Vol. 57 No. 1 (260) (2023)

DOI: https://doi.org/10.46991/PYSUA.2023.57.1
Published: 2023-05-01

Mathematics

  • Mathematics

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

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

    The present work is devoted to deriving closed form expressions for the elements of the Moore-Penrose inverse of tridiagonal real skew-symmetric matrices. In the first part of the work we obtain results, concerning matrices of even order. A calculation approach for the generalized inverses of odd order matrices is provided.

    References

    Balonin N.A., Sergeev M.B. Special Matrices: Pseudoinverse, Orthogonal, Hadamard and Cretan. Saint-Petersburg, Polytechnika Publ. (2019), 196 (in Russian). https://doi.org/10.25960/7325-1155-0

    Bunch J.R. A Note on the Stable Decomposition of Skew-symmetric Matrices. Math. Comp. 38 (1982), 475-479. https://doi.org/10.2307/2007283

    Benner P., Byers R., et al. Cholesky-like Factorization of Skew-symmetric Matrices. Electron. Trans. Numer. Anal. 11 (2000), 85-93. https://etna.ricam.oeaw.ac.at/vol.11.2000pp85-93.dir/pp85-93.pdf

    Heinig G., Rost K. Fast Algorithms for Skew-symmetric Toeplitz Matrices. Operator Theory: Advances and Applications 135 (2002), 193-208. https://doi.org/10.1007/978-3-0348-8199-9_12

    Greif C., Varah J.M. Iterative Solution of Skew-symmetric Linear Systems. SIAM J. Matrix Anal. Appl. 31 (2009), 584-601. https://doi.org/10.1137/080732390

    Meyer C. Matrix Analysis and Applied Linear Algebra. Philadelphia, SIAM (2000), 705.

    Strang G. Linear Algebra and Its Applications. Academic Press (1976).

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

  • Mathematics

    LINEARITY OF $n$-ARY ASSOCIATIVE ALGEBRAS

    David N. Harutyunyan
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    Abstract

    In this paper n-ary regular division associative algebras are discussed.  It is shown that every operation in n-ary regular division associative algebra will be endo-linearly represented over the same binary group. Schauffler like theorem will be proved for those algebras.

    References

    Davidov S., Krapez A., Movsisyan Yu. Functional Equations with Division and Regular Operations. Asian-Eur. J. Math. 11 (2018), 1850033. https://doi.org/10.1142/S179355711850033X

    Schauffler R. Eine Anwendung Zyklischer Permutationen and Ihretheorie. Ph.D. Thesis. Marburg University (1948). https://doi.org/10.1142/12796

    Schauffler R. Über die Bildung von Codewörtern. Arch. Elekt. Übertragung 10 (1956), 303-314.

    Schauffler R. Die Associativität im Ganzen. Besonders bei Quasigruppen 67 (1957), 428-435.

    Movsisyan Yu. Hyperidentities: Boolean and De Morgan Structures. World Scientific (2022), 560. https://doi.org/10.1142/12796

    Movsisyan Yu. Introduction to the Theory of Algebras with Hyperidentities. Yerevan, YSU Press (1986) (in Russian).

    Movsisyan Yu. Hyperidentities and Hypervarieties in Algebras. Yerevan, YSU Press (1990) (in Russian).

    Movsisyan Yu. On a Theorem of Schauffler. Math. Notes 53 (1993), 172-179. https://doi.org/10.1007/BF01208322

    Movsisyan Yu. Hyperidentities in Algebras and Varieties. Russ. Math. Surv. 53 (1998), 57-108. https://doi.org/10.1070/RM1998v053n01ABEH000009

    Ushan Ya. Globally Associative Systems of $n$-ary Quasigroups (Constructions of $iA$-systems. A generalization of the Hossu-Gluskin Theorem). Publ. Inst. Math. 19 (1975), 155-165 (in Russian).

    Ushan Ya., Zhizhovich M. $n$-Ary Analog of Schauffler's Theorem. Publ. Inst. Math. 19 (1975), 167-172 (in Russian).

Physics

  • Physics

    STABILITY AND CONDUCTIVITY OF BILAYER LIPID MEMBRANE IN PRESENCE $\mathrm{Al_2O_3}$ NANOPARTICLES

    Tsovinar M. Jomardyan, Gayane V. Ananyan, Valery B. Arakelyan
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    Abstract

    The effect of aluminum oxide nanoparticles (Al2O3) on the stability and conductivity of BLM (the bilayer lipid membrane) in solution was studied.  It has been shown that Al2O3 nanoparticles increase the stability of BLM in an electric field, and BLM becomes more stable with increasing their concentration. The experimental data are analyzed in terms of the well-known theory of BLM stability, which is based on the concept of spontaneous formation of microscopic pores on the BLM, the development of which leads to the loss of BLM stability. It is shown that aluminum oxide nanoparticles increase the value of the coefficient of linear tension of the pore edge, and with an increase in the concentration of nanoparticles, the linear tension also increases. It has been shown that the presence of nanoparticles in the solution leads to a decrease in BLM conductivity.

