Mathematical structures and modeling ¹1(45)

 Mathematical structures and modeling. - Omsk : OmSU, 2018. ¹1(45), 171 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 fulfillment of the central limit theorem for symmetric functions of random variables, in which the scale normalization is carried out by regularly varying sequences of arbitrary positive order are obtained in this article. These conditions include the so-called minimal conditions of the weak dependence.
Keywords: symmetric functions of random variables, central limit theorem, minimal conditions of the weak dependence

A.V. Levichev, A.Yu. Palyanov
Analysis in Spacetime Bundles Based on Groups U(1,1) and U(2): The Case of SU(2,2)-Actions in Their 2- and 4-Covers

In this paper (which is the direct continuation of A.V.Levichev and A.Yu.Palyanov, Mathematical structures and modeling, 2016, N. 4(40), 24-38) we continue to develop mathematical tools which are necessary in order to perform analysis of homogeneous vector bundles on the basis of the parallelizing group U(1,1). In particular, the compatibility of the (introduced by Paneitz and Segal) coherence condition with the SU(2,2)-action on finite covers of U(1,1) and of U(2) was studied: such a compatibility is violated in case of 2-covers but it holds in case of 4-covers. In terms of the U(2)-approach, we have corrected the proof of one (fundamental for the implementation of any induced representation) Paneitz-Segal formula given in Corollary 4.1.2 of S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47(1982), 78-142.
Keywords: parallelizations of space-time bundles, conformal SU(2,2)-actions in U(1,1), U(2), 2- and 4-covers, coherence condition and induced representations

The article is devoted to philosophical and mathematical aspects of use of the computer in the formalized mathematical proof. The problem of computer proofs reliability is reflected in a demand of experimental mathematics in scientific knowledge.
Keywords: philosophy of mathematical proofs, crisis of recomplexity of proofs, trust to computer proofs

Article considers problem-oriented training in the higher mathematics, problematic approach to specific objectives and the idea of methodological pragmatism. The methodological pragmatism is expressed in criticism of the idea of absolute justification of mathematics and the educational argument.
Keywords: methodological pragmatism, problem training, philosophy of mathematical education

Ethical problems associated with the work of a quantum time machine are investigated.
Keywords: quantum time machine, ethical problems, intertemporal transitions

Can Mass Be Negative?

Overcoming the force of gravity is an important part of space travel and a significant obstacle preventing many seemingly reasonable space travel schemes to become practical. Science fiction writers like to imagine materials that may help to make space travel easier. Negative mass -- supposedly causing anti-gravity -- is one of the popular ideas in this regard. But can mass be negative? In this paper, we show that negative masses are not possible -- their existence would enable us to create energy out of nothing, which contradicts to the energy conservation law.
Keywords: negative mass, equivalence principle, anti-gravity, energy conservation law

J. McClure, O. Kosheleva, V. Kreinovich
The Sums of $$\mathbf{m_i\cdot v_i}$$ and $$\mathbf{m_i\cdot v_i^2}$$ Are Preserved, Why not Sum of $$\mathbf{m_i\cdot v_i^3}$$: A Pedagogical Remark

Students studying physics sometimes ask a natural question: the momentum -- sum of $$m_i\cdot v_i$$ -- is preserved, the energy -- one half of the sum of $$m_i\cdot v_i^2$$ -- is preserved, why not sum of $$m_i\cdot v_i^3$$? In this paper, we give a simple answer to this question.
Keywords: momentum conservation law, energy conservation law, teaching physics

Applied Mathematics and Modeling

L.A. Volodchenkova,A.K. Guts
Modeling the Equilibrium Evolution of the Forest Biocenosis on Solid Felling

It is shown that the restoration of the forest after felling can be described as a Nash equilibrium.
Keywords: restoration of the forest, evolution, the Nash equilibrium

For the class of systems specified in the title we prove necessary and sufficient criteria for exponential stability and dichotomy of solutions of the Cauchy problem in terms of the zeros $$\left(\lambda,\mu\right)$$ of determinant of a matrix pencil -- symbol of the functional-differential operator in the left part of the system. An illustrative example is considered.
Keywords: transition to the difference Cauchy problem, characterization of the spectrum of resolving operator

Calculation algorithm of a block criterion of the interval forecastability for dynamic indicator based on Tarsitano-Lombardo’s nonparametric correlation coefficient was proposed and tested. It was experimentally shown that the value of the block criterion of the interval forecastability, which was calculated from prehistory of dynamic indicator values, makes it possible to quantitatively estimate the appropriateness of performing interval forecasting of the dynamic indicator based on its statistical properties. The interval forecasting is to determine the interval from two predetermined intervals in which the future value of the indicator will be located. The forecasting is based on probability estimates of these events. In this case, the separation division of the intervals is set by the calculation method based on statistical characteristics of the dynamic indicator. The proposed calculation algorithm of the block criterion of the dynamic indicator interval forecastability was implemented in the programming language "R".
Keywords: interval forecasting, interval predictability estimating, dynamic indicator, nonparametric correlation, interval forecasting accuracy

