Skip to main content
SpringerLink
Log in

The Hexagonal Geometrical Structure of the N-Coverage Networks in the G2-Lie Algebra Framework

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

In this paper, we investigate an interplay between the hexagonal network models and the root system of a particular class of rank two Lie algebras, called \(G_2\). In this work, we show that the hexagonal cells explored in telecommunication systems are associated with the nonzero roots of \(G_2\)-generalized Lie algebras. More precisely, the \(G_2\) hexagons are analyzed in some details and they are shown to be linked to the equation defining the co-channel reuse ratio. Using root systems and Dynkin diagrams technics, we reveal that such a equation can be converted into an algebraic relation in terms of the two simple roots describing \(G_2\)-generalized Lie algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Alouini, M., & Goldsmith, A. J. (1999). Area spectral efficiency of cellular mobile radio systems. IEEE Transactions on Vehicular Technology, 48(4), 1047–1066.

    Article  Google Scholar 

  2. Althaus, E., Calinescu, G., Mandoiu, I. I., Prasad, S., Tchervenski, N., & Zelikovsky, A. (2006). Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks. Wireless Networks, 12, 287–299.

    Article  Google Scholar 

  3. Anang, K. A., Rapajic, P. B., Wu, R., Bello, L., & Eneh, T. I. (2012). Cellular system information capacity change at higher frequencies due to propagation loss and system parameters. Progress in Electromagnetics Research B, 44, 191–221.

    Article  Google Scholar 

  4. Alam, S. M., & Haas, Z. J. (2006). Coverage and connectivity in three-dimensional networks. In Proceedings of the twelfth annual international conference on mobile computing and networking (ACM Mobicom06), September 23–29, 2006, Los Angeles, CA, USA.

  5. Burde, D., & Dekimpe, K. (2006). Novikov structures on solvable Lie algebras. Journal of Geometry and Physics, 56(9), 1837–1855.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, X., Kumar, S., Xuan, D., Yun, Z., & Lai, T. H. (2006). Deploying wireless sensors to achieve both coverage and connectivity. In Proceedings of the 7th ACM international symposium on mobile ad hoc networking and computing (MobiHoc06) (pp. 131–142), May 22–25, 2006, Florence, Italy.

  7. Belhaj, A. (2012). Symétrie en Physiques: Algèbres de Lie, Théorie des groupes et Représentations, arXiv:1205.3335.

  8. Belhaj, A., Boya, L. J., & Segui, A. (2013). On hexagonal structures in higher dimensional theories. International Journal of Theoretical Physics, 52(1), 130–136.

    Article  MathSciNet  MATH  Google Scholar 

  9. Rubio, R. (2013). \(B_n\)-generalized geometry and \(G_2^2\)-structures. Journal of Geometry and Physics, 73, 150–156.

    Article  MathSciNet  MATH  Google Scholar 

  10. Belhaj, A. (2002). Manifolds of G2 holonomy from N = 4 sigma model. Journal of Physics: A Mathematical and General, 35(42), 8903–8912.

    MathSciNet  MATH  Google Scholar 

  11. Humphreys, J. E. (1978). Introduction to Lie algebras and representation theory. New York: Sauteur-Verlag.

    MATH  Google Scholar 

  12. Huang, K., & Andrews, J. G. (2011). A stochastic-geometry approach to coverage in cellular networks with multi-cell cooperation. In Global telecommunications conference (pp 1–5).

  13. Huang, C. F., Tseng, Y. C., & Wu, H. L. (2007). Distributed protocols for ensuring both coverage and connectivity of a wireless sensor network. ACM Transactions on Sensor Networks, 3, 1–24.

    Article  Google Scholar 

  14. Huang, C., & Tseng, Y. (2005). The coverage problem in a wireless sensor network, In Proceedings of the international conference on wireless sensor networks and applications (WSNA) (pp. 519–528).

  15. Iyengar, R., Kar, K., & Mukherjee, S. (2005). Low-coordination topologies for redundancy in sensor networks. In Proceedings of the 7th ACM international symposium on Mobile ad hoc networking and computing (MobiHoc05) (pp. 332–342), May, 2005, Urbana-Champaign, IL, USA.

  16. Ghosh, A., & Das, S. K. (2008). Coverage and connectivity issues in wireless sensor networks: A survey. Pervasive and Mobile Computing, 4, 303–334.

    Article  Google Scholar 

  17. Kochan, D. (2004). Grassmann electrodynamics and general relativity. Journal Geometry and Physics, 51(2), 196–210.

    Article  MathSciNet  MATH  Google Scholar 

  18. Katiyar, S., Jain, R. K., & Agrawal, N. K. (2012). R.F. Pollution reduction in cellular communication. International Journal of Scientific & Engineering Research, 3(3), 1–6.

