Vol. 57 No. 2 (261) (2023)

DOI: https://doi.org/10.46991/PYSUA.2023.57.2
Published: 2023-07-12

Mathematics

  • Mathematics

    THE MOORE-PENROSE INVERSE OF TRIDIAGONAL SKEW-SYMMETRIC MATRICES. II

    Yuri R. Hakopian, Avetik H. Manukyan, Hamlet V. Mikaelyan
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    Abstract

    This article is the second part of the work started in the previous publication by the authors [1]. The results presented here relate to deriving closed form expressions for the elements of the Moore-Penrose inverse of tridiagonal real skew-symmetric matrices of odd order. On the base of the formulas obtained, an algorithm that is optimal in terms of the amount of computational efforts is constructed.

    References

    Hakopian Yu.R., Manukyan A.H., Mikaelyan H.V. The Moore-Penrose Inverse of Tridiagonal Skew-Symmetric Matrices. I.

    Proc. of the YSU. Phys. and Math. Sci. 57 (2023), 1-8. https://doi.org/10.46991/PYSU:A/2023.57.1.001

    Ben-Israel A., Greville T.N.E. Generalized Inverses: Theory and Applications. New-York, Springer (2003).

    Hakopian Yu.R., Manukyan A.H. Analytical Inversion of Tridiagonal Hermitian Matrices. Mathematical Problems of Computer Science 58 (2022), 7-19. https://doi.org/10.51408/1963-0088

  • Mathematics

    ON CORRECT SOLVABILITY OF DIRICHLET PROBLEM IN A HALF-SPACE FOR REGULAR EQUATIONS WITH NON-HOMOGENEOUS BOUNDARY CONDITIONS

    Mikayel A. Khachaturyan
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    Abstract

    In this paper we consider the following Dirichlet problem with non-homogeneous boundary conditions in a multianisotropic Sobolev space $W_2^{\mathfrak{M}}(R^2 \times R_+)$ $$\begin{cases} P(D_x, D_{x_3}) u = f(x, x_3), \quad x_3 > 0, \quad x \in R^2, \\ D_{x_3}^s u \big\rvert_{x_3 = 0} = \varphi_s(x),\quad s = 0, \dots, m-1. \end{cases} $$ It is assumed that $P(D_x, D_{x_3})$ is a multianisotopic regular operator of a special form with a characteristic polyhedron $\mathfrak{M}$. We prove unique solvability of the problem in the space $W_2^{\mathfrak{M}}(R^2 \times R_+)$, assuming additionally, that $f(x, x_3)$ belongs to $L_2(R^2 \times R^+)$ and has a compact support, boundary functions $\varphi_s$ belong to special Sobolev spaces of fractional order and have compact supports.

    References

    Karapetyan G.A., Petrosyan H.A. Correct Solvability of the Dirichlet Problem in the Half-space for Regular Hypoelliptic Equations. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 54 (2019), 45-69. https://doi.org/10.3103/S1068362319040022

    Ghazaryan H.G. The Newton Polyhedron, Spaces of Differentiable Functions and General Theory of Differential Equations. Armenian Journal of Mathematics 9 (2017), 102-145.

    Karapetyan G.A. Integral Representations of Functions and Embedding Theorems for Multianisotropic Spaces on the Plane with One Anisotropy Vertex. Journal of Contemporary Mathematical Analysis 51 (2016), 269-281. https://doi.org/10.3103/S1068362316060017

    Karapetyan G.A. Integral Representation of Functions and Embedding Theorems for Multianisotropic Spaces for the Three-dimensional Case. Eurasian Mathematical Journal 7 (2016), 19-39.

    Karapetyan G.A., Arakelyan M.K. Estimation of Multianisotropic Kernels and their Application to the Embedding Theorems. Transactions of A. Razmadze Mathematical Institute 171 (2017), 48-56.

    Karapetyan G.A. Integral Representations of Functions and Embedding Theorems for $n$-dimensional Multianisotropic Spaces with One Anisotropy Vertex. Siberian Mathematical Journal 58 (2017), 445-460. https://doi.org/10.1134/S0037446617030089

    Karapetyan G.A., Petrosyan H.A. Embedding Theorems for Multianisotropic Spaces with Two Vertices of Anisotropicity. Proc. of the YSU. Physical and Mathematical Sci. 51 (2017), 29-37. https://doi.org/10.46991/PYSU:A/2017.51.1.029

    Karapetyan G.A. An Integral Representation and Embedding Theorems in the Plane for Multianisotropic Spaces. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 52 (2017), 267-275. https://doi.org/10.3103/S1068362317060024

    Karapetyan G.A., Arakelyan M.K. Embedding Theorems for General Multianisotropic Spaces. Matematical Notes 104 (2018), 422-438. https://doi.org/10.1134/S0001434618090092

    Khachaturyan M.A., Hakobyan A.R. On Traces of Functions from Multianisotropic Sobolev Spaces. Vestnik RAU. Phys.-Math. Est. Nauki 1 (2021), 56-77 (in Russian).