E.V. Rabinovich, P.I. Vaynmaster, G.S. Shefel
Elimination of Data Redundancy in Seismic Monitoring of Hydraulic Fracturing

The important stage of hydraulic fracturing monitoring is constructing a graphical model of fracturing area. However, the real data recorded in the process of hydraulic fracturing are characterized by high intensity of seismic activity in the monitoring area, which hinders correct constructing of such model. Therefore, the problem of eliminating the redundancy of seismic data is important. The original algorithm is considered in this article, which allows eliminating redundancy by detection of local hypocenters of seismic activity. The algorithm is based on the methods of cluster and factor analysis. Factor analysis is used to identify similar characteristics and reduce the number of variables used to describe a seismic source. Cluster analysis is used to detect clouds of seismic emission and find their local hypocenters. The results obtained are used to construct a three-dimensional graphic fracture model.
Keywords: hydraulic fracturing, fracture, factor and cluster analysis

V.R. Shagiev, A.M. Akhtyamov
Identification of Pipe Fastening Using the Minimum Number of Natural Frequencies

The oscillations of a pipeline with a liquid are considered. Previously, it was shown that if the liquid does not flow through the pipeline, then by all its frequencies of flexural vibrations of the pipeline, the type of fixing of the pipeline is uniquely determined up to permutations of fastenings at its ends. The problem of identifying boundary conditions was also solved for nine natural frequencies. In this paper, the number of eigenvalues with which one can uniquely reduce the boundary conditions, up to permutations of the fastenings at its ends, is reduced to five. The number of spectral data was reduced due to the fact that if a linear system of 9 equations was previously solved, then in the present paper a system of five nonlinear equations is solved with respect to four unknown coefficients from the boundary conditions reduced to the canonical form. An example of solving this inverse problem is presented. Two counterexamples are also given in which it is shown that fewer eigenvalues for identification are generally insufficient. In the first counterexample, it is shown that four nonzero eigenfrequencies are still not enough to identify the type of anchorage of the pipeline. The second counterexample shows that in some cases information is needed about whether zero is an eigenvalue.
Keywords: boundary conditions, natural frequencies, eigenvalues, clamping, free support, floating fixing, free end, pipeline

Transcript of the remote presentation at the conference "Reflective Theater of the Situation Center"\ 29.11.2016. It has been edited by the author.
Keywords: subjective psychology, psychophysics, behaviorism, cognitive psychology, reflexive approach

The issues of creating intelligent automated training system with the function of reflection are considered. Using the reflexive games theory of V.A. Lefebvre, the structure of the scheduler of training system introduces a reflexive intelligent agent (RIA), which provides trade-offs in the interaction models of a student, teacher, subject tutor and practitioner. The basic components of the reflexive perception model of RIA of student and learning situation are described, taking into account the specifics of decision-making for different stages of learning.
Keywords: modeling, individualization of learning, automated learning, reflexive governance, reflexive intelligent agent

The constructor of subjects of reflexive games is considered as a component of support for reflexive control in the project ”Gen-Guru”. The project is implemented with the use of cross-technologies of situational center. Key ideas of V. A. Lefebvre’s model system are analyzed. List for the options of detail models to improve the accuracy of modeling situations is suggested. We consider the specifics of modeling such subjects as individuals, and robots.
Keywords: reflexive control, models of V. A. Lefebvre, details of the models, subjects

Computer Sciences

E.A. Tyumentsev
About Formal Definition of an Abstraction

The article proposes a formal definition of abstraction as a mapping of a set of entities into a set of subwords of all words of some alphabet, a hierarchy of abstractions as relations on the set of subwords of a language and interpretation. Also received a number of limitations, which has an abstraction mechanism.
Keywords: abstraction, abstarction barrier, inheritance, Grady Booch, Barbara Liskov, Robert Martin

In the article ”About The Formalization Of The Software Development Process”, the definitions of the software development process as a process of editing the text of the program, the speed of the process, and a sufficient condition for linear speed are formulated. In determining the speed of the development process, the existence of a limit is required. However, the required limit may not exist. In this paper, we define the speed of software development process, which does not require the existence of a limit, and reformulates a sufficient condition for linear velocity in accordance with the new definition.
Keywords: formalization, development process, software, productivity

T.V. Vahniy, A.K. Guts, I.Yu. Pahotin
Determining the Optimal Set of Tools for Computer System Security by Monte-Carlo Method

The article presents a software application that allows based on game theory to find the most optimal set of tools for computer information protection by Monte-Carlo method.
Keywords: information security, theory of games, hacker attacks, optimal strategy, software product, Monte-Carlo method

The article is devoted to the analysis of the themes of the unified state exam in informatics and the construction on its basis of the structural model of the course for the vocational guidance school of the Faculty of Computer Science. The analysis of information resources, which are sources of tasks for preparation, has been carried out. The structural model is supplemented with topics that ensure consistency and coherence of topics.
Keywords: structural model, computer science, information and telecommunication technologies, unified state examination, vocational guidance, additional education