Vol. 56 No. 2 (258) (2022)

DOI: https://doi.org/10.46991/PYSUA.2022.56.2
Published: 2022-07-25

Mathematics

  • Mathematics

    POWERS OF SUBSETS IN FREE PERIODIC GROUPS

    Varouzhan S. Atabekyan, Hayk T. Aslanyan, Satenik T. Aslanyan
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    Abstract

    It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

    References

    Chang M.-Ch. Product Theorems in SL2 and SL3. J. Inst. Math. Jussieu, 7 (2008), 1--25. https://doi.org/10.1017/S1474748007000126

    Safin S.R. Powers of Subsets of Free Groups. Mat. Sb., 202 (2011), 97--102. https://doi.org/10.4213/sm7811

    Razborov A.A. A Product Theorem in Free Groups. Ann. of Math., 179 (2014), 405--429. https://doi.org/10.4007/annals.2014.179.2.1

    Terence T., Van V. Additive Combinatorics. V. 105. In: Cambridge Studies in Advanced Mathematics. Cambridge, Cambridge University Press (2006).

    Adian S.I. The Burnside Problem and Identities in Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. V. 95. Berlin, Springer--Verlag (1979).

    Adian S.I., Lysenok I.G. On Groups all of whose Proper Subgroups are Finite Cyclic. Izv. Akad. Nauk SSSR. Ser. Mat., 55 (1991), 933--990 (in Russian); Izv. Math., 39 (1992), 905--957 (in English). https://doi.org/10.1070/IM1992v039n02ABEH002232

    Atabekian V.S. On Subgroups of Free Burnside Groups of Odd Period n ≥ 1003. Izv. Ross. Akad. Nauk Ser. Mat., 73 (2009), 3--36 (in Russian); Izv. Math., 73 (2009), 861--892 (in English). https://doi.org/10.4213/im2633

    Atabekian V.S. Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period. Mat. Zametki, 85 (2009), 516--523 (in Russian); Math. Notes, 85 (2009), 496--502 (in English). https://doi.org/10.1134/S0001434609030213

    Atabekian V.S. Monomorphisms of Free Burnside Groups. Math. Notes, 86 (2009), 457--462. https://doi.org/10.1134/S0001434609090211

  • Mathematics

    ON AUTOMORPHISM GROUPS OF ENDOMORPHISM SEMIGROUPS OF FINITE ELEMENTARY ABELIAN GROUPS

    Arman A. Bayramyan
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    Abstract
    In this article, we explore the automorphisms of endomorphism semigroups and automorphism groups
    of the finite elementary Abelian groups. In particular, we prove that
    $\mathrm{Aut}(\mathrm{End}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p))$ can be canonically embedded into
    $\mathrm{Aut}(\mathrm{Aut}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p))$ using an elementary approach based
    on matrix operations. We also show that all automorphisms of $\mathrm{End}(\Z_p\oplus\Z_p\oplus\cdots\oplus\Z_p)$
    are inner.

    References

    Plotkin B. Seven Lectures on the Universal Algebraic Geometry. Preprint (2002). https://doi.org/10.48550/arXiv.math/0204245

    Formanek E. A Question of B. Plotkin about the Semigroup of Endomorphisms of a Free Group. Proc. Amer. Math. Soc., 130 (2002), 935--937. https://doi.org/10.2307/2699537

    Atabekyan V.S. The Automorphisms of Endomorphism Semigroups of Free Burnside Groups. Int. J. Algebra Comput., 25 (2015), 669--674. https://doi.org/10.1142/S0218196715500149

    Atabekyan V.S., Aslanyan H.T. The Automorphisms of Endomorphism Semigroups of Relatively Free Groups. Int. J. Algebra Comput., 28 (2018), 207--215. https://doi.org/10.1142/S0218196718500108

    Gluskin L.M. Automorphisms of Multiplicative Semigroups of Matrix Algebras. Uspehi Mat. Nauk (N.S.), 11 (1956), 199--206 (in Russian).

    Halezov E.A. Automorphisms of Matrix Subgroups. Dokl. Akad. Nauk SSSR, 96 (1954), 245--248 (in Russian).

    Waterhouse W.C. Two Generators for the General Linear Groups over Finite Fields. Linear Multilinear Algebra, 24 (1988), 227--230. https://doi.org/10.1080/03081088908817916

    Dieudonne J. On the Automorphisms of the Classical Groups. Mem. Amer. Math. Soc., 2 (1951). https://doi.org/10.1090/S0002-9939-1951-0040426-0

    Waterhouse W.C. Automorphisms of $GL_n(R)$. Proc. Am. Math. Soc., 79 (1980), 347--351. https://doi.org/10.2307/2043063

  • Mathematics

    PROOF COMPLEXITIES ON A CLASS OF BALANCED FORMULAS IN SOME PROPOSITIONAL SYSTEMS

    Anahit A. Chubaryan
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    Abstract

    In this paper four proof complexity characteristics for some class of balanced tautologies are investigated in two proof systems of propositional logic. One of the considered systems is based on determinative disjunctive normal form, the other on the generalization of splitting method. The optimal upper and lower bounds by logarithmic scale for all main proof complexity characteristics of considered tautologies are obtained in both systems.