Mechanics

  • Mechanics

    ON OPTIMAL STABILIZATION OF PART OF VARIABLES OF ROTARY MOVEMENT OF A RIGID BODY WITH ONE FIXED POINT IN THE CASE OF SOPHIA KOVALEVSKAYA

    Smbat G. Shahinyan
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    Abstract

    An optimal stabilization problem for part of variables of rotary movement of a rigid body with one fixed point in the Sophia Kovalevskaya's case is discussed in this work. The differential equations of motion of the system are given and it is shown that the system may rotate around Ox with a constant angular velocity. Taking this motion as unexcited, the differential equations for the corresponding excited motion were drawn up. Then the system was linearized and a control action was introduced along one of the generalized coordinates. The optimal stabilization problem for part of the variables was posed and solved. The graphs of optimal trajectories and optimal control were constructed.

    References

    Buchholz N.N. The Main Course of Theoretical Mechanics. Part 2. Moskow, Nauka, (1972), 332 (in Russian).

    Shahinyan S.G., Avetisyan L.M. On One Problem of Optimal Stabilization of the Rotational Motion of a Rigid Body Having a Fixed Point. Proceedings of the IX International Conference ``Problems of the Dynamics of Interaction of Deformable Media. Armenia, Goris (2018), 306-310.

    Malkin I.G. Theory of Motion of Stability. Moskow, Nauka (1966), 475-514 (in Russian).

    Krasovsky N.N. Some Tasks of the Theory of Motion Control. Moskow, Nauka (1968), 475 (in Russian).

    Rumyantsev V.V., Oziraner A.S. Stability and Stabilization of Movement in Relation to Part of the Variables. Moskow, Nauka (1987), 256 (in Russian).

    Krasovsky N.N. Problems of Stabilization of Controlled Motions. In Book: Theory of Motion of Stability (ed. I.G. Malkin Add. 4). Moskow, Nauka (1966), 475-515 (in Russian).

    Al'brecht E.G., Shelementiev G.S. Lectures on the Theory of Stabilization. Sverdlovsk (1972), 274 (in Russian).

Physics

  • Physics

    CIRCULAR POLARIZATION IN A $2\mathrm{D}$ PERIODIC ARTIFICIAL ANISOTROPIC DIELECTRIC MEDIUM

    Armen N. Sargsyan
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    Abstract

    The phenomenon of polarization rotation in dielectric periodic structures is quite interesting. A similar structure was used in a sample with a 2D artificial periodic structure, where the plane wave propagation method was applied. Anisotropic dielectric constants were obtained, which were used for polarization rotation, and polarization rotation in an anisotropic medium was studied based on the dielectric permittivity of the effective medium. Those relationships, quantities, and parameters depending on the dielectric medium, dielectric permittivity, and 2D periodic structure have been studied and analyzed, which provide control of the degree of anisotropy, i.e. provide a distinction between fast and slow modes, which in turn provide the best polarization rotation per unit length.

    References

    Tuovinen T., Salonen E.T., Berg M. An Artificially Anisotropic Antenna Substrate for the Generation of Circular Polarization. IEEE Trans. Antenna Propag. 64 (2016), 4937-4942. https://doi.org/10.1109/TAP.2016.2602381

    Berg M., Tuovinen T., Salonen E. Artificial Anisotropic Dielectric Material for Antenna Polarization Rotation. In: Electromagnetics Research Symposium - Spring (PIERS). Russia, St Petersburg (2017). https://doi.org/10.1109/PIERS.2017.8262414

    Bardi I., Remski R., et al. Plane Wave Scattering from Frequency Selective Surfaces by the Finite-element Method. IEEE Trans. Magn. 38 (2002), 641-644. https://doi.org/10.1109/20.996167

    Hitoshi Ohsato, Jeong-Seog Kim, et al. Millimeter-Wave Dielectric Properties of Cordierite/Indialite Glass Ceramics.

    Japanese Journal of Applied Physics 50 (2011), 09NF01. https://doi.org/10.1143/JJAP.50.09NF01

    Chen Ding, Kwai-Man Luk A Wideband High-Gain Circularly-Polarized Antenna Using Artificial Anisotropic Polarizer.

    IEEE Trans. Antenna Propag. 67 (2019). https://doi.org/10.1109/TAP.2019.2923739