Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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FIXED POINT THEOREMS FOR DIGITAL IMAGES

  • HAN, SANG-EON (Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University)
  • Received : 2015.11.16
  • Accepted : 2015.12.08
  • Published : 2015.12.25
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Abstract

In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

Keywords

References

  1. H. Arslan, I. Karaca, A. Oztel, Homology groups of n-dimensional digital images XXI. Turkish National Mathematics Symposium (2008) 1-13.
  2. S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math. 3 (1922) 133-181. https://doi.org/10.4064/fm-3-1-133-181
  3. L. Boxer, A classical construction for the digital fundamental group, Jour. of Mathematical Imaging and Vision 10 (1999) 51-62. https://doi.org/10.1023/A:1008370600456
  4. L. Boxer, I. Karaca, A. Oztel, Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications 11 (2011) 109-140.
  5. L. Boxer, O. Ege, I. Karaca, J. Lopez, Digital Fixed Points, Approximate Fixed Points, and Universal Functions, preprint, available at http://arxiv.org/abs/1507.02349.
  6. R.F. Brown, The Nielsen number of a fiber map, Ann. Math. 85 (1967) 483-493. https://doi.org/10.2307/1970354
  7. O. Ege, I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theorey and Applications (2013) 2013:253; doi:10.1186/1687-1812-2013-253.
  8. O. Ege, I. Karaca, Applications of the Lefschetz Number to Digital Images, The Bulletin of the Belgian Mathematical Society-Simon Stevin 21(5) (2014) 823-839.
  9. O. Ege, I. Karaca, Banach fixed point theorem for digital images, Journal of Nonlinear Sciences and Applications 8 (2015) 237-245.
  10. O. Ege, I. Karaca, M. E. Ege, Relative homology groups of digita images, Appl. Math. Inf. Sci. 8(5) (2014), 2337-2345. https://doi.org/10.12785/amis/080529
  11. S.E. Han, Non-product property of the digital fundamental group, Information Sciences 171(1-3) (2005) 73-91. https://doi.org/10.1016/j.ins.2004.03.018
  12. S.E. Han, On the simplicial complex stemmed from a digital graph, Honam Mathematical Journal 27(1) (2005) 115-129.
  13. S.E. Han, Connected sum of digital closed surfaces, Information Sciences 176(3) (2006) 332-348. https://doi.org/10.1016/j.ins.2004.11.003
  14. S.E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences 177(16) (2006) 3314-1329. https://doi.org/10.1016/j.ins.2006.12.013
  15. S.E. Han, The k-homotopic thinning and a torus-like digital image in $Z^n$, Journal of Mathematical Imaging and Vision 31(1) (2008) 1-16. https://doi.org/10.1007/s10851-007-0061-2
  16. S.E. Han, KD-($k_{0},\;k_{1}$)-homotopy equivalence and its applications, Journal of Korean Mathematical Society 47(5) (2010) 1031-1054. https://doi.org/10.4134/JKMS.2010.47.5.1031
  17. S.E. Han, Ultra regular covering spaces and its automorphism group, Int. J. Appl. Math. Comput. Sci. 20(4)(2010), 699-710. https://doi.org/10.2478/v10006-010-0053-z
  18. S.E. Han, Personal communication with the authors of [4] on Euler characteristic for digital images, (June 28, 2015).
  19. S.E. Han, Digital version of the fixed point theory, Proceedings of 11th ICFPTA (Abstracts) (2015) p.60.
  20. S.E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Sciences and Applications 9(3) (2016) 895-905. https://doi.org/10.22436/jnsa.009.03.19
  21. S.E. Han, The Lefschetz number for digital images with the same digital homotopy type, Fixed Point Theorey and Applications, submitted.
  22. S.E. Han and B.G. Park, Digital graph ($k_{0},\;k_{1}$)-isomorphism and its applications, http://atlas-conferences.com/c/a/k/b/36.htm (2003).
  23. E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conferences on Systems, Man, and Cybernetics (1987) 227-234.
  24. T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
  25. S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28(1) (1926) 1-49. https://doi.org/10.1090/S0002-9947-1926-1501331-3
  26. S. Lefschetz, On the fixed point formula, Ann. of Math. 38(4) (1937) 819-822. https://doi.org/10.2307/1968838
  27. A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters 4 (1986) 177-184. https://doi.org/10.1016/0167-8655(86)90017-6
  28. E.H. Spanier, Algebraic Topology, McGraw-Hill Inc., New York, 1966.

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