    References

    Cook S.A., Reckhow A.R. The Relative Efficiency of Propositional Proof Systems. Journal of Symbolic Logic, 44 (1979), 36--50. https://doi.org/10.2307/2273702

    Strasburger L. Extension without Cut. Annals of Pure and Applied Logic, 163 (2012), 1995--2007.

    Chubaryan A. Relative Efficiency of Some Proof Systems for Classical Propositional Logic . Proceedings of NASA RA, 37 (2002); Journal of CMA (AAS), 37 (2002), 71--84.

    Chubaryan An., Chubaryan Arm. Bounds of Some Proof Complexity Characteristics in the System of Splitting Generalization. Otechestv. Nauka v Epokhu Izmeneniy, 10 (2015), 11--14 (in Russian).

    Chubaryan A., Hovhannisyan S., Gasparyan G. About Some Properties of a Propositional System of Generalized Splittings.

    Vestnik RAU, 2 (2019), 34--42 (in Russian).

    Chubaryan A., Hovhannisyan S., Gasparyan G. Comparison of Two Propositional Proof Systems by Lines and by Sizes, ASL, ESM. Logic Colloquium-2021. Book of Abstracts. Poznan (2021), 166 p.

    Filmus Y., Lauria M., Nordstrom J., Thapen N., Ron-Zewi N. Space Complexity in Polynomial Calculus. 2012 IEEE Conference on Computational Complexity (CCC) (2012), 334--344.

    Chubaryan A., Mnatsakanyan A. On the Bounds of the Main Proof Measures in Some Propositional Proof Systems. Scholar Journal of Phis. Math. and Stat., 1 (2014), 111--117.

Physics

  • Physics

    IMAGING OF MICROWAVE NEAR-FIELD DISTRIBUTION OF GPS PATCH ANTENNA

    Zhirayr A. Baghdasaryan
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    Abstract

    Microwave near-field distribution of the GPS patch antenna was visualized by a thermo-elastic optical indicator microscopy (TEOIM) technique at $1.575~GHz$. Visualization of the antenna radiation is realized to describe the electromagnetic field intensity and distribution depending on the distance from the antenna surface and optical indicator. Experimental data was compared and confirmed with simulation results, which are in good agreement. Possible applications of the TEOIM system were discussed.

    References

    Fan W., Kyosti P., Rumney M., et al. Over-the-air Radiated Testing of Millimeter-wave Beam-steerable Devices in a Cost-effective Measurement Setup. IEEE Commun. Mag., 56 (2018), 64--71. https://doi.org/10.1109/MCOM.2018.1701006

    Xue W., Chen X., Zhang M., et al. Statistical Analysis of Antenna Efficiency Measurements with Non-Reference Antenna Methods in a Reverberation Chamber. IEEE Access, 8 (2020), 113967--113980. https://doi.org/10.1109/ACCESS.2020.3003530

    Pozar D.M., Kaufman B. Comparison of Three Methods for the Measurement of Printed Antenna Efficiency. IEEE Trans. Antennas Propag., 36 (1988), 136--139. https://doi.org/10.1109/8.1084

    Luo Q., Zhou Y., Qi Y., et al. Rapid Test Method for Multi-beam Profile of Phased Array Antennas. Sensors, 22 (2022). https://doi.org/10.3390/s22010047}

    Coq Le L., Mézières N., Leroy P., Fuchs B. Some Contributions for Antenna 3D far Field Characterization at Terahertz. Sensors, 21 (2021), 1--11. https://doi.org/10.3390/s21041438

    Rodríguez Varela F., López Morales M.J., Tena Sánchez R., et al. Multi-Probe Measurement System Based on Single-Cut Transformation for Fast Testing of Linear Arrays. Sensors, 21 (2021), 1744. https://doi.org/10.3390/s21051744

    Capozzoli A., Curcio C., Liseno A. Different Metrics for Singular Value Optimization in Near-field Antenna Characterization. Sensors, 21 (2021), 1--15. https://doi.org/10.3390/s21062122

    D'agostino F., Ferrara F., Gennarelli C., et al. Reconstruction of the Far-field Pattern of Volumetric Auts from a Reduced Set of Near-field Samples Collected along a Planar Spiral with a Uniform Step. Sensors, 21 (2021), 1--13. https://doi.org/10.3390/s21051644

    Arakelyan S., Lee H., Babajanyan A., et al. Antenna Investigation by a Thermoelastic Optical Indicator Microscope: Defects Measurement and 3D Visualization of Electromagnetic Fields. IEEE Antennas Propag. Mag., 61 (2019), 27--31. https://doi.org/10.1109/MAP.2019.2895667

