Vol. 58 No. 2 (264) (2024)

DOI: https://doi.org/10.46991/PYSUA.2024.58.2
Published: 2024-10-30

Mathematics

  • Mathematics

    PROBABILISTIC IDENTITIES IN BURNSIDE GROUPS OF EXPONENT 3

    Arman R. Fahradyan
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    Abstract

    Burnside groups B(m,n) are relatively free groups that are factor groups of the absolutely free group Fm of rank m by its subgroup, generated by n-th degrees of all the elements. They are the largest groups of fixed rank that have the exponent equal to n. In this work we compute the commuting probability for free Burnside groups B(m,3) of exponent 3 and rank m ≥ 1.

    References

    Amir G., Blachar G., et al. Probabilistic Laws on Infinite Groups (2023). https://doi.org/10.48550/arXiv.2304.09144

    Atabekyan V.S., Bayramyan A.A. Probabilistic Identities in n-Torsion Groups. Journal of Contemporary Mathematical Analysis 59 (2024).

    Gustafson W.H. What is the Probability That Two Group Elements Commute? The Amer. Math. Monthly 80 (1973), 1031-1034. https://doi.org/10.2307/2318778

    Antolin Y., Martino A., Ventura Capell E. Degree of Commutativity of Infinite Groups. Proc. Amer. Math. Soc. 145 (2017), 479-485. http://doi.org/10.1090/proc/13231

    Hall M. The Group Theory. ISBN 978-0821819678.

    Atabekyan V.S., Aslanyan H.T., et al. Analogues of Nielsen's and Magnus's Theorems for Free Burnside Groups of Period 3. Proc. of the YSU. Phys. and Math. Sci. 51 (2017), 217-223. https://doi.org/10.46991/PYSU:A/2017.51.3.217

  • Mathematics

    VERTEX DISTINGUISHING PROPER EDGE COLORINGS OF THE CORONA PRODUCTS OF GRAPHS

    Tigran K. Petrosyan
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    Abstract

    A proper edge coloring of a graph $G$ is a mapping $f:E(G)\longrightarrow \mathbb{Z}_{\geq 0}$ such that $f(e)\not=f(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. A proper edge coloring $f$ of a graph $G$ is called vertex distinguishing if for any different vertices $u,v \in V(G)$, $S(u,f) \ne S(v,f)$, where $S(v,f) = \{f(e) \ | \ e = uv \in E(G)\}$. The minimum number of colors required for a vertex distinguishing proper coloring of a graph $G$ is denoted by $\chi'_{vd}(G)$ and called vertex distinguishing chromatic index of $G$. In this paper we provide lower and upper bounds on the vertex distinguishing chromatic index of the corona products of graphs.

    References

    West D.B. Introduction to Graph Theory. New Jersey, Prentice-Hall (2001). https://dwest.web.illinois.edu/igt/

    Burris A.C., Schelp R.H. Vertex-Distinguishing Proper Edge-colorings. J. Graph Theory 26 (1997), 73-82.

    Cerny J., Hornak M., Sotak R. Observability of a Graph. Math. Slovaca 46 (1996), 21-31. https://eudml.org/doc/34424>

    Burris A.C. Vertex-Distinguishing Edge-Colorings. Ph.D. Thesis. Tennessee, Memphis, Memphis State University (1993). https://www.memphis.edu/msci/people/pbalistr/vdecg.pdf

    Hornak M., Sotak R. Observability of Complete Multipartite Graphs with Equipotent Parts. Ars Comb. 41 (1995), 289-301. https://dblp.org/rec/journals/arscom/HornakS95.html

    Balister P.N., Bollobas B., Schelp R.H. Vertex Distinguishing Coloring of Graphs with Δ(G)=2. Discrete Math. 252 (2002), 17-29. https://doi.org/10.1016/S0012-365X(01)00287-4

    Frucht R.W., Harary F. On the Corona of Two Graphs. Aequ. Math. 4 (1970), 322-325. https://eudml.org/doc/136091

    Baril J.-L., Kheddouci H., Togni O. Vertex Distinguishing Edge- and Total-Colorings of Cartesian and Other Product Graphs. Ars Comb. 107 (2012), 109-127. http://jl.baril.u-bourgogne.fr/AV1.pdf

  • Mathematics

    SOME BOUNDS ON THE NUMBER OF COLORS IN INTERVAL EDGE-COLORINGS OF GRAPHS

    Petros A. Petrosyan, Levon N. Muradyan
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    Abstract

    An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval \lb $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A vertex $v$ of a graph $G=(V,E)$ is called a dominating vertex if $d_{G}(v)=|V|-1$, where $d_{G}(v)$ is the degree of $v$ in $G$. In this paper we prove, that if $G$ is a graph with the dominating vertex $u$ and it has an interval $t$-coloring, then $t\leq |V|+2\Delta(G-u)-1$, where $\Delta(G)$ is the maximum degree of $G$. We also show, that if a $k$-connected graph $G=(V,E)$ admits an interval $t$-coloring, then $t\leq 1+\left(\left\lfloor \dfrac{|V|-2}{k}\right\rfloor+2\right)(\Delta(G)-1)$. Moreover, if $G$ is also bipartite, then this upper bound can be improved to $t\leq 1+\left(\left\lfloor \dfrac{|V|-2}{k}\right\rfloor+1\right)(\Delta(G)-1)$. Finally, we discuss the sharpness of the obtained upper bounds on the number of colors in interval edge-colorings of these graphs.

