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  5. VOL 40, NO 2 (2018)
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  7. 2015 год
  8. 37-2

Electronic Modeling

VOL 37, NO 2 (2015)

CONTENTS

Mathematical Modeling and Computation Methods

  SWAPNA N., UDAYA KUMAR S., MURTY K.N.
Best Least Square Solution of Boundary Value Problems Associated with a System of First Order Matrix DifferentialEquation
3-16
  KALINOVSKY Ya.A., TURENKO A.S., BOYARINOVA Yu.E., KHITSKO Ya.V.
A Research of Properties of the Generalized Quaternions and Their Relationships with Grassmann-Clifford Procedure of DoublingProperties of the Generalized Quaternions and Their Relationships with Grassmann-Clifford Procedure of Doubling
17-26
  KRAVCHENKO Yu.V., RAKUSHEV M.Yu.
The Stability of the Differential Spectrum for Linear Ordinary Differential Equations SystemOrdinary Differential Equations System
27-40

Computational Processes and Systems

  MINAEV Yu.N., FILIMONOVA O.Yu., MINAEVA Yu.I.
Influence of Hierarchic Structure of Fuzzy Set Granules on Computation Procedures of Fuzzy Mathematics.Fuzzy Set Granules on Computation Procedures of Fuzzy Mathematics.
41-58

Parallel Computations

  LUNTOVSKYY A.O., MELNYK I.V.
Up-to-Date Fog Computing Systems and Methods of Their DesignTheir Design
59-76
  FARHADZADE E.M., MURADALIYEV A.Z., FARZALIYEV Yu.Z.
Estimation of Expediency of Classification of Multivariate Data To the Set AttributeClassification of Multivariate Data To the Set Attribute
77-86

Application of Modeling Methods and Facilities

  KRASIL’NIKOV A.I., BEREGUN V.S.
Using of the Generalized Summation Methods at Approximation of Probability Density Functionof Probability Density Function
87-100
  BUDASHKO V.V., YUSHKOV Y.A.
Mathematical Modeling of All-range Controllers of Speed of Thrusters for Ship Power Plants in Combined Propulsion complexesSpeed of Thrusters for Ship Power Plants in Combined Propulsion complexes
101-114

Short Notes

  SHINKARENKO N.V.
Simulation of Direction Patterns of Transmission Impulse X–Ray Tubes
114-120

 

 

Best Least Square Solution of Boundary Value Problems Associated with a System of First Order Matrix Differential Equation (10)

N. Swapna, S. Udaya Kumar
Dept of Computer Science and Engineering
K.N. Murty
Dept of Mathematics Geethanjali College
of Engineering and Technology,
(Hyderabad (A.P.) India, å-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.)

ABSTRACT

The best least square solution of the boundary value problem is constructed via modified QR algorithm and also RQ algorithm. As a result of this work one has a choice of effective methods for finding solutions to two point boundary value problems in the non invertible case. Further these results are exemplified with suitable examples to highlight the modified QR and RQ algorithms.

KEYWORDS

boundary value problems, least square solution, QR and RQ algorithms, overdetermined systems, underdetermined systems.

REFERENCES

1. Cole, R.H. (1986), Theory of Ordinary Differential Equations, Appleton-Century Grafts, Norwalk.
2. Rice, J.R. (1966), Experiments of Gram-Schmidt orthogonalization, Math-Comp., Vol. 20, pp. 325-328.
3. Sreedharan, V.P. (1988), A Note on modified Gram-Schmidt process, Math-Comp., Vol. 24, pp. 277-290.
4. Sastry, B.R., Murty, K.N. and Balaram, V.V.V.S.S. (2007), General first order matrix difference system-existence and uniqueness via new lattice based cryptographic construction, Elektronnoe modelirovanie, Vol. 29, no. 2, pp. 245-259.

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PROPERTIES OF THE GENERALIZED QUATERNIONS AND THEIR RELATIONSHIP WITH GRASSMANN—CLIFFORD PROCEDURE OF DOUBLING

Ya. Kalinovsky, Yu. Boyarinova, A.Turenko, Y. Khitsko

ABSTRACT

The class of non-commutative hypercomplex number systems of the 4th dimension constructed by using of non-commutative procedure of Grassman—Clifford doubling of 2-dimensional systems has been investigated in the article and their relationships with the generalized quaternions has been established. Algorithms of operations performance and methods of algebraic characteristics calculation in them, such as conjugation, normalization, a type of zero divisors are investigated.

KEYWORDS

quaternion, generalized quaternion, hypercomplex number system, zero divisor, pseudonorm, conjugation, Grassman—Clifford procedure of doubling.