    Doust E.G., Clénet M., Hemmati V., Wight J. An Aperture-coupled Circularly Polarized Stacked Microstrip Antenna for GPS Frequency Bands L1, L2, and L5. IEEE Int. Symp. Antennas Propag. Usn. Natl. Radio Sci. Meet. APSURSI, 1 (2008), 25--28. https://doi.org/10.1109/APS.2008.4619440

    Zhang F., Qiao N., Li J. A PCB Photoelectric Image Edge Information Detection Method. Optik (Stuttg), 144 (2017), 642--646. https://doi.org/10.1016/j.ijleo.2017.07.002

    Kacprzak D., Taniguchi T., Nakamura K., et al. Novel Eddy Current Testing Sensor for the Inspection of Printed Circuit Boards. IEEE Trans. Magn. 37 (2001), 2010--2012. https://doi.org/10.1109/20.951037

    Lee H., Arakelyan S., Friedman B., Lee K. Temperature and Microwave Near Field Imaging by Thermo-elastic Optical Indicator Microscopy. Sci. Rep., 6 (2016), 1--11. https://doi.org/10.1038/srep39696

    Yoshikawa N. Fundamentals and Applications of Microwave Heating of Metals. J. Microw. Power Electromagn. Energy, 44 (2010), 4--13. https://doi.org/10.1080/08327823.2010.11689772

    Bosman H., Lau Y.Y., Gilgenbach R.M. Microwave Absorption on a Thin Film. Appl. Phys. Lett., 82 (2003), 1353--1355. https://doi.org/10.1063/1.1556969

    Lee H., Baghdasaryan Z., Friedman B., Lee K. Detection of a Conductive Object Embedded in an Optically Opaque Dielectric Medium by the Thermo-Elastic Optical Indicator Microscopy. IEEE Access, 7 (2019), 46084--46091. https://doi.org/10.1109/ACCESS.2019.2908885

  • Physics

    MINIATURIZED ANTIPODAL VIVALDI ANTENNA BASED ON MAGNETODIELECTRIC MATERIALS

    Ararat H. Stepanyan, Hovhannes S. Haroyan
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    Abstract

    Miniaturization of antipodal Vivaldi antenna (AVA) based on high dielectric constant and magnetodielectric ferrite materials is studied. The basic parameters of antipodal Vivaldi antenna based on high dielectric and magnetodielectric materials (MDM) are compared. To miniaturize the size of the antenna by a higher factor, MDM ferrite with paremeters $\varepsilon_r=\mu_r=8$ is used. The analysis shows that MDM based Vivaldi antenna surface is minimized by a factor of 4 compared to that of air-based Vivaldi antenna. The sizes of AVA are $120 \times 120 \times 1~(mm).$

    References

    Dixit A.S., Kumar S. A Survey of Performance Enhancement Techniques of Antipodal Vivaldi Antenna. IEEE Access, (2020), 45774--45796. url{https://doi.org/10.1109/ACCESS.2020.2977167}

    Fisher J. Design and Performance Analysis of a 1-40 GHz Ultra-Wideband Antipodal Vivaldi Antenna. German Radar Symposium GRS 2000 (2010), 1--5.

    Stuart G.H.R., Pilwerbetsky A. Electrically Small Antenna Elements Using Negative Permittivity Resonator. IEEE Trans. Antennas Propag., 54 (2006), 1644--1653. url{https://doi.org/10.1109/APS.2005.1551411}

    Hien Chu Ba, Hiroshi Shirai, Chien Dao Ngoc Analysis and Design of Antipodal Vivaldi Antenna for UWB Applications. 2014 IEEE Fifth International Conference on Communications and Electronics (ICCE) (2014), 391--394. url{https://doi.org/10.1109/CCE.2014.6916735}

    Andreou E., Zervos T., Alexandridis A.A., Fikioris G. Magnetodielectric Materials in Antenna Design: Exploring the Potentials for Reconfigurability. IEEE Antennas and Propagation Magazine, 61 (2019), 29--40. url{https://doi.org/10.1109/MAP.2018.2883029}

    Ikonen P., Rozanov K., Osipov A., Alitalo P., Tretyakov S. Magnetodielectric Substrates in Antenna Miniaturization: Potential and Limitation. IEEE Trans. Antennas Propogat., 54 (2006), 3391--3398. url{https://doi.org/10.1109/TAP.2006.884303}

    Hansen R.C., Burke M. Antennas with Magneto-dielectrics. Microwave and Optical Tech. Lett., 26 (2000), 75--78. url{https://doi.org/10.1002/1098-2760(20000720)26:2%3C75::AID-MOP3%3E3.0.CO;2-W}

    Zongliang Zheng, Xu Wu A Miniaturized UHF Vivaldi Antenna With Tailored Radiation Performance Based on Magneto-Dielectric Ferrite Materials. IEEE Transactions on Magnetics, 56 (2020). url{https://doi.org/10.1109/TMAG.2019.2962030}

    Amin M. Abbosh Directive Antenna for Ultrawideband Medical Imaging Systems. International Journal of Antennas and Propagations, (2008), 1--6. url{https://doi.org/10.1155/2008/854012}