Ya.А. Кalinovskiy ¹, Yu.E. Boiarinova ^1,2

¹ Institute for Information Recording NAS of Ukraine
  Shpaka str, 2, 03113 Kyiv, Ukraine
  e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

² National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
  Peremogy pr., 37, 03113 Kyiv, Ukraine

Èlektron. model. 2021, 43(6):03-18

https://doi.org/10.15407/emodel.43.06.003

ABSTRACT

The structure of method for constructing a representation of an exponential function in hypercomplex number systems (HNS) by the method of solving an associated system of linear differential equations is considered. Brief information about the hypercomplex computing software (HCS) is given. With the use of HCS, the necessary cumbersome operations on symbolic expressions were performed when constructing the representation of the exponent in the fifth-dimensional HNS. Fragments of programs in the environment of HCS and results of symbolic calculations are resulted.

KEYWORDS

hypercomplex number system, representation of functions, exponent, characteristic number, computer algebra systems, algebraic operation, Keli table.

REFERENCES

1. Hamilton, W.R. (1848), “Researches respecting quaternions: first series”, Transactions of the Royal Irish Academy, Vol. 21, no. 1, pp. 199–296.
2. Brackx, F. (1979), “The exponential function of a quaternion variable”, Applicable Analysis, Vol. 8, pp. 265– 276.
   https://doi.org/10.1080/00036817908839234
3. Scheicher, K., Tichy, R.F. and Tomantschger, K.W. (1997), “Elementary Inequalities in Hypercomplex Numbers”, Anzeiger, Vol. 2, no. 134, pp. 3–10.
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   https://doi.org/10.1007/BF02366787
5. Kalinovskiy, Ya.A., Roenko, N.V. and Sinkov, M.V. (1994), “Building nonlinear functions in quaternion and other hypercomplex number systems for the solution of applied mecanics problem”, of the First Int. Conf. On parallel processing and appl. Math., Poland, pp. 170–177.
6. Sinkov, M.V., Kalinovskiy, Ya.A., Boiarinova, Yu.E. and Fedorenko, A.F. (2006), “On differential equations defining functions of a hypercomplex variable”, Data Recording, Storage & Processing, Vol. 8, no. 3, pp. 20-24.
7. Sinkov, M.V., Kalinovskiy, Ya.A., Boiarinova, Yu.E. and Fedorenko, A.F. (2008), “Imaging non-linearities in scanned-dynamic hypercomplex number systems”, Dopovidi NANU, Vol. 8, pp. 43-51.
8. Sinkov, M.V., Boiarinova, Yu.E. and Kalinovskiy, Ya.A. (2010), Konechnomernyye giperkompleksnyye chislovyye sistemy. Osnovy teorii. Primeneniya [Finite-dimensional hypercomplex number systems. Foundations of the theory. Applications], Infodruk, Kyiv, Ukraine.
9. Klimenko, V.P., Lyahov, A.L. and Gvozdik, D.N. (2011), “Modern features of the development of computer algebra systems”, Matematychni mashyny i systemy, Vol. 2, pp. 3-18.
10. Тatarnikov, O. (2006), “Overview of Symbolic Mathematics Programs”, available at: https://compress.ru/article.aspx?id=16152.
11. Kalinovskiy, Ya.A, Boiarinova, Yu.E. and Hitsko, Ya.V. (2020), Giperkompleksnyye vychisleniya v Maple [Hypercomplex calculates in Maple], IPRI NANU, Kyiv, Ukraine,  ISBN 978-966-02-8879-9.
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13. Kalinovskiy, Ya.A. and Boiarinova, Yu.E. (2012), Vysokorazmernyye izomorfnyye giperkompleksnyye chislovyye sistemy i ikh primeneniya [High-dimensional isomorphic hypercomplex number systems and their applications], IPRI NANU, Kyiv, Ukraine.
14. Korn, G. (1974), Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov [A guide to mathematics for scientists and engineers], Nauka, Moscow, USSR.