REFERENCES

1. Hamilton, W.R. (1844), “On a new species of imaginary quantities connected with a theory of quaternions», Proceedings of the Royal Irish Academy, Vol. 2, pp. 424-434.
2. Gorberashvili, M. (2014), “Split quaternions and particles in (2+1)-space”, Eur. Phys. J., pp. 3199-3207.
3. Sinkov, M.V., Boyarinova, Yu.E. and Kalinovsky, Ya.A. (2010), Konechnomernyie giperkompleksnyie chislovyie sistemy. Osnovy teorii. Primeneniya [Finite-dimensional hypercomplex number systems. Fundamentals of the theory. Applications], Infodruk, Kiev, Ukraine.
4. Toyoshima, H. (2002), “Computationally efficient implementation of hypercomplex digital filters”, IEICE Trans. Fundamentals, Vol. E85-A, no. 8, pp. 1870-1876.
5. Godel, C. (1949), “An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation”, Rev. Mod. Phys, Vol. 21, no. 3, pp. 447-450.
6. Szeto, G. (1980), “On generalized quaternion algebras”, Internat. I. Math. And Math. Sci., Vol. 3, no. 2, pp. 237-245.
7. Cai Yong-yu (2002), “On the first-degree algebraic equation of the generalized quaternion”, Chinese Quarterly Journal of Mathematics, Vol. 17, no. 2, pp. 59-64.
8. Flaut, C. and Shpakivskyi, V. (2014), “An efficient method for solving equations in generalized quaternion and octonion algebras”, available at: http://arxiv.org/pdf/1405.5652.pdf. 2014 (accessed June 4, 2014).
9. Jafari, M. and Yayli, Y. (2012), “Generalized quaternion and rotation in 3-space”, available at: http://arxiv.org/abs/1204.2476 (accessed April 11, 2012).
10. Mamagami, A.B. and Jafari, M. (2013), “On properties of generalized quaternion algebra”, Journal of Novel Applied Sciences, Vol. 2, no. 12, pp. 683-689.
11. Mamagami, A.B. and Jafari, M. (2013), “Some notes on matrix of generalized quaternion», International Research Journal of Applied and Basic Sciences, Vol. 7, no. 14, pp. 1164-1171.
12. Kalinovsky, Ya.O., Lande, D.V., Boyarinova, Yu.E. and Turenko, A.S. (2014), “Computing characteristics of one class of non-commutative hypercomplex number systems of 4-dimension”, available at : http://arxiv.org/ftp/arxiv/papers/1409/1409.3193.pdf (accessed September 9, 2014).
13. Kantor, I.L. and Solodovnikov, A.S. (1973) Giperkompleksnyie chisla [Hypercomplex numbers], Nauka, Moscow, Russia.
14. Kalinovsky, Ya.A. and Boyarinova, Yu.E. (2012) Vysokorazmernyie izomorfnyie giperkompleksnyie chislovyie sistemy i ikh ispolzovaniye dlya povysheniya efektivnosti vychisleniy [High-dimensional isomorphic hypercomplex number systems and their use to increase calculation efficiency], Infodruk, Kiev, Ukraine.
15. Boyarinova, Yu.E. (2011), “Non-canonical hypercomplex number systems of dimension 2 and their isomorphisms”, Data recording, storage and processing, Vol. 13, no. 1, pp. 29-38.
16. Kalinovsky,Ya.A. (2003), “Research of properties of isomorphism of quadriplex and bicomplex number systems”, Data recording, storage and processing, Vol. 5, no. 1, pp. 69-73.

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THE STABILITY OF THE DIFFERENTIAL SPECTRUM FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS SYSTEM

Yu.V. Kravchenko, M.Yu. Rakushev

ABSTRACT

The majorant estimations of dependence of recursive calculations errors of T-spectrum for the system of linear ordinary differential equations with constant coefficients on representation errors of real numbers in a computer with a floating point were obtained. Proceeding from T-spectrum stability condition the limitations on T-discrete number were defined which are calculated with allowance for the mantissa length of the numbers which used in calculations.

KEYWORDS

differential transformation, T-spectrum, stability range, linear ordinary differential equations system.

