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Yıl 2022, Cilt: 71 Sayı: 4, 1029 - 1043, 30.12.2022

Bilal Demir , Mustafa Karataş

https://doi.org/10.31801/cfsuasmas.1089480

Öz

Kaynakça

  • Beardon, A. F., Algebra and Geometry, Cambridge University Press, 2005, https://dx.doi.org/10.1017/CBO9780511800436.
  • Beardon, A. F., Hockman, M., Short, I., Geodesic continued fractions, The Michigan Mathematical Journal, 61 (1) (2012), 133–150, https://dx.doi.org/10.1307/mmj/1331222851.
  • Şahin, R., On the some normal subgroups of the extended modular group, Applied Mathematics and Computation, 218 (3) (2011), 1025–1029, https://dx.doi.org/10.1016/j.amc.2011.03.074.
  • Şahin, R., İkikardeş, S., Squares of congruence subgroups of the extended modular group, Miskolc Mathematical Notes, 14 (3) (2013), 1031–1035, https://dx.doi.org/10.18514/MMN.2013.780.
  • Demir, B., Özgür, N. Y., Koruoğlu, Ö., Relationships between fixed points and eigenvectors in the group gl(2, r), Fixed Point Theory and Applications, 2013 (1) (2013), 55, https://dx.doi.org/10.1186/1687-1812-2013-55.
  • Fine, B., Trace classes and quadratic forms in the modular group, Canadian Mathematical Bulletin, 37 (2) (1994), 202–212, https://dx.doi.org/10.4153/CMB-1994-030-1.
  • Hayat, U., Farid, G., Karapinar, E., Relationships between tropical eigenvectors and tropical fixed points of the group gl (2, r), Italian Journal of Pure and Applied Mathematics (2020), 291.
  • İkikardeş, S., Koruoğlu, O., Şahin, R., Cangül, I. N., One relator quotients of the extended modular group, Advanced Studies in Contemporary Mathematics, 17 (2) (2008), 203–210.
  • Jones, G., Singerman, D., Wicks, K., The modular group and generalized Farey graphs, London Mathematical Society Lecture Note Series (160) (1991), 316–341, https://dx.doi.org/10.1017/CBO9780511661846.006.
  • Jones, G., Thornton, J., Automorphisms and congruence subgroups of the extended modular group, Journal of the London Mathematical Society, 2 (1) (1986), 26–40, https://dx.doi.org/10.1112/jlms/s2-34.1.26.
  • Jones, G. A., Singerman, D., Complex Functions: An Algebraic and Geometric Viewpoint, Cambridge University Press, 1987, https://dx.doi.org/10.1017/CBO9781139171915.
  • Koruoğlu, Ö., Şahin, R., Generalized Fibonacci sequences related to the extended Hecke groups and an application to the extended modular group, Turkish Journal of Mathematics, 34(3)(2010), 325–332, https://dx.doi.org/10.3906/mat-0902-33.
  • Koruoğlu, Ö., Şahin, R., İkikardeş, S., Trace classes and fixed points for the extended modular group, Turkish Journal of Mathematics, 32 (1) (2008), 11–19, https://dx.doi.org/10.3906/mat-0609-3.
  • Koruoğlu, Ö., Sarıca, S¸. K., Demir, B., Kaymak, A. F., Relationships between cusp points in the extended modular group and Fibonacci numbers, Honam Math. J. (2019), https://dx.doi.org/10.5831/HMJ.2019.41.3.569.
  • Mushtaq, Q., Hayat, U., Horadam generalized Fibonacci numbers and the modular group, Indian Journal of Pure and Applied Mathematics, 38 (5) (2007), 345.
  • Mushtaq, Q., Hayat, U., Pell numbers, Pell–Lucas numbers and modular group, Algebra Colloquium, 14 (2007), 97–102, https://dx.doi.org/10.1142/S1005386707000107.
  • Newman, M., A complete description of the normal subgroups of genus one of the modular group, American Journal of Mathematics, 86 (1) (1964), 17–24, https://dx.doi.org/10.2307/2373033.
  • Newman, M., Normal subgroups of the modular group which are not congruence subgroups, Proceedings of the American Mathematical Society, 16 (4) (1965), 831–832, https://dx.doi.org/10.1090/S0002-9939-1965-0181618-X.
  • Newman, M., Classification of normal subgroups of the modular group, Transactions of the American Mathematical Society, 126 (2) (1967), 267–277, https://dx.doi.org/10.2307/1994453.
  • Newman, M., Maximal normal subgroups of the modular group, Proceedings of the American Mathematical Society, 19 (5) (1968), 1138–1144.
  • Newman, M., Free subgroups and normal subgroups of the modular group, Illinois Journal of Mathematics, 8 (2) (1964), 262–265, https://dx.doi.org/10.1215/ijm/1256059670.
  • Özdemir, N., İskender, B. B., Özgür, N. Y., Complex valued neural network with Mobius activation function, Communications in Nonlinear Science and Numerical Simulation, 16 (12) (2011), 4698–4703, https://dx.doi.org/10.1016/j.cnsns.2011.03.005.
  • Rankin, R., Subgroups of the modular group generated by parabolic elements of constant amplitude, Acta Arithmetica, 1 (18) (1971), 145–151.
  • Ressler, W., On binary quadratic forms and the Hecke groups, International Journal of Number Theory, 5 (08) (2009), 1401–1418, https://dx.doi.org/10.1142/S1793042109002730.
  • Rosen, D., A class of continued fractions associated with certain properly discontinuous groups, Duke Mathematical Journal, 21 (3) (1954), 549–563, https://dx.doi.org/10.1215/S0012-7094-54-02154-7.
  • Şahin, R., İkikardeş, S., Koruoğlu, Ö., On the power subgroups of the extended modular group, Turkish Journal of Mathematics, 28 (2) (2004), 143–152, https://dx.doi.org/10.3906/mat-0301-2.
  • Schmidt, T. A., Sheingorn, M., Length spectra of the Hecke triangle groups, Mathematische Zeitschrift, 220 (1) (1995), 369–397.
  • Series, C., The modular surface and continued fractions, Journal of the London Mathematical Society, 2 (1) (1985), 69–80, https://dx.doi.org/10.1112/jlms/s2-31.1.69.
  • Short, I., Walker, M., Even-integer continued fractions and the Farey tree, In Symmetries in Graphs, Maps, and Polytopes Workshop (2014), Springer, pp. 287–300, https://dx.doi.org/10.1007/978-3-319-30451-9 15.
  • Short, I., Walker, M., Geodesic rosen continued fractions, The Quarterly Journal of Mathematics (2016), 1–31, https://dx.doi.org/doi.org/10.1093/qmath/haw025.
  • Yılmaz, N., Cangul, I. N., The normaliser of the modular group in the Picard group, Bulletin-Institue of Mathematics Academia Sinica, 28 (2) (2000), 125–130.
  • Yılmaz Özgür, N., Generalizations of Fibonacci and Lucas sequences, Note di Matematica, 21 (1) (2002), 113–125, https://dx.doi.org/10.1285/i15900932v21n1p113.
  • Yılmaz Özgür, N., On the power subgroups of the modular group, Ars Combinatoria, 89 (2) (2009), 112–125.
  • Yılmaz Özgür, N., On the two-square theorem and the modular group, Ars Combinatoria, 94 (1) (2010), 225–239.