    References

    West D.B. Introduction to Graph Theory. New Jersey, Prentice-Hall (2001). https://dwest.web.illinois.edu/igt/

    Asratian A.S., Kamalian R.R. Interval Colorings of Edges of a Multigraph. Appl. Math. 5 (1987), 25-34 (in Russian). https://arxiv.org/abs/1401.8079

    Asratian A.S., Kamalian R.R. Investigation on Interval Edge-Colorings of Graphs. J. Combin. Theory Ser. B 62 (1994), 34-43.

    https://doi.org/10.1006/jctb.1994.1053

    Kamalian R.R. Interval Colorings of Complete Bipartite Graphs and Trees. Yerevan, Comp. Cen. of Acad. Sci. of Armenian SSR (1989) (in Russian). https://arxiv.org/pdf/1308.2541

    Petrosyan P.A. Interval Edge-Colorings of Complete Graphs and n-Dimensional Cubes. Discrete Math. 310 (2010), 1580-1587. https://doi.org/10.1016/j.disc.2010.02.001

    Petrosyan P.A., Khachatrian H.H., Tananyan H.G. Interval Edge-Colorings of Cartesian Products of Graphs I. Discuss. Math. Graph Theory 33 (2013), 613-632. https://doi.org/10.7151/dmgt.1693

    Giaro K. The Complexity of Consecutive Δ-Coloring of Bipartite Graphs: 4 is Easy, 5 is Hard. Ars Combinatoria 47 (1997), 287-298. https://combinatorialpress.com/ars/vol47/

    Sevastjanov S.V. Interval Colorability of the Edges of a Bipartite Graph. Metodi Diskret. Analiza 50 (1990), 61-72 (in Russian).

    Petrosyan P.A., Khachatrian H.H. Interval Non-Edge-Colorable Bipartite Graphs and Multigraphs. J. Graph Theory 76 (2014), 200-216. https://doi.org/10.1002/jgt.21759

    Giaro K., Kubale M., Malafiejski M. Consecutive Colorings of the Edges of General Graphs. Discrete Math. 236 (2001), 131-143. https://doi.org/10.1016/S0012-365X(00)00437-4

    Kamalian R.R. Interval Edge Colorings of Graphs. Doctoral Thesis. Novosibirsk (1990).

    Axenovich M.A. On Interval Colorings of Planar Graphs. Congr. Numer. 159 (2002), 77-94. https://www.math.kit.edu/iag6/~axenovich/media/interval_coloring.pdf

    Axenovich M, Girão A., et al. A Note on Interval Colourings of Graphs. European Journal of Combinatorics 120 (2024). Article Number 103956. https://doi.org/10.1016/j.ejc.2024.103956

    Hambardzumyan A., Muradyan L. Upper Bounds on the Number of Colors in Interval Edge-Colorings of Graphs.

    Discrete Math. 348 (2025). Article Number 114229. https://doi.org/10.1016/j.disc.2024.114229

    Kamalian R.R., Petrosyan P.A. A Note on Upper Bounds for the Maximum Span in Interval Edge-Colorings of Graphs. Discrete Math. 312 (2012), 1393-1399. https://doi.org/10.1016/j.disc.2012.01.005

    Casselgren C.J., Khachatrian H.H., Petrosyan P.A. Some Bounds on the Number of Colors in Interval and Cyclic Interval Edge Colorings of Graphs. Discrete Math. 341 (2018), 627-637. https://doi.org/10.1016/j.disc.2017.11.001

    Muradyan L. On Interval Edge-Colorings of Complete Multipartite Graphs. Proc. of the YSU. Phys. and Math. Sci. 56 (2022), 19-26. https://doi.org/10.46991/PYSU:A/2022.56.1.019

Physics

  • Physics

    MICROWAVE RESONANCE IN A SYSTEM OF INTERACTING CONDUCTING RINGS AND ITS APPLICATIONS

    Narek G. Margaryan
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    Abstract

    The interaction between standing Sommerfeld microwaves within a system comprising two closely spaced conducting rings gives rise to pronounced resonance phenomena. The behavior of this system depends on the relative arrangement of the receiving and transmitting points. Specifically, it leads to a sharp reduction or enhancement of signal output within a narrow frequency range. Remarkably, this structure can serve dual roles: acting as both a band-stop filter and a band-pass filter, all within the same restricted frequency band.

