-completeness is a sufficient but not a necessary condition for a problem to be maximally complex; how to come up with a more adequate notion of complexity is still an open problem.
Keywords: NP-complete, divide-and-conquer approach, Ladner's result, maximally complex problems.

Applied Mathematics and Modeling

The electromagnetic field in electrical exploration problems is often represented as integrals with a fast-oscillating nucleus. When calculating these integrals on a computer, it is necessary to deform the contour of integration into the plane of the complex variable. The article studies the allowable deformation region of the integration contour in the case of a non-uniform medium. The source of the field is a vertical dipole. A similar problem was solved for a horizontally layered medium with a horizontal harmonious dipole as a source.
Keywords: Electrical exploration, electromagnetic field of vertical electric dipole, fast-oscillating integrals, deformation contour, complex plane, absence of singular points, deformation domain.

V.A. Shovin.
Factor Analysis Based on the $$k$$-means Method.

Actual problems and research methods are analysis and search for hidden dependencies in experimental data. The definition of such dependencies allows them to construct a model of phenomenon or object that would best fit the experimental data and at the same time have a minimally complex version. Known mathematical and software tools for the automatic construction of such models is a factor analysis. Various generalizations and modernizations of the methods of factor analysis are in demand. The article proposes a new approach to factor analysis based on the $$k$$-means data clustering method and subsequent factor rotation. Factor analysis identifies from a set of initial indicators $$k$$ main components or factors - with the greatest accuracy approximating the scatter and distribution of the initial data. These main components form the factorial structure of the source data. Various characteristic points of the original data structure can be used as the directions and aspects of the main components. In this paper, we propose to use centers of raw data clusters. There is a large number of clustering methods for dividing data points into classes. The most popular is the $$k$$-means method. As a result, the $$k$$-means method allows finding the factor structure in the original multidimensional data space from the positions of the $$k$$ centers of the selected clusters. The subsequent factor rotation according to the original criterion of interpretability allows us to find a simple factor structure. Conducting numerical experiments showed good agreement with the results of this method of factor analysis with previously known methods. The proposed method of factor analysis has good and superior efficiency compared with other methods of factor analysis. It has limited capabilities and resources needed to determine the factor structure.
Keywords: k-means method, factor analysis, factor rotation.

Computer Science

D.N. Lavrov, M.A. Kharlamova, E.A. Kostushina.
Representation of the Corpus of Medium Irtysh Folk Dialect.

The article discusses ways of representing dialect entries in the regional corpus. The focus is on models for the presentation of thematic, structural and partly phonetic markings. Particular attention is paid to the presentation model of extralinguistic data. The proposed solutions are based on the representation of relational databases and XML format.
The reported study was funded by RFBR according to the research project ¹ 18-012-00519.
Keywords: thematic markup, metatext markup, XML format, regional dialect body.

Information Security

Signs of large-scale systems are described. Possible metric characteristics of these signs are defined. Large-scale complex IT systems and their access control subsystems are considered. Belonging of such access control subsystems to a class of larged-scale systems is proved. The automation need of the management processes by the access control subsystem in the large-scale complex IT system is explained.
Keywords: larged-scale systems, access control, management, automation.

A.V. Bazhenov, A.K. Guts.
Software for Modeling of Network and Simulation of Computer Network Attacks.

The article presents a software application that simulates different computer networks and attacks on these networks.
Keywords: software application, modeling, network attack, smurfing.

S.V. Belim, D.E. Vilkhovskiy.
Koch-Zhao Algorithm Steganalysis.

The analysis of the Koch-Zhao steganographic algorithm was carried out. The possibility of an attack on the detection of messages is considered. An algorithm for calculating the boundaries of the embedded message based on the analysis of the discrete cosine transform coefficients is proposed. A computer experiment is conducted. Embedding parameters that allow the attack are defined.
Keywords: steganography, steganalysis, Koch-Zhao algorithm, discrete cosine transform.

S.V. Belim, P.G. Cherepanov.
Selection of Blocks in a Video Stream for Embedding Digital Watermarks

The article proposes a method for selecting blocks for embedding digital watermarks in video files. The video stream is represented as a three-dimensional object. For embedding digital watermarks, the Koch-Zhao method is used, which is generalized to the three-dimensional case. The main goal of the proposed method is to reduce visible distortions from embedding data. For the selection of blocks, their statistical analysis is performed. The variance of the color intensity of individual pixels is chosen as the main parameter. A computer experiment is conducted. To assess the level of visibility, the ratio of noise to the main signal is calculated. The threshold value of the variance above which the embedding is the least noticeable is experimentally determined.
Keywords: digital watermarks, video steganography, discrete cosine transform.

In this paper the possibility of representing of some types of role-based access control models by object-oriented discretionary access control model is considered. The role-based security model without hierarchy and the role-based security model with hierarchy with inheritance "from above" are considered. The permissions of the role-based access control model are represented as a set of pairs of object and access right. A hierarchy of classes of the object-oriented HRU model based on the role-based access control policy is constructed. Commands of the object-oriented HRU model corresponding to the reassignment of roles in the original role-based model are described.
Keywords: role-based access control model, object-oriented discretionary access control model, role hierarchy.

As a rule, in computer networks information security is considered from the point of view of three properties: confidentiality, integrity and availability. Currently, it is important to consider such property of information as anonymity. To ensure the latter, a tool such as TOR is used. The TOR network gives us the illusion of anonymity on the Internet and lulls the vigilance of the end user. But at the output node of the TOR network, the traffic is decrypted and its contents may become known to the owner of this output node. If the owner of a node or a series of nodes is an authoritarian government, or a foreign agent, then he can not only know the contents of the message being transmitted, but also try to modify it or completely interrupt the communication channel. Confidentiality of the message can be protected by encryption, but one can only hope that it is secure. Accessibility and integrity remain at risk. Here, availability means a guarantee of delivering a complete message to a destination. The article discusses the principles that should be the basis of the data transfer protocol with guaranteed delivery. The idea of the protocol is based on the existence in the network of several independent delivery routes and the use of cryptographic $$(k, n)$$-threshold secret separation schemes in the $$n$$ network flows of different routes. This will allow not only to further anonymize traffic, but also in the case of control by the authoritarian government of the output nodes (lower than $$k$$ threshold) will provide additional protection for the confidentiality of traffic.
This research was done as part of a research project NIOKTR number 01-06/683.
Keywords: secret sharing, routed network, anonymity, availability.

The article discusses the role-based security policy implemented in the Oracle DBMS. The role model in a DBMS can be quite complex, contain a large number of roles and authorities. This information is stored in a large number of related tables. A mechanism has been implemented to extract data mapped to role-based security policies in an Oracle DBMS and to present this information in GraphML format. It also implemented the ability to import role-based security policy, presented in GraphML format, in Oracle DBMS.
Keywords: role-based model, DBMS, graph representation language.

S.V. Belim, Y.S. Rakitskiy.
Chaotic Masking Scheme for Messages Based on Orthogonal Functions.

The method of chaotic masking of a discrete signal is proposed in the article. The proposed approach does not require the synchronization of chaotic generators and is resistant to noise in the communication channel. These properties remove the main problem of chaotic masking associated with the desynchronization of control parameters. The basic idea is to use message coding using orthogonal functions. Next, a simple summation of a hidden message with a chaotic signal is applied. The extraction of the message can be performed on the basis of the orthogonality property without subtraction of the chaotic component.