Farey graph and rational fixed points of the extended modular group

Yıl 2022, Cilt: 71 Sayı: 4, 1029 - 1043, 30.12.2022

Bilal Demir , Mustafa Karataş

https://doi.org/10.31801/cfsuasmas.1089480

Öz

Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group ¯¯¯¯ΓΓ¯ is the group of 2×22×2 matrices with integer entries and determinant ±1±1. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set ^Q=Q∪{∞}Q^=Q∪{∞}. In this study, we consider the elements in ¯¯¯¯ΓΓ¯ that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in $\overline{\Gamma}$ that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.

Anahtar Kelimeler

Extended modular group fixed points Farey sequence Farey graph

Kaynakça

  • Beardon, A. F., Algebra and Geometry, Cambridge University Press, 2005, https://dx.doi.org/10.1017/CBO9780511800436.
  • Beardon, A. F., Hockman, M., Short, I., Geodesic continued fractions, The Michigan Mathematical Journal, 61 (1) (2012), 133–150, https://dx.doi.org/10.1307/mmj/1331222851.
  • Şahin, R., On the some normal subgroups of the extended modular group, Applied Mathematics and Computation, 218 (3) (2011), 1025–1029, https://dx.doi.org/10.1016/j.amc.2011.03.074.
  • Şahin, R., İkikardeş, S., Squares of congruence subgroups of the extended modular group, Miskolc Mathematical Notes, 14 (3) (2013), 1031–1035, https://dx.doi.org/10.18514/MMN.2013.780.
  • Demir, B., Özgür, N. Y., Koruoğlu, Ö., Relationships between fixed points and eigenvectors in the group gl(2, r), Fixed Point Theory and Applications, 2013 (1) (2013), 55, https://dx.doi.org/10.1186/1687-1812-2013-55.
  • Fine, B., Trace classes and quadratic forms in the modular group, Canadian Mathematical Bulletin, 37 (2) (1994), 202–212, https://dx.doi.org/10.4153/CMB-1994-030-1.
  • Hayat, U., Farid, G., Karapinar, E., Relationships between tropical eigenvectors and tropical fixed points of the group gl (2, r), Italian Journal of Pure and Applied Mathematics (2020), 291.
  • İkikardeş, S., Koruoğlu, O., Şahin, R., Cangül, I. N., One relator quotients of the extended modular group, Advanced Studies in Contemporary Mathematics, 17 (2) (2008), 203–210.
  • Jones, G., Singerman, D., Wicks, K., The modular group and generalized Farey graphs, London Mathematical Society Lecture Note Series (160) (1991), 316–341, https://dx.doi.org/10.1017/CBO9780511661846.006.
  • Jones, G., Thornton, J., Automorphisms and congruence subgroups of the extended modular group, Journal of the London Mathematical Society, 2 (1) (1986), 26–40, https://dx.doi.org/10.1112/jlms/s2-34.1.26.
  • Jones, G. A., Singerman, D., Complex Functions: An Algebraic and Geometric Viewpoint, Cambridge University Press, 1987, https://dx.doi.org/10.1017/CBO9781139171915.
  • Koruoğlu, Ö., Şahin, R., Generalized Fibonacci sequences related to the extended Hecke groups and an application to the extended modular group, Turkish Journal of Mathematics, 34(3)(2010), 325–332, https://dx.doi.org/10.3906/mat-0902-33.
  • Koruoğlu, Ö., Şahin, R., İkikardeş, S., Trace classes and fixed points for the extended modular group, Turkish Journal of Mathematics, 32 (1) (2008), 11–19, https://dx.doi.org/10.3906/mat-0609-3.
  • Koruoğlu, Ö., Sarıca, S¸. K., Demir, B., Kaymak, A. F., Relationships between cusp points in the extended modular group and Fibonacci numbers, Honam Math. J. (2019), https://dx.doi.org/10.5831/HMJ.2019.41.3.569.
