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  7. 2017 год
  8. 39-4

Electronic Modeling

Vol 39, No 4 (2017)

https://doi.org/10.15407/emodel.39.04

CONTENTS

Mathematical Modeling and Computation Methods

  LISTROVOY S.V., SIDORENKO A.V., LISTROVAYA E.S.
Method of Enumeration of Maximum Independent Sets in Nonoriented Graphs
3-18
  KRASILNIKOV A.I.
Analysis of the Kurtosis Coefficient of Contaminated Gaussian Distributions
19-30
  GAMZAEV Kh.M.
About one Stefan Inverse Problem for Phase Transformations in Solids
31-42

Computational Processes and Systems

  MINAEV Yu.N., GUZIY N.N., FILIMONOVA O.Yu., MINAEVA J.I.
Analysis of Self-Similarity of Multivariate Time Series (Ts) on the Basis of the Methods of Intellectual Analysis of the Data
43-68
  SAPOZHNIKOV V.V., SAPOZHNIKOV Vl.V., EFANOV D.V.
Modulo Weighted Codes with Summation with the Minimum Number of Undetectable Errors in Data Vectors
69-88 

Application of Modeling Methods and Facilities

  FATTAKHOVA M.I., VELIDZHANOVA G.M., KADYROV H.A.
Analysis of Schemes for Partition of Channels in Cellular Networks without Queue
89-104
  POLISSKY Yu.D.
On Certain Approaches to Implementation of Some Problem Operations in the Residue Class System
105-114

METHOD OF ENUMERATION OF MAXIMUM INDEPENDENT SETS IN NONORIENTED GRAPHS

S.V. Listrovoy, A.V. Sidorenko, E.S. Listrovaya

Èlektron. model. 2017, 39(4):03-18
https://doi.org/10.15407/emodel.39.04.003

ABSTRACT

Based on the rank approach the authors propose a method of enumeration of maximum independent sets of nonoriented connected graph with time complexity that does not exceed, at an average, O (n6), where n is the number of vertices in the graph, for the graphs which do not contain separating vertices, which dimension does not exceed n=125.

KEYWORDS

maximal independent set, click, vertex cover.

REFERENCES

1. Merrifield, R.E. and Simmons, H.E. (1989), Topological methods in chemistry, New York, John Wiley & Sons, USA.
2. Hosoya, H. (1971),Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn., Vol. 44, no. 9, pp. 2332-2339.
3. Miller, R.E. and Muller, D.E. (1960), The problem of the maximum consistent subsets, IBM Research Report RC-240, J. T. Watson Research Center, Yorktown Heights, New York, USA. Moon J.W., Moser L. On cliques in graphs, Israel J. Math.,Vol. 3, p. 23-28.
4. Watson J.T. Research Center, Yorktown Heights, N.Y. Moon J.W., Moser L. On cliques in graphs, Israel J. Math. Vol. 3, p. 23-28.
5. Harley, E., Bonner, A. and Goodman, N. (2001), Uniform integration of genome mapping data using intersection graphs, Bioinformatics, Vol. 17, pp. 487-494.
6. Moon, J.W. and Moser, L. (1965), On cliques in graphs, Israel J. Math., Vol. 3, pp. 23-28.
7. Tomita, E., Tanaka, A. and Takahashi, H. (2006), The worst-case time complexity for generating all maximal cliques and computational experiments, Theoretical Computer Science, Vol. 363, pp. 28-42.
8. Prodinger, H. and Tichy, R.F. (1982), Fibonacci numbers of graphs, Fibonacci Quart, Vol. 20, no. 1, pp.16-21.
9. Listrovoy, S.V. and Minukhin, S.V. (2010), General approach to solving optimization problems in distributed computing systems and theory of intelligence systems construction, Journal of automation and information sciences, Vol. 42, no. 3, pp. 30-46.
10. Listrovoy, S.V. and Minukhin, S.V. (2010), “A general approach to the solution of optimization problems in distributed computing systems and the theory of constructing intellectual systems”, Problemy upravleniya i informatika, no. 2, pp.65-82.
11. Listrovoy, S.V. (2014), “The method of enumeration of maximal independent sets in arbitrary non-oriented graphs”, Elektronnoe modelirovanie, Vol. 36, no. 1, pp. 3-17.
12. Listrovoy, S.V., Listrovaya, E.S., Panchenko, S.V., Moiseenko, V.I. and Kamenev, A.U. (2017), Mathematical models in computer control systems RAILWAYS and parallel computing: Monograph, FOP Brovin O., Kharkiv, Ukraine.

