American Institute of Mathematical Sciences

Advanced
May  2014, 8(2): 223-239. doi: 10.3934/amc.2014.8.223

A reduction point algorithm for cocompact Fuchsian groups and applications

Pilar Bayer 1, and  Dionís Remón 1,

1. 

Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain, Spain

Received  October 2013 Published  May 2014

In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations. We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group $\mathbf{SL}(2, \mathbb{Z})$ and whose roots go back to [9]. The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.
Keywords: fundamental domains, Fuchsian codes, SISO channels., reduction point algorithm, Quaternionic Fuchsian groups.
Mathematics Subject Classification: 11F06, 11Y16, 14G50, 30F35, 65Y04, 94B4.
Citation: Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223
References:
[1]

M. Alsina and P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves,, Amer. Math. Soc., (2004).   Google Scholar

[2]

P. Bayer, Contributions to Shimura curves,, in Win-Women in Numbers: Research Directions in Number Theory, (2011), 15.   Google Scholar

[3]

I. Blanco-Chacón, C. Hollanti and D. Remón, Fuchsian codes for AWGN channels,, in International Workshop on Coding and Cryptography 2013, (2013), 496.   Google Scholar

[4]

E. Brandani da Silva, M. Firer, S. Costa and R. Palazzo, Signal constellations in the hyperbolic plane: a proposal for new communication systems,, J. Franklin Institute, 343 (2006), 69.  doi: 10.1016/j.jfranklin.2005.09.001.  Google Scholar

[5]

E. D. Carvalho, A. A. Andrade, R. Palazzo and J. Vieira, Arithmetic Fuchsian groups and space time codes,, Comput. Appl. Math., 30 (2011), 485.  doi: 10.1590/S1807-03022011000300001.  Google Scholar

[6]

D. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL(2,Z),, Exp. Math., 1 (1992), 275.   Google Scholar

[7]

S. Katok, Fuchsian Groups,, Univ. Chicago Press, (1992).   Google Scholar

[8]

A. Lascurain, Some presentations for $\overline{\Gamma}_0(N)$,, Conform. Geom. Dyn., 6 (2002), 33.  doi: 10.1090/S1088-4173-02-00073-5.  Google Scholar

[9]

J.-P. Serre, A Course in Arithmetic,, Springer-Verlag, (1973).   Google Scholar

[10]

F. Strömberg, Maass waveforms on $(\Gamma_0(N), \chi)$ (computational aspects),, in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology (eds. J. Bolt and F. Steiner), (2012), 187.   Google Scholar

[11]

J. Voight, Computing fundamental domains for Fuchsian groups,, J. Théor. Nombres Bordeaux, 21 (2009), 467.   Google Scholar

show all references

References:
[1]

M. Alsina and P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves,, Amer. Math. Soc., (2004).   Google Scholar

[2]

P. Bayer, Contributions to Shimura curves,, in Win-Women in Numbers: Research Directions in Number Theory, (2011), 15.   Google Scholar

[3]

I. Blanco-Chacón, C. Hollanti and D. Remón, Fuchsian codes for AWGN channels,, in International Workshop on Coding and Cryptography 2013, (2013), 496.   Google Scholar

[4]

E. Brandani da Silva, M. Firer, S. Costa and R. Palazzo, Signal constellations in the hyperbolic plane: a proposal for new communication systems,, J. Franklin Institute, 343 (2006), 69.  doi: 10.1016/j.jfranklin.2005.09.001.  Google Scholar

[5]

E. D. Carvalho, A. A. Andrade, R. Palazzo and J. Vieira, Arithmetic Fuchsian groups and space time codes,, Comput. Appl. Math., 30 (2011), 485.  doi: 10.1590/S1807-03022011000300001.  Google Scholar

[6]

D. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL(2,Z),, Exp. Math., 1 (1992), 275.   Google Scholar

[7]

S. Katok, Fuchsian Groups,, Univ. Chicago Press, (1992).   Google Scholar

[8]

A. Lascurain, Some presentations for $\overline{\Gamma}_0(N)$,, Conform. Geom. Dyn., 6 (2002), 33.  doi: 10.1090/S1088-4173-02-00073-5.  Google Scholar

[9]

J.-P. Serre, A Course in Arithmetic,, Springer-Verlag, (1973).   Google Scholar

[10]

F. Strömberg, Maass waveforms on $(\Gamma_0(N), \chi)$ (computational aspects),, in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology (eds. J. Bolt and F. Steiner), (2012), 187.   Google Scholar

[11]

J. Voight, Computing fundamental domains for Fuchsian groups,, J. Théor. Nombres Bordeaux, 21 (2009), 467.   Google Scholar

[1]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[2]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
[3]

Rodolfo Gutiérrez-Romo. A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle. Journal of Modern Dynamics, 2019, 14: 227-242. doi: 10.3934/jmd.2019008
[4]

Dmitri Burago, Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with Abelian fundamental groups. Journal of Modern Dynamics, 2008, 2 (4) : 541-580. doi: 10.3934/jmd.2008.2.541
[5]

Carolyn Mayer, Kathryn Haymaker, Christine A. Kelley. Channel decomposition for multilevel codes over multilevel and partial erasure channels. Advances in Mathematics of Communications, 2018, 12 (1) : 151-168. doi: 10.3934/amc.2018010
[6]

A.V. Borisov, A.A. Kilin, I.S. Mamaev. Reduction and chaotic behavior of point vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 100-109. doi: 10.3934/proc.2005.2005.100
[7]

Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027
[8]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[9]

Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381
[10]

Olav Geil, Carlos Munuera, Diego Ruano, Fernando Torres. On the order bounds for one-point AG codes. Advances in Mathematics of Communications, 2011, 5 (3) : 489-504. doi: 10.3934/amc.2011.5.489
[11]

Chuangqiang Hu, Shudi Yang. Multi-point codes from the GGS curves. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020020
[12]

John B. Little. The ubiquity of order domains for the construction of error control codes. Advances in Mathematics of Communications, 2007, 1 (1) : 151-171. doi: 10.3934/amc.2007.1.151
[13]

Cristóbal Camarero, Carmen Martínez, Ramón Beivide. Identifying codes of degree 4 Cayley graphs over Abelian groups. Advances in Mathematics of Communications, 2015, 9 (2) : 129-148. doi: 10.3934/amc.2015.9.129
[14]

Thomas Feulner. The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Advances in Mathematics of Communications, 2009, 3 (4) : 363-383. doi: 10.3934/amc.2009.3.363
[15]

Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the P*NCP. Journal of Industrial & Management Optimization, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569
[16]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[17]

Elvira Zappale. A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains. Evolution Equations & Control Theory, 2017, 6 (2) : 299-318. doi: 10.3934/eect.2017016
[18]

Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419
[19]

Evgeny I. Veremey, Vladimir V. Eremeev. SISO H-Optimal synthesis with initially specified structure of control law. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 121-138. doi: 10.3934/naco.2017009
[20]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences