Mathematical structures and modeling. - Omsk : OmSU, 2017. �4(44), 144 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

V.V. Varlamov
About Axiom System of Nonlocal Quantum Theory

An axiom system of nonlocal quantum theory is defined within Heisenberg-Fock conception. A notion of single quantum system lies in the ground of axiomatics. A generating kernel of this system is an abstract \(C^\ast\)-algebra. It is shown that different concrete realizations of the operator algebra depend on the structure of generators of the group of a fundamental symmetry (these generators are joined to energy operator). In the case of generators of the conformal group, we have a state spectrum defined within Rumer-Fet representation, that leads to a group-theoretical description of Mendeleev periodic system.
Keywords: single quantum system, Heisenberg-Fock conception, operator algebra, complex envelopes, conformal group, Mendeleev periodic system.

A.G. Grin'
On the Attraction of Symmetric Functions from Dependent Variables to the Normal Law

The necessary and sufficient conditions for attracting a certain class of functions from dependent random variables to a normal law are obtained. These conditions include the so-called minimal conditions of weak dependence.
Keywords: symmetric functions of random variables, attraction to the normal law, minimal conditions of weak dependence.

I.A. Eganova, W. Kallies
Foundation of Minkowski's World as a Mathematical Structure: to the Answer for Riemann's Question

In the given article B. Riemann's gnoseological question about an internal cause of initiation of metric relations in space with respect to the World of events (space-time) is raised. On the whole, the relevance of Riemann's question for space-time is shown: its decision visualizes the physical foundation of Minkowski's World, gives new notations about interconnections in the World of events and appreciably new methods for experimental investigations. An executed in the article conceptual review of historical development of A.D. Alexandrov's approach to the Riemann's question is made according to the following plan. 1. Previously, it was shown that Alexandrov's approach is completing the definite sequence of investigations, namely of H. Minkowski, A.A. Friedman, and N.A. Kozyrev, in the World of events history. 2. Further, along Alexandrov's line, it is determined that on the basis of Friedman's and J.L. Synge's analysis of methods of temporal intervals measurements and G.J. Whitrow's decision of the problem of the standard clock choice an innate (a priori) interconnection of simultaneous world events, that is able to create the space-time metric, is revealed. 3. Then we are taking into consideration that the essence of physical processes, which are used for the temporal intervals measurements, indicates in fact some way to finding a possibility of observations of a certain physical phenomenon which is connected with the a priori interconnection of simultaneous world events. Some mathematical relation indicating that as this phenomenon figures the revealed and successively learned by Kozyrev distance-type action of external irreversible processes changing the internal state of any complex, structured systems, that can be in various internal states, is obtained and analyzed. This phenomenon, in principle, justifies the known I. Prigogine's notations about a constructive function of irreversible processes in the appearance, existence, and evolution of complex systems. 4. And finally, some results of astronomical observations, which testify about properties of the Minkowski's World structure, are considered and new possibilities for investigations of various branches of science are enumerated.
Keywords: Minkowski's World, World of events, space-time, space-time metric, measurements of temporal intervals, standard clock problem, irreversible processes.

A.N. Kabanov
The Upper Hypercentral Series of the Group of Unitriangular Automorphisms of a Free Leibniz Algebra

The upper hypercentral series of the group of unitriangular automorphisms of a free Leibniz algebra over an arbitrary field is described. The length of this series is obtained.
Keywords: Leibniz algebra, unitriangular automorphism, hypercenter.

N.V. Michailova
Problem-Oriented Justification of the Modern Mathematical Analysis

Formation of the mathematical analysis revealed a contradiction between its ability to receive concrete mathematical results and specific difficulties of its justification. Article considers problem-oriented justification of the mathematical analysis.
Keywords: justification of the modern mathematical analysis, problem-oriented approach, philosophy of mathematics aspects.

V.I. Strunin, V.V. Shkurkin, A.A. Lyahov
Software and Hardware Equipment for Probe Diagnostics of Low-Temperature Plasmas

Automated system for probe measurements in low-temperature plasmas is considered. To acquire of current-voltage characteristics a multifunctional data-acquisition board is used. The double differentiation of characteristics is provided by the programmed digital filter. The test probe characteristic of air lower pressure plasma was obtained and the electron energy distribution function was recovered from it. The influence of filter's parameter on the accuracy of electron mean energy measurements is also discussed.
Keywords: Langmuir probe, Savitsky-Goley filter, differentiation of experimental curves, automated measurements.

R. Martinez, V. Kreinovich
Does the Universe Really Expand Faster than the Speed of Light: Kinematic Analysis Based on Special Relativity and Copernican Principle

In the first approximation, the Universe�s expansion is described by the Hubble�s law v = H � R, according to which the relative speed v of two objects in the expanding Universe grows linearly with the distance R between them. This law can be derived from the Copernican principle, according to which, cosmology-wise, there is no special location in the Universe, and thus, the expanding Universe should look the same from every starting point. The problem with the Hubble�s formula is that for large distance, it leads to nonphysical larger-than-speed-of-light velocities. Since the Universe�s expansion is a consequence of Einstein�s General Relativity Theory (GRT), this problem is usually handled by taking into account GRT�s curved character of spacetime. In this paper, we consider this problem from a purely kinematic viewpoint. We show that if we take into account special-relativistic effects when applying the Copernican principle, we get a modified version of the Hubble�s law, in which all the velocities are physically meaningful � in the sense that they never exceed the speed of light.
Keywords: cosmological expansion, Copernican principle, special relativity, faster-than-speed-of-light.