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ANALYSIS OF THE KURTOSIS COEFFICIENT OF CONTAMINATED GAUSSIAN DISTRIBUTIONS

A.I. Krasilnikov

Èlektron. model. 2017, 39(4):19-30
https://doi.org/10.15407/emodel.39.04.019

ABSTRACT

A formula for finding the kurtosis coefficient of symmetric contaminated Gaussian distributions has been obtained. The dependence of the kurtosis coefficient on the parameters of the model of contaminated distributions has been studied. Examples of contaminating with uniform and logistic distributions have been considered. The obtained results allow the author to analyze non-Gaussian random variables described by the model of contaminated Gaussian distributions.

KEYWORDS

contaminated distributions, Tukey-Huber model, mixtures of distributions, kurtosis coefficient, cumulant analysis.

REFERENCES

1. Aivazian, S.A., Eniukov, I.S. and Meshalkin, L.D. (1983), Prikladnaya statistika: Osnovy modelirovaniia i pervichnaia obrabotka dannykh. Spravochnoe izd. [Applied statistics: bases of modeling and initial data processing. Reference edition], Finansy i statistika, Moscow, USSR.
2. Mukha, V.S. (2009), Statisticheskie metody obrabotki dannykh: Uchebnoe posobie [Statistical methods of data processing: Tutorial], Izdatelskiy tsentr BGU, Minsk, Belarus.
3. Lemeshko, B.Yu., Lemeshko, S.B., Postovalov, S.N. and Chimitova, E.V. (2011), Statisticheskii analiz dannykh, modelirovanie i issledovanie veroiatnostnykh zakonomernostei. Kompiuternyi podkhod [Statistical data analysis, simulation and study of probability regularities. Computer approach], Izdatelstvo NGTU, Novosibirsk, Russia.
4. Tukey, J.W. (1960), A survey of sampling from contaminated distributions. Contributions to Probability and Statistics, Ed. I. Olkin, Stanford University Press, Stanford, UK.
5. Huber, P.J. (1984), Robastnost v statistike [Robust statistics], Translated by I.A. Makhova and V.I. Khokhlov, Ed. I.G. Zhurbenko, Mir, Moscow, USSR.
6. Hampel, F., Ronchetti, E., Rousseeuw, P. and Stahel, W. (1989), Robastnost v statistike. Podkhod na osnove funktsii vliianiia [Robust statistics. The approach based on influence functions], Translated by V.M. Zolotarev, Mir, Moscow, USSR.
7. Figueiredo, F. and Gomes, M.I. (2016), The total median statistic to monitor contaminated normal data, Journal Quality Technology & Quantitative Management, Vol. 13, pp. 1-16, available at: http://www.tandfonline.com/doi/abs/10.1080/16843703.2016.1139840
8. Punzo, A. and McNicholas, P.D. (2016), Parsimonious mixtures of multivariate contaminated normal distributions, Preprint submitted to arXiv 1305.4669, May 20, 2016. - pp. 1-28, available at: https://arxiv.org/pdf/1305.4669.pdf
9. Marchuk, V.I. and Tokareva, S.V. (2009), Sposoby obnaruzheniia anomalnykh znachenii pri analize nestatsionarnykh sluchainykh protsessov: Monografiia [Methods for detecting anomalous values in the analysis of non-stationary random processes: Monograph], Yuzhnorossiiskii gosudarstvennyi universitet ekonomiki i servisa, Shakhty, Russia.
10. Denisov, V.I. and Timofeev, V.S. (2011), “Stable distributions and estimation of parameters of regression dependencies”, Izvestiya Tomskogo politekhnicheskogo instituta, Vol. 318, no. 2, pp. 10-15.
11. Osadchiy, I.S. (2015), “Method for estimating the distribution parameters of Gaussian noise for the operation of a pulse signal system”, Zhurnal radioelektroniki: elektronnyy zhurnal, no. 4, pp. 1-27, available at: http://jre.cplire.ru/jre/apr15/1/text.html
12. Robbins, H. (1948), Mixture of distributions, Ann. Math. Stat, Vol. 19, pp. 360-369.
https://doi.org/10.1214/aoms/1177730200
13. Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985), Statistical analysis of finite mixture distributions, John Wiley & Sons, New York, USA.
14. McLachlan, G. and Peel, D. (2002), Finite mixture models, John Wiley & Sons, New York, USA.
15. Korolev, V.Yu. (2004), Smeshannye gaussovskie veroiatnostnye modeli realnykh protsessov [The mixed Gaussian probabilistic models of real processes], Maks Press, Moscow, Russia.
16. Korolev, V.Yu. (2008), Veroyatnostno-statisticheskii analiz khaoticheskikh protsessov s pomoshchiu smeshannykh gaussovskikh modelei. Dekompozitsiia volatilnosti finansovykh indeksov i turbulentnoi plazmy [Probabilistic-statistical analysis of chaotic processes using mixed Gaussian models. Decomposition of volatility of financial indices and turbulent plasma], Izdatelstvo Instituta problem informatiki RAN, Moscow, Russia.
17. Krasilnikov, A.I. and Pilipenko, K.P. (2007), “Unimodal two-componental Gaussian mixture. Excess kurtosis”, Elektronika i sviaz, no. 2 (37), pp. 32-38.
18. Krasilnikov, A.I. (2017), “Class of non-Gaussian symmetric distributions with zero coefficient of kurtosis”, Elektronnoe modelirovanie, Vol. 39, no. 1, pp. 3-17.
19. Vadzinskii, R.N. (2001), Spravochnik po veroiatnostnym raspredeleniiam [Reference book on probabilistic distributions], Nauka, St. Petersburg, Russia.