REFERENCES

1. Samarsky, A.A. and Gulin, A.V. (1989), Chislennyie metodyi: Ucheb. posobie dlya vuzov [Numerical Methods: Studies. manual for schools], Nauka, Moscow, Russia.
2. Semagina, E.P. (1986), “Differential transformations and their ability to solve the problems of dynamics”, Elektronnoe modelirovanie, no. 4, pp. 44-50.
3. Pukhov, G.E. (1990), Differentsialnye spektry i modeli [Differential spectrum and model], Naukova dumka, Kiev, Ukraine.
4. Koval, N.V. and Semagina, E.P. (1985), “On the stability of algorithms for solving ordinary differential equations systems by the method of differential transformation”, Teoreticheskaya elektrotekhnika, Vol. 39, pp. 108-118.
5. Rakushev, M.Yu. (2007), “Approximation and stability of the method of shifted differential-taylorivskyh transformations to solve the Cauchy problem”, Visnik ZhDTU, Vol. 42, no. 3, pp. 128-132.
6. Semagina, E.P. (1981), “On the efficiency of T-transformations in the numerical solution of differential equations”, Elektronnoe modelirovanie, no. 4, pp. 103-104.
7. Rakushev, M.Yu. (2012), “Computational scheme for integration of stiff ordinary differential equations on the basis of shifted differential transformations”, Problemy upravleniya i informatiki, no. 6, pp. 87-96.
8. Voevodin, V.V. (1977), Vyichislitelnye osnovy lineynoy algebry [Numerical linear algebra basics], Nauka, Moscow, Russia.
9. Sigorskiy, V.P. (1977), Matematicheskiy apparat inzhenera. Izd. 2-e stereotip [Mathematical apparatus engineer], Tekhnika, Kiev, Ukraine.
10. Stepanov, A.V. (1985), “Approximation implicit variant T-scheme of numerical integration”, Teoreticheskaya elektrotehnika, Iss. 39, pp. 123-126.

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INFLUENCE OF HIERARCHIC STRUCTURE OF FUZZY SET GRANULES ON COMPUTATION PROCEDURES OF FUZZY MATHEMATICS

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

ABSTRACT

It is proposed to consider a fuzzy set as an object provided with hierarchic structure obtained with the help of the methods of hierarchic clusterization. Anotion of structure belonging has been introduced which is determined depending on the hierarchic structure of the universal set and considers the norm of the 2-adic number equivalent to the hierarchic structure. It has been proposed to consider the structured FS as the union of the standard FS with the norm of 2-adic number obtained on the basis of binary tree being a result of hierarchic FS clusterization. A generalized indicator of FS structurization - structurally dephadzificated number

KEYWORDS

fuzzy set, hierarchic clusterization, p-adic analysis, structure matrix, binary tree.

REFERENCES

1. Baruah, H. K. (2011), “The Theory of Fuzzy Sets: Beliefs and Realities”, International Journal of Energy, Information and Communications, Vol. 2, issue 2, pp. 1-22.
2. Simon, H.A. (1996), The Sciences of the Artificial, MIT PRESS, Cambridge.
3. Delgado, M., Gomez-Skarmeta, A.F. and Vila, A. (1996), “On the Use of Hierarchical Clustering in Fuzzy Modeling”, Intern. J. of Approximate Reasoning, no. 14, pp. 237-257.
4. Murtagh, F. (2004), “Quantifying ultrametricity”, In J. Antoch, ed., COMPSTAT 2004 – Proceedings in Computational Statistics, Prague, Czech Republic, Springer-Verlag, Berlin, pp. 1561-1568.
5. Khrennikov, A.Y. (2003), Nearkhimedov analiz i yego prilozheniya [Non-Archimedean analysis and its applications], Fizmatli, Moscow, Russia.
6. Murtagh, F. (2008), “From Data to the p-Adic or Ultrametric Model”, available at: arXiv:0805.2744v1 [stat.ML], May 18, 2008.
7. Kurkina, M.V. and Slavskii, V.V. “The inverse problem in the theory of fuzzy relations of equivalence”, available at: http:://www. uctt.ru›download/32/
8. Gundyrev, I.A. (2008), “Similarly homogeneous locally compact spaces with an intrinsicmetric”, available at: http://www. ksu. ru/journals/izv_vuz/.
9. Minayev, Yu.N., Filimonova, O.Yu. and Minaeva, Yu.I. (2012), “Hierarchical clustering of fuzzy data”, Elektronnoe modelirovanie, Vol. 34, no. 4, pp. 3-22.
10. Minayev, Yu.N., Filimonova, O.Yu. and Minaeva, Yu.I. (2013), “Tensor models of FSgranules and their application to solving problems of fuzzy arithmetic”, Iskusstvennyy intellekt, no. 2, pp. 18-32.
11. Minayev, Yu.N., Filimonova, O.Yu. and Minaeva, Yu.I. (2014), “Structured FS-granules in granular computing tasks”, Elektronnoe modelirovanie, Vol. 36, no. 6, pp. 77-96.
12. Khrennikov, A.Yu. (2004), Modelirovaniye protsessov myshleniya v ð-àdicheskikh sistemakh koordinat [Modeling of thinking processes in p-adic coordinate systems], Fizmatlit, Moscow, Russia.
13. Murtagh, F. (2004), “On ultrametricity, data coding, and computation”, J. Classification, Vol. 21, pp. 167-184.
14. Bradley, P.E. “Mumford dendrograms”, The Computer Journal, To appear. Arxiv: 0707. 3540 [cs.DM].
15. Gomez, S., Fernandez, A., Montiel, J. and Torre, D. (2012), “Multidendrograms: variable-group agglomerative hierarchical clustering», available at :http://deim.urv.cat/sgomez/multidendrograms.php (assessed Jan.8, 2012).

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