    Google Scholar 

  19. Kershner, R. (1939). The number of circles covering a set. American Journal of Mathematics, 61(3), 665–671.

    Article  MathSciNet  MATH  Google Scholar 

  20. Kumar, S., Lai, T. H., & Balogh, J. (2004). On k-coverage in a mostly sleeping sensor network. In Proceedings of the 10th annual international conference on Mobile computing and networking (pp.144–158), Philadelphia, PA, USA.

  21. Madhusudhanan, P., Restrepo, J. G., Liu, Y., Brown, T. X., & Baker, K. R. (2011). Multi-tier network performance analysis using a shotgun cellular system. In IEEE Globecom (pp. 1–6).

  22. Megerian, S., Koushanfar, F., Qu, G., Veltri, G., & Potkonjak, M. (2002). Exposure in wireless sensor networks: Theory and practical solutions. Wireless Networks, 8, 443–454.

    Article  MATH  Google Scholar 

  23. Meguerdichian, S., Koushanfar, F., Potkonjak, M., & Srivastava, M. (2001). Coverage problems in wireless ad-hoc sensor networks. In Proceedings of the 20th IEEE international conference on Computer communications (INFOCOM01) (pp. 115–121), April, 2001, Anchorage, AK.

  24. Slate, P. B. (2005). Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems. Journal Geometry and Physics, 53, 74–97.

    Article  MathSciNet  Google Scholar 

  25. Steane, A. M. (2002). Quantum computer architecture for fast entropy extraction. Quantum Information and Computation, 2(4), 297–306.

    MATH  Google Scholar 

  26. Stravroulakis, P. (2003). Interference analysis and reduction for wireless system. Norwood: Artech House.

    Google Scholar 

  27. Stojanovic, M. (2008). Design and capacity analysis of cellular-type underwater acoustic networks. IEEE Journal of Oceanic Engineering, 33(2), 171–181.

    Article  Google Scholar 

  28. Stojanovic, M., & Peleato, B. M. (2008). A channel sharing scheme for underwater cellular networks. Sea Technology, 49(5), 39–42.

    Google Scholar 

  29. Shakkottai, S., Srikant, R., & Shroff, N. B. (2005). Unreliable sensor grids: Coverage, connectivity and diameter. Ad Hoc Networks, 3, 702–716.

    Article  Google Scholar 

  30. Veltri, G., Huang, Q., Qu, G., & Potkonjak, M. (2003). Minimal and maximal exposure path algorithms for wireless embedded sensor networks. In Proceedings of the 1st international conference on Embedded networked sensor systems (Sensys03) (pp. 40–50), November, Los Angeles, CA.

  31. Wang, Y., Hu, C., & Tseng, Y. (2005). Efficient deployment algorithms for ensuring coverage and connectivity of wireless sensor networks. In Proceedings of the first international conference on wireless internet (WICON) (pp. 114–121).

  32. Wang, X., Xing, G., Zhang, Y., Lu, C., Pless, R., & Gill, C. (2003). Integrated coverage and connectivity configuration in wireless sensor networks. In Proceedings of the 1st international conference on embedded networked sensor systems (Sensys03) (pp. 28–39), November, Los Angeles, CA.

  33. Younis, M., & Akkaya, K. (2008). Strategies and techniques for node placement in wireless sensor networks: A survey. Ad Hoc Networks, 6, 621–655.

    Article  Google Scholar 

  34. Zhang, H., & Hou, J. C. (2005). Maintaining sensing coverage and connectivity in large sensor networks. Ad Hoc and Sensor Wireless Networks, 1(1–2), 125–157.

    Google Scholar 

  35. Zhang, H., & Hou, J. (2004). On deriving the upper bound of \(\alpha\)-lifetime for large sensor networks. In Proceedings of the 5th ACM international symposium on Mobile ad hoc networking and computing (pp. 121–132). May 24-26, 2004, Roppongi Hills, Tokyo, Japan.

  36. Zhuang, Y., Pan, J. (2011). Random distances associated with hexagons. The Computing Research Repository (CoRR).

Download references

Author information

Authors and Affiliations

  1. BOSS team, École Nationale des Sciences Appliquées, Université Ibn Tofail, Kenitra, Morocco

    Aouatif Amine & Moulay Brahim Sedra

  2. Département de Physique, LIRST, Faculté Polydisciplinaire, Université Sultan Moulay Slimane, Béni Mellal, Morocco

    Adil Belhaj

  3. SIMO, Faculté des Sciences, Université Ibn Tofail, Kenitra, Morocco

    Moulay Brahim Sedra

Authors
  1. Aouatif Amine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adil Belhaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Moulay Brahim Sedra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aouatif Amine.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amine, A., Belhaj, A. & Sedra, M.B. The Hexagonal Geometrical Structure of the N-Coverage Networks in the G2-Lie Algebra Framework. Wireless Pers Commun 89, 319–330 (2016). https://doi.org/10.1007/s11277-016-3267-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-016-3267-z

Keywords

Search

Navigation