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ABOUT ONE STEFAN INVERSE PROBLEM FOR PHASE TRANSFORMATIONS IN SOLIDS

Kh.M. Gamzaev

Èlektron. model. 2017, 39(4):31-42
https://doi.org/10.15407/emodel.39.04.031

ABSTRACT

The diffusion phase transformation process described by a nonlinear system of partial differential equations with a moving boundary has been considered. The inverse problem is formulated to determine the solute concentration on the external surface of the volume under consideration, which ensures the moving boundary displacement according to a given law. Applying the methods of fronts rectification and difference approximation, the problem posed is reduced to two difference problems. A computational algorithm is proposed for solving the obtained difference problems.

KEYWORDS

diffusion phase transformation, moving phase boundary, forward rectification method, boundary inverse problem, difference method.

REFERENCES

1. Lyubov, B.Ya. (1969), Kineticheskaya teoriya fazovykh prevrashcheniy [Kinetic theory of phase transformations], Metallurgiya, Moscow, USSR.
2. Lyubov, B.Ya. (1981), Diffuzionnyie protsessy v neodnorodnykh tvyordykh sredakh [Diffusion processes in inhomogeneous solid media], Nauka, Moscow, USSR.
3. Merer, Kh. (2011), Diffuziya v tvyordykh telakh [Diffusion in solids], Izdatelskiy dom «Intellect», Dolgoprudny, Russia.
4. Bokshtein, B.S. (1978), Diffuziya v metallakh [Diffusion in metals], Metallurgiya, Moscow, USSR.
5. Rubinshtein, L.I. (1967), Problema Stefana [The problem of Stefan], Zvaygzne, Riga, USSR.
6. Samarskiy, A.A. and Vabishchevich, P.N. (2003), Vychislitelnaya teploperedacha [Computing heat transfer], Editorial, Moscow, Russia.
7. Alifanov, O.M., Artyukhin, E.A. and Rumyantsev, S.V. (1988), Ekstremalnyie metody resheniya nekorrektnykh zadach [Extreme methods of the solution of incorrect problems], Nauka, Moscow, USSR.
8. Samarskiy, A.A. and Vabishchevich, P.N. (2009), Chislennyie metody resheniya obratnykh zadach matematicheskoi fiziki [Numerical methods of the solution of the inverse problems of mathematical physics], Izdatelstvo LKI, Moscow, Russia.
9. Goldman, N.L. (2002), “Classical and generalized solution of the two-phase boundary inverse Stefan problem”, Vychislitelnyiemetody i programmirovanie,Vol. 3, no. 1, pp. 133-143.
10. Goldman, N.L. (2003), “Properties of solutions of the inverse Stefan problem”, Differentsialnyie uravneniya, Vol. 39, no. 1, pp. 66-72.
11. Gamzaev, Kh.M. (2015), “Numerical solution of the problem of unsaturated filtration with a moving boundary”, Elektronnoe modelirovanie, Vol. 37, no. 1, pp.15-24.