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V.I. Havrysh

Èlektron. model. 2021, 43(6):19-33

https://doi.org/10.15407/emodel.43.06.019

ABSTRACT

Nonlinear mathematical models for the analysis of temperature regimes in a thermosensitive isotropic plate heated by locally concentrated heat sources have been developed. For this purpose, the heat-active zones of the plate are described using the theory of generalized functions. Given this equation of thermal conductivity and boundary conditions contain discontinuous and singular right parts. The original nonlinear equations of thermal conductivity and nonlinear boundary conditions are linearized by Kirchhoff transformation. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, their analytical solutions in the images were determined. The inverse integral Fourier transformtransform was applied to these solutions, which made it possible to obtain analytical expressions for determining the Kirchhoff variable. As an example, the linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, analytical relations were obtained to determine the temperature in the heat-sensitive plate. The given analytical solutions are presented in the form of improper convergent integrals. According to Newton’s method (three eighths), numerical values of these integrals are obtained with a certain accuracy for given values of plate thickness, spatial coordinates, specific power of heat sources, thermal conductivity of structural materials of the plate and geometric parameters of the heat-active zone. The material of the plate is silicon and germanium. To determine the numerical values of temperature in the structure, as well as the analysis of heat transfer processes in the middle of the plate due to local heating, developed software, using which geometric mapping of temperature distribution depending on spatial coordinates, thermal conductivity, specific heat flux density. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in the thermosensitive plate with local heating to the real physical process.

KEYWORDS

temperature field; isotropic thermosensitive plate; thermal conductivity; heat-insulated surface; perfect thermal contact, local heating.

REFERENCES

1. Carpinteri, A. and Paggi, M. (2008), “Thermoelastic mismatch in nonhomogeneous beams”, Journal of Engineering Mathematics, Vol. 61, no. 2-4, pp. 371-384, available at:
   https://doi.org/10.1007/s10665-008-9212-8
2. Noda, N. (1991), “Thermal stresses in materials with temperature-dependent properties”, Applied Mechanics Reviews, Vol. 44, pp. 383-397, available at: 
   https://doi.org/10.1115/1.3119511
3. Otao, Y., Tanigawa, O. and Ishimaru, O. (2000), “Optimization of material composition of functionality graded plate for thermal stress relaxation using a genetic algorithm”, Journal of Thermal Stresses, Vol. 23, pp. 257-271, available at: 
   https://doi.org/10.1080/014957300280434
4. Tanigawa, Y., Akai, T. and Kawamura, R. (1996), “Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material pro­perties”, Journal of Thermal Stresses, Vol. 19, no. 1, pp. 77-102, available at: 
   https://doi.org/10.1080/01495739608946161
5. Tanigawa, Y. and Otao, Y. (2002), “Transient thermoelastic analysis of functionally graded plate with temperature-dependent material properties taking into account the thermal radiation”, Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu, Vol. 2, pp. 133-134, available at: https://doi.org/10.1299/jsmemecjo.2002.2.0_133
6. Yangian, Xu and Daihui, Tu. (2009), “Analysis of steady thermal stress in a ZrO2/FGM/ Ti-6Al-4V composite ECBF plate with temperature-dependent material properties by NFEM”, 2009-WASE Int. Conf. on Informa. Eng, Vol. 2, pp. 433-436, available at: https://doi.org/109/ICICTA.2009.842.
7. Dovbnia, K.M. and Dundar, O.D. (2016), “Stationary heat exchange of thin flat isotropic shells, which are under the action of heat sources concentrated in a two-dimensional region”, Visnyk DonNU. Ser. A: Pryrodnychi nauky, 1-2, pp. 107-112.
8. Azarenkov, V.I. (2012), “Research and development of a thermal model and methods for analyzing temperature fields of electronic equipment structures”, Technology audit and production reserves, 3/1, no. 5, pp. 39-40.
   https://doi.org/10.15587/2312-8372.2012.4739
   
9. Havrysh, V.I. and Fedasjuk, D.V. (2012), Modelyuvannya temperaturnykh rezhymiv u kuskovo-odnoridnykh strukturakh [Modelling of temperature regimes in piecewise-homogeneous structures], Vydavnytstvo Lvivskoyi politekhniky, Lviv, Ukraine.
10. Havrysh, V.I., Baranetskiy, Ya.O. and Kolyasa, L.I. (2018), “Investigation of temperature modes in thermosensitive non-uniform elements of radioelectronic devices”, Radio electronics, computer science, management, Vol. 3, no. 46, pp.7-15, available at: 
    https://doi.org/10.15588/1607-3274-2018-3-1
    