  • Mushtaq, Q., Hayat, U., Horadam generalized Fibonacci numbers and the modular group, Indian Journal of Pure and Applied Mathematics, 38 (5) (2007), 345.
  • Mushtaq, Q., Hayat, U., Pell numbers, Pell–Lucas numbers and modular group, Algebra Colloquium, 14 (2007), 97–102, https://dx.doi.org/10.1142/S1005386707000107.
  • Newman, M., A complete description of the normal subgroups of genus one of the modular group, American Journal of Mathematics, 86 (1) (1964), 17–24, https://dx.doi.org/10.2307/2373033.
  • Newman, M., Normal subgroups of the modular group which are not congruence subgroups, Proceedings of the American Mathematical Society, 16 (4) (1965), 831–832, https://dx.doi.org/10.1090/S0002-9939-1965-0181618-X.
  • Newman, M., Classification of normal subgroups of the modular group, Transactions of the American Mathematical Society, 126 (2) (1967), 267–277, https://dx.doi.org/10.2307/1994453.
  • Newman, M., Maximal normal subgroups of the modular group, Proceedings of the American Mathematical Society, 19 (5) (1968), 1138–1144.
  • Newman, M., Free subgroups and normal subgroups of the modular group, Illinois Journal of Mathematics, 8 (2) (1964), 262–265, https://dx.doi.org/10.1215/ijm/1256059670.
  • Özdemir, N., İskender, B. B., Özgür, N. Y., Complex valued neural network with Mobius activation function, Communications in Nonlinear Science and Numerical Simulation, 16 (12) (2011), 4698–4703, https://dx.doi.org/10.1016/j.cnsns.2011.03.005.
  • Rankin, R., Subgroups of the modular group generated by parabolic elements of constant amplitude, Acta Arithmetica, 1 (18) (1971), 145–151.
  • Ressler, W., On binary quadratic forms and the Hecke groups, International Journal of Number Theory, 5 (08) (2009), 1401–1418, https://dx.doi.org/10.1142/S1793042109002730.
  • Rosen, D., A class of continued fractions associated with certain properly discontinuous groups, Duke Mathematical Journal, 21 (3) (1954), 549–563, https://dx.doi.org/10.1215/S0012-7094-54-02154-7.
  • Şahin, R., İkikardeş, S., Koruoğlu, Ö., On the power subgroups of the extended modular group, Turkish Journal of Mathematics, 28 (2) (2004), 143–152, https://dx.doi.org/10.3906/mat-0301-2.
  • Schmidt, T. A., Sheingorn, M., Length spectra of the Hecke triangle groups, Mathematische Zeitschrift, 220 (1) (1995), 369–397.
  • Series, C., The modular surface and continued fractions, Journal of the London Mathematical Society, 2 (1) (1985), 69–80, https://dx.doi.org/10.1112/jlms/s2-31.1.69.
  • Short, I., Walker, M., Even-integer continued fractions and the Farey tree, In Symmetries in Graphs, Maps, and Polytopes Workshop (2014), Springer, pp. 287–300, https://dx.doi.org/10.1007/978-3-319-30451-9 15.
  • Short, I., Walker, M., Geodesic rosen continued fractions, The Quarterly Journal of Mathematics (2016), 1–31, https://dx.doi.org/doi.org/10.1093/qmath/haw025.
  • Yılmaz, N., Cangul, I. N., The normaliser of the modular group in the Picard group, Bulletin-Institue of Mathematics Academia Sinica, 28 (2) (2000), 125–130.
  • Yılmaz Özgür, N., Generalizations of Fibonacci and Lucas sequences, Note di Matematica, 21 (1) (2002), 113–125, https://dx.doi.org/10.1285/i15900932v21n1p113.
  • Yılmaz Özgür, N., On the power subgroups of the modular group, Ars Combinatoria, 89 (2) (2009), 112–125.
  • Yılmaz Özgür, N., On the two-square theorem and the modular group, Ars Combinatoria, 94 (1) (2010), 225–239.
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Bilal Demir 0000-0002-6638-6909