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ANALYSIS OF SELF-SIMILARITY OF MULTIVARIATE TIME SERIES (TS) ON THE BASIS OF THE METHODS OF INTELLECTUAL ANALYSIS OF THE DATA

Yu.N. Minaev, N.N. Guziy, O.Yu Filimonova., J.I. Minaeva

Èlektron. model. 2017, 39(4):43-68
https://doi.org/10.15407/emodel.39.04.043

ABSTRACT

Calculation methods have been proposed for the Hurst factor for univariate and multivariate TS on the basis of the main diagonals of TS tensor models. It is shown that the problem complexity determines the joint use of several mathematical theories, in particular, the tensor and multivariate matrix analysis. The examples of using the proposed methods are presented.

KEYWORDS

tensor, multivariate time series, intellectual analysis of the data, 3D matrices, diagonal 3D matrices, matrix development, self-similarity, Hurst parameter.

REFERENCES

1. Time series: Advanced methods IIa. Multivariate time series, available at: www.ucl.ac.uk/jdi/events/int-CIA-conf/ICIAC11_ Sli-des/ ICIAC11_1E_ LTompson.
2. Cichocki, A., Mandic, D., Phan, A.-H., Caiafa, C., Tensor decompositions for signal processing applications. From two-way to multiway component analysis, available at: http://www. commsp. ee.ic.ac.uk/~mandic/SPM-Cichocki-Mandic-DeLathauwer. pdf 
3. Sokolov, N.P. (1960), Prostranstvennyie matritsy i ikh prilozheniya [Space matrice and their application], Gosudarstvennoe izdatelstvo fiz.-mat. literatury, Moscow, USSR.
4. Claude, Z.B., Introduction to the general multidimensional matrix in mathematics, available at: www.ijera.com/pages/v3no6.html.
5. Solo, A., Multidimensional matrix mathematics: Notation, representation, and simplification. Parts: 1-6, Proceedings of the World Congress on Engineering (3), available at: www.ijera.com/ papers/Vol.3_issue6/ U36123129.pdf.
6. De Lathauwer, L. and Moor, B. (1998), From matrix to tensor: Multilinear algebra and signal proce-ssing, Proceedings of the 4-h IMA Int. Conf. on Mathematics in Signal Processing, Oxford, United Kingdom, Selected papers presented at pp. 1-15, J. McWhirter (Ed.), Mathematics in Signal Processing IV, Oxford University Press.
7. Skillicorn, D., Understanding complex datasets : data mining with matrix decomposi- tions. Chapman & Hall/ CRC Taylor & Francis Group 6000 Broken Sound Parkway NW. Suite 300 Boca Raton, FL 33487, 2742.
8. Cichocki, A. (2013), Tensor decompositions: A new concept in brain data analysis?, available at: arXiv:1305.0395v1 [cs.NA] May 2, 2013.
9. Lim, L.-H. (2005), Singular values and eigenvalues of tensors: A variational approach, Proceedings of the 1-st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), December 13-15, 2005, pp. 129-132.
10. Liqun, Qi. (2007), Eigenvalues and invariants of tensors, J. Math. Anal. Appl., Vol. 325, pp. 1363-1377, available at: www. elsevier. com/ locate/jmaa.
11. Kolda, T.G. and Bader, B.W. (2009), Tensor decompositions and applications, SIAM Review, Vol. 51, no. 3, pp. 455-500.
12. Kamalja, K.K. and Khangar, N.V. (2013), Singular value decomposition for multidimensional matrices, Int. Journal of Engineering Research and Applications, Vol. 3, Iss. 6, pp. 123-129.
13. Bader, B.W. and Kolda, T.G. Tensor decompositions, the MATLAB tensor toolbox, and applications to data analysis, available at: www.sandia.gov/~tgkolda/ TensorToolbox.
14. Bader, B.W. and Kolda, T.G. (2006), Multilinear operators for higher-order decompositions: Technical report SAND 2006-2081, Sandia National Laboratories, available at: pubs/pubfiles/SAND2007-6702.pdf.
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16. Tensor toolbox is software for working with multidimensional arrays, available at: http://csmr. ca. sandia.gov/~tgkolda.
17. Kindlmann, G., Tensor invariants and their gradients, available at: This email address is being protected from spambots. You need JavaScript enabled to view it.. edu. 
18. Bozhokin, S.V. and Parshin, D.A. (2001), Fraktaly i multifraktaly [Fractals and multi fractals], NITs Regulyarnaya i khaoticheskaya dinamika, Izhevsk, Russia.
19. Shelukhin, O.I. (2011), Multifraktaly. Infokomunikatsionnyie prilozheniya [Multifractals. Infocommunication applications], Goryachaya liniya – Telekom, Moscow, Russia.

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