    References

    Molnar D., Schaich T., et al. Interaction between Surface Waves on Wire Lines. Proc. R. Soc. A 477 (2021). Article Number 20200795. https://doi.org/10.1098/rspa.2020.0795

    Shen X., Jun Cui T. Planar Plasmonic Metamaterial on a Thin Film with Nearly Zero Thickness. Appl. Phys. Lett. 102 (2013). Article Number 211909. https://doi.org/10.1063/1.4808350

    Tang W.X., Zhang H.C., et al. Concept, Theory, Design, and Applications of Spoof Surface Plasmon Polaritons at Microwave Frequencies. Adv. Opt. Mater. 7 (2019). Article Number 1800421. https://doi.org/10.1002/adom.201800421

    Chen L., Gao F., et al. Editorial: Recent Progress in Surface Electromagnetic Modes. Front. Phys. 9 (2021). https://doi.org/10.3389/fphy.2021.684584

    Goubau G. Single-Conductor Surface-Wave Transmission Lines. Proc. IRE 39 (1951), 619-624. https://doi.org/10.1109/JRPROC.1951.233782

    Vaughn B.J., Peroulis D., Fisher A. Mid-Range Wireless Power Transfer Based on Goubau Lines. Microwave Symposium. IEEE/MTT-S International (IMS, IEEE) (2018), 968-971. https://doi.org/10.1109/MWSYM.2018.8439372

    Li J., Zhang Q., et al. Pulse Transmission Performance of Goubau Lines and Spoof Surface Plasmon Polaritons Transmission Lines. 2020 IEEE Asia-Pacific Microwave Conference (APMC 2020) (2020), 795-797. https://doi.org/10.1109/APMC47863.2020.9331408

    Schaich T., Dinc E., et al. Advanced Modeling of Surface Waves on Twisted Pair Cables: Surface Wave Stopbands. IEEE Trans. Microw. Theory Tech. 70 (2022), 2541-2552. https://doi.org/10.1109/TMTT.2022.3160708

    Vaughn B., Peroulis D. An Updated Applied Formulation for the Goubau Transmission Line. J. Appl. Phys. 126 (2019). Article Number 194902. https://doi.org/10.1063/1.5125141

    Smirnov Y., Smolkin E., Shestopalov Y. Surface Waves in a Goubau Line Filled with Nonlinear Anisotropic Inhomogeneous Medium. Appl. Anal. 101 (2022), 6172-6190. https://doi.org/10.1080/00036811.2021.1919645

    Ge S., Zhang Q., et al. Analysis of Asymmetrically Corrugated Goubau-Line Antenna for Endfire Radiation. IEEE Trans. Antennas Propag. 67 (2019), 7133-7138. https://doi.org/10.1109/TAP.2019.2927633

    Laurette S., Treizebre A., Bocquet B. Corrugated Goubau Lines to Slow Down and Confine THz Waves. IEEE Trans. Terahertz Sci. Technol. 2 (2012), 340-344. https://doi.org/10.1109/TTHZ.2012.2189207

    Akalin T., Treizebre A., Bocquet B. Single-Wire Transmission Lines at Terahertz Frequencies. IEEE Trans. Microw. Theory Tech. 54 (2006), 2762-2767. https://doi.org/10.1109/TMTT.2006.874890

    Wagih M. Broadband Low-Loss On-Body UHF to Millimeter-Wave Surface Wave Links Using Flexible Textile Single Wire Transmission Lines. IEEE Open J. Antennas Propag. 3 (2022), 101-111. https://doi.org/10.1109/OJAP.2021.3136654

    Chen W.-C., Mock J.J., et al. Controlling Gigahertz and Terahertz Surface Electromagnetic Waves with Metamaterial Resonators. Phys. Rev. X. 1 (2011). Article Number 021016. https://doi.org/10.1103/PhysRevX.1.021016

    Horestani A.K., Withayachumnankul W., et al. Metamaterial-Inspired Bandpass Filters for Terahertz Surface Waves on Goubau Lines. IEEE Trans. Terahertz Sci. Technol. 3 (2013), 851-858. https://doi.org/10.1109/TTHZ.2013.2285556

    Maier S.A., Andrews S.R., et al. Terahertz Surface Plasmon-Polariton Propagation and Focusing on Periodically Corrugated Metal Wires. Phys. Rev. Lett. 97 (2006). Article Number 176805. https://doi.org/10.1103/PhysRevLett.97.176805

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    Tang X.-L., Zhang Q., et al. Continuous Beam Steering through Broadside Using Asymmetrically Modulated Goubau Line Leaky-Wave Antennas. Sci. Rep. 7 (2017). Article Number 11685. https://doi.org/10.1038/s41598-017-12118-8

    Liu L.W., Kandwal A., et al. Non-Invasive Blood Glucose Monitoring Using a Curved Goubau Line. Electronics 8 (2019), 662. https://doi.org/10.3390/electronics8060662

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