Applied Mathematics and Modeling

S.L. Deryabin, A.S. Kiryanova
Mathematical Modeling of Fluid Flows Under Gravity is Appearing as a Result Of Damage to the Dam

The paper examines the three-dimensional isentropic flows of a polytropic gas under the action of gravity. As a mathematical model, a system of equations of gas dynamics for a polytropic gas is used. With the polytropic index \( \gamma=7.02 \), this system describes the movement of water. The system introduces an orthogonal curvilinear coordinate system. To formulate the problem of the disintegration of a special discontinuity, the system makes a degenerate change of variables, namely: dependent and independent variables change roles. In new variables, an initial-boundary value problem with data on the sound characteristic and an additional condition is posed for the system. The solution of the initial-boundary value problem is constructed in the form of power series. The convergence of the constructed series in the region from the surface of a weak discontinuity to the gas-vacuum boundary is proved. To determine the law of motion of the gas-vacuum boundary, a quasilinear system of partial differential equations is written out, which, with the help of the characteristic parameter, reduces to a system of ordinary differential equations.
Keywords: polytropic gas, vacuum, force of gravity, the gas dynamics equations, gas-vacuum boundary, initial-boundary value problem, Riemann problem, centered wave.

A.B. Krohaleva, V.M. Belov
The Human Factor in the System of Socially Significant Activity

In article the entity and a role of a human factor in an assessment of quality of socially important activities are considered. The evolution of this category is provided, since the period of operations of neoclassics until now. Determination of a human factor in the context of socially important activities is offered. For the description of component composition of an integral index of a human factor the classification by different bases is provided: the form of the embodiment, the nature of interaction in system, the content of knowledge, the subject of a human factor. Basic approaches to an assessment of a human factor are analyzed. The two-component model of an assessment of a human factor in socially important activities, consisting of units of qualitative and quantitative analysis of data with use of linguistic, mark and interval scales, is offered. The quantitative component represents an integral assessment of the measured characteristics by means of a complex of medical instruments and devices, combinations of psychophysiological tests. The qualitative component contains an assessment of difficult formalized characteristics in the conditions of uncertainty of basic data.
Keywords: human factor, fuzzy logic, interval analysis, quality assessment, socially significant activity.

A.A. Senkovskaya
Modeling the Distribution Process of the Department's Teaching Load Using a Greedy Algorithm

The main reasons for the need to automate the process of distribution of the academic load of the department are presented. The mathematical formulation of the load distribution problem of the department is formulated. The main algorithms realizing the assigned task are considered, in particular greedy algorithms aimed at optimization of this automated process.
Keywords: optimization, modeling, teaching load, greedy algorithm.

A.M. Pownuk, O. Kosheleva, V. Kreinovich
Towards Decision Making under General Uncertainty

There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.
Keywords: decision making, general uncertainty, fuzzy uncertainty, probabilistic uncertainty.

F. Zapata, O. Kosheleva, V. Kreinovich
Why Stable Teams Are More Efficient in Education

It is known that study groups speed up learning. Recent studies have shown that stable study groups are more efficient than shifting-membership groups. In this paper, we provide a theoretical explanation for this empirical observation.
Keywords: study groups, stable groups, shifting-membership groups.

O. Kosheleva, V. Kreinovich
The Onsager Conjecture: A Pedagogical Explanation

In 1949, a Nobelist Lars Onsager considered liquid flows with velocities changing as \(r^\alpha\) for spatial points at distance \(r\), and conjectured that the threshold value \(\alpha=1/3\) separates the two possible regimes: for \(\alpha>1/3\) energy is always preserved, while for \(\alpha<1/3\) energy is possibly not preserved. In this paper, we provide a simple pedagogical explanation for this conjecture.
Keywords: Euler equations, hydrodynamics, Onsager conjecture.

Computer Science

Heuristic Network for Chatbot Program

An efficient algorithm for the heuristic network, with some properties similar to a recurrent neural network, for a virtual companion program is proposed in the article. The algorithm is improved with the use of an explanatory dictionary of the Russian language. A generator of new responses on the basis of statistical information knowledge base is introduced in the heuristic network.
Keywords: chatbot, virtual companion, recurrent neural network, heuristic network.

L. Brunoni, O. Beaudet-Labrecque
Smart Contracts and Cybercrime: a Game Changer?

The purpose of this paper is to provide a brief explanation regarding the authors� current research in the field of the possible uses of smart contracts in cybercrime, focusing in particular on how the technology could provide a substitute for trust both in client-criminal transactions and in transactions taking place within criminal organizations. The authors share the conviction put forward by Alharby and Moorsel [1] in their 2017 analysis of blockchain-based smart contracts that there is a �lack of studies on criminal activities in smart contracts�: while quality research does exist, including a paper by Juels et al. [2] detailing three types of such activities that can be facilitated by the technology, it is evident that the subject deserves a more widespread attention. Quality research, in fact, could play an important role in aiding authorities and regulators to understand the issue and react accordingly.
Keywords: smart contracts, blockchain, cybercrime, cybersecurity, criminal behavior.