    References

    Syrma E.I. Physical Properties of Nanoparticles and Their Biological Effects. Integrative Anthropology 1 (2013), 30-33 (in Russian).

    Andreev G.B., Minashkin V.M., et al. Materials Produced by Nanotechnologies: Potential Risk in Obtaining and Using. Ros. Chem. and (J. Russian Chemical Society Named After D.I. Mendeleev) LII (2008), 32-38 (in Russian).

    Bondarenko O., Juganson K., Ivask A. Toxicity of Ag, CuO and ZnO Nanoparticles to Selected Environmentally Relevant Test Organisms and Mammalian Cells in vitro: A Critical Review. Archives of Toxicology 87 (2013), 1181-1200. https://doi.org/10.1007/s00204-013-1079-4

    Böhme S., Stärk H., et al. Quantification of Al2O3 Nanoparticles in Human Cell Lines Applying Inductively Coupled Plasma Mass Spectrometry (neb-ICP-MS, LA-ICP-MS) and Flow Cytometry-based Methods. J. Nanopart. Res. 16 (2014). https://doi.org/10.1007/s11051-014-2592-y

    Sarapultsev A.P., Rempel C.V., et al. Interaction of Nanoparticles with Biological Objects. Bulletin of the Ural Medical Academic Science 3 (2016), 97-110 (in Russian).

    Ivkov V.G., Berestovsky G.N. Lipid Bilayer of Biological Membranes. Moscow, Nauka (1982), 224 (in Russian).

    Zaitseva N.V., Zemlyanova M.A., et al. Evaluation of Toxicity and Potential Hazard of Aluminum Oxide Nanoparticles for Human Health. Human Ecology 5 (2018), 9-15 (in Russian).

    Mueller P., Rudin D.O., et al. Methods for the Formation of Single Bimolecular Lipid Membranes in Aqueous Solution. J. Phys. Chem. 67 (1963), 534-535.

    Abidor I.G., Arakelyan V.B., et al. Electric Breakdown of Bilayer Lipid Membranes I. The Main Experimental Facts and Their Qualitative Discussion. Bioelectrochemistry and Bioenergetics 6 (1979), 37-52.

    Melikyan G.B., Matinyan N.S., Arakelian V.B. The Influence of Gangliosides on the Hydrophilic Pore Edge Line Tension and Monolayer Fusion of Lipid Membranes. Biochim. Biophys. Acta 1030 (1990), 11-15.

    Torosyan A.L., Arakelyan V.B. Influence of H2TOEtPyP4 Porphyrin on the Stability and Conductivity of Bilayer Lipid Membranes. Eur. Biophys. J. 44} (2015), 745-750. https://doi.org/10.1007/s00249-015-1074-1

    Ivkov V.G., Berestovsky G.N. Dynamic Structure of the Lipid Bilayer. Moscow, Nauka (1981) (in Russian).

    Chernomordik L.V., Kozlov M.M., et al. Shape of Lipid Molecules and Monolayer Membrane Fusion. Biological Membranes 1 (1984), 411-427 (in Russian).

    Sokolov A.V. Study of the Effects of Products of the Visual Cycle on Bilayer Lipid Membranes. Ph.D. Thesis. Moscow (2009), 23 (in Russian).

    Chizmadzhev Yu.A., Chernomordik L.V., et al. Electrical Breakdown of Bilayer Lipid Membranes. In: Results of Science and Technology. Biophysics of Membranes 2 (1982), 161-266 (in Russian).

    Markin V.S., Kozlov M.M. Pore Statistics in Bilayer Lipid Membranes. Biological Membranes 2 (1985), 205-223 (in Russian).

    Ghosh S., Mashayekhi H., et al. Colloidal Behavior of Aluminum Oxide Nanoparticles as Affected by pH and Natural Organic Matter. Langmuir 24 (2008), 12385-12391. https://doi.org/10.1021/la802015f

    Mui J., Ngo J., Kim B. Aggregation and Colloidal Stability of Commercially Available Al2O3 Nanoparticles in Aqueous Environments. Nanomaterials 6 (2016), 90-104. https://doi.org/10.3390/nano6050090

    Pedersen U., Leidy C., et al. The Effect of Calcium on the Properties of Charged Phospholipid Bilayers. Biochimica et Biophysica Acta (BBA). Biomembranes 1758 (2006), 573-582. https://doi.org/10.1016/j.bbamem.2006.03.035

    Shevchenko E.V., Antonov V.F. Influence of Divalent Ions on the Physical Properties of Bilayer Lipid Membranes from Zviterrionic and Acidic Phospholipids. Siberian Medical Journal (Irkutsk) 3 (1995), 5-8 (in Russian).