Mustafa Karataş 0000-0002-1147-0169

Yayımlanma Tarihi 30 Aralık 2022
Gönderilme Tarihi 18 Mart 2022
Kabul Tarihi 23 Mayıs 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 71 Sayı: 4

Kaynak Göster

APA Demir, B., & Karataş, M. (2022). Farey graph and rational fixed points of the extended modular group. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 1029-1043. https://doi.org/10.31801/cfsuasmas.1089480
AMA Demir B, Karataş M. Farey graph and rational fixed points of the extended modular group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Aralık 2022;71(4):1029-1043. doi:10.31801/cfsuasmas.1089480
Chicago Demir, Bilal, ve Mustafa Karataş. “Farey Graph and Rational Fixed Points of the Extended Modular Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, sy. 4 (Aralık 2022): 1029-43. https://doi.org/10.31801/cfsuasmas.1089480.
EndNote Demir B, Karataş M (01 Aralık 2022) Farey graph and rational fixed points of the extended modular group. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 1029–1043.
IEEE B. Demir ve M. Karataş, “Farey graph and rational fixed points of the extended modular group”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 71, sy. 4, ss. 1029–1043, 2022, doi: 10.31801/cfsuasmas.1089480.
ISNAD Demir, Bilal - Karataş, Mustafa. “Farey Graph and Rational Fixed Points of the Extended Modular Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (Aralık 2022), 1029-1043. https://doi.org/10.31801/cfsuasmas.1089480.
JAMA Demir B, Karataş M. Farey graph and rational fixed points of the extended modular group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:1029–1043.
MLA Demir, Bilal ve Mustafa Karataş. “Farey Graph and Rational Fixed Points of the Extended Modular Group”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 71, sy. 4, 2022, ss. 1029-43, doi:10.31801/cfsuasmas.1089480.
Vancouver Demir B, Karataş M. Farey graph and rational fixed points of the extended modular group. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):1029-43.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

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