11. Havrysh, V.I., Kolyasa, L.I., and Ukhanska, O.M. (2019), “Determination of temperature field in thermally sensitive layered medium with inclusions”, Naukovyi Visnyk NHU, Vol. 1, pp. 94-100, available at: https://doi.org/10.29202/nvngu/2019-1/5
12. Podstrigach, Ia.S., Lomakin, V.A. and Koliano, Iu.M. (1984), Termouprugost tel neodnorodnoi struktury [Thermoelasticity of bodies of inhomogeneous structure], Nauka, Moscow, USSR.
13. Koliano, Iu.M. (1992), Metody teploprovodnosti i termouprugosti neodnorodnogo tela [Methods of thermal conductivity and thermoelasticity of an inhomogeneous body], Naukova dumka, Kyiv, Ukraine, available at: https://doi.org/10.1192/bjp.161.2.280 b.
14. Korn, G. and Korn, T. (1977), Spravochnik po matematike dlia nauchnykh rabotnikov i inzhenerov [A guide to mathematics for scientists and engineers], Nauka, Moscow, USSR.
15. Kikoina, I.K. (1976), Tablitcy fizicheskikh velichin. Spravochnik [Tables of physical quantities. Directory], Atomizdat, Moscow, USSR.

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S.I. Klipkov

Èlektron. model. 2021, 43(6):34-49

https://doi.org/10.15407/emodel.43.06.034

ABSTRACT

Mathematical properties of imaginary exponential functions that can be used to represent the elements of two-dimensional algebraic systems constructed by introducing anticommutativity in the laws of composition of the elements of the basis of two-dimensional canonical numerical systems are considered.

KEYWORDS

exponential functions, imaginary units, complex numbers, double numbers, dual numbers, algebraic systems.

REFERENCES

1. Kantor, I.L. and Solodovnikov, A.S. (1984), Giperkompleksnyye chisla [Hypercomplex numbers], Mir, Moscow, USSR.
2. Rozenfeld, B.A. (1955), Neyevklidovy geometrii [Non-euclidean geometries], GITTL, Moscow, USSR.
3. Kalinovsky, Ya.A. and Boyarinova, Yu.E. (2012), Vysokorazmernyje izomorfnyje giperkompleksnyie chislovyje systemy i ikh ispolzovanije dlya povysheniya effektivnosti vychis­leniy [High-dimensional isomorphic hypercomplex number systems and their use for efficiency increase of calculations], Infodruk, Kyiv, Ukraine.
4. Klipkov, S.I. (2021), “Generalized Analysis of Division Algebras of Dimension 2”, Elekt­ronne modelyuvannya, Vol. 43, no. 4, pp. 5-21.

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Y.N. Gruts, P.O. Holotiuk

Èlektron. model. 2021, 43(6):50-60

https://doi.org/10.15407/emodel.43.06.050

ABSTRACT

Considered the future development of the computer procedure for stereo interpolation, implemented on the basis of two cosine waves with a variable period. The proposed procedure is designed to create and visualize a 3D curve passing through three arbitrary points in the stereo vision space, specified by the researcher using a stereo cursor. The feature of this method is the implemented mode in which the tangent drawn to the curve at the midpoint is always parallel to the line segment connecting the end points. The two cosine waves merge harmoniously around the midpoint, and this gives the researcher the feeling that the synthesized curve stretches for the midpoint as it moves.

KEYWORDS

wireframe models, stereography, stereo operator, stereo indicator, stereointerpolator.

REFERENCES

1. Rogers, L. and Adams, J. (1980), Matematicheskiye osnovy mashinnoy grafiki [Mathematical foundations of machine graphics], Mashinostroyeniye, Moscow, USSR.
2. Gruts, Yu.N. (1989), Stereoskopicheskaya mashinnaya grafika [Stereoscopic computer graphics], Naukova Dumka, Kyiv, USSR, ISBN 5-12-001188-8.
3. Evdokimov, V.F. and Gruts, Yu.N. (2003), “Graphic stereo editor for working with three-dimensional objects of skeletal type”, IPME NAN Ukrayiny. Zbirnyk naukovykh prats, Vol. 20, pp. 105–112.
4. Gruts, Y.N. (1999), “Stereointerpolation Procedure”, Engineering Simulation, Vol. 17, pp. 117-125.
   https://doi.org/10.1061/(ASCE)0733-9445(1999)125:2(117)
5. Yarovoy, R.V. and Gruts, Yu.N. (2010), “Algorithm for increasing the accuracy of the interpolation curve synthesis”, Elektronnoye modelirovaniye, Vol. 32, no. 5, pp. 105-111.
6. Korn, G. and Korn, T. (1968), Spravochnik po matematike [Handbook of mathematics], Nauka, Moscow, USSR.
7. Gruts, Y.N., Jung-Young, Son and Donghoon, Kong (2001), “Stereoscopic operators and their Application”, Journal of Optical Society of Korea, Vol. 5, no. 3, pp. 90-92.
   https://doi.org/10.3807/JOSK.2001.5.3.090
   

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