Digital fixed points, approximate fixed points, and universal functions

Laurence Boxer

United States

Niagara University

Professor and past Chair of Computer and Information Sciences; also, Research Professor of Computer Science and Engineering at SUNY - Buffalo

Ozgur Ege

Turkey

Celal Bayar University

Research Assistant, Department of Mathematics

Ismet Karaca

Turkey

Ege University

Professor of Mathematics

Jonathan Lopez

United States

Canisius College

Assistant Professor of Mathematics

Joel Louwsma

United States

Niagara University

Assistant Professor of Mathematics

Submitted: 2016-02-16

|

Accepted: 2016-07-17

|

Published: 2016-10-03

DOI: https://doi.org/10.4995/agt.2016.4704
Funding Data

Downloads

Keywords:

digital image, digitally continuous, digital topology, fixed point

Supporting agencies:

This research was not funded

Abstract:

A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).
Show more Show less

References:

H. Arslan, I. Karaca, and A. Oztel, Homology groups of $n$-dimensional digital images, XXI Turkish National Mathematics Symposium (2008), B, 1-13.

K. Borsuk, Theory of Retracts, Polish Scientific Publishers, Warsaw, 1967.

L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839. https://doi.org/10.1016/0167-8655(94)90012-4

L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456

L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), 19-26. https://doi.org/10.1007/s10851-005-4780-y

L. Boxer, Digital products, wedges, and covering spaces, Journal of Mathematical Ima-ging and Vision 25 (2006), 159-171. https://doi.org/10.1007/s10851-006-9698-5

L. Boxer, Fundamental groups of unbounded digital images, Journal of Mathematical Imaging and Vision 27 (2007), 121-127. https://doi.org/10.1007/s10851-007-0778-y

L. Boxer, Continuous maps on digital simple closed curves, Applied Mathematics 1 (2010), 377-386. https://doi.org/10.4236/am.2010.15050

L. Boxer, I. Karaca, and A u{O}ztel, Topological invariants in digital images, Journal of Mathematical Sciences: Advances and Applications 11, no. 2 (2011), 109-140.

G. Chartrand and L. Lesniak, Graphs $&$ digraphs, 2nd ed., Wadsworth, Inc., Belmont, CA, 1986.

L. Chen, Gradually varied surfaces and its optimal uniform approximation, SPIE Proceedings 2182 (1994), 300-307 https://doi.org/10.1117/12.171078

L. Chen, Discrete surfaces and manifolds, Scientific Practical Computing, Rockville, MD, 2004

E. Demir and I. Karaca, Simplicial homology groups of certain digital surfaces, Hacettepe Journal of Mathematics and Statistics, 44, no. 5 (2015), 1011-1022. https://doi.org/10.15672/HJMS.2015449657

O. Ege and I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theory and Applications 2013, 2013:253 (texttt{http://www.fixedpointtheoryandapplications.com/content/2013/1/253}). https://doi.org/10.1186/1687-1812-2013-253

O. Ege and I. Karaca, Fundamental properties of digital simplicial homology groups, American Journal of Computer Technology and Application, 1, no. 2 (2013), 25-42.

O. Ege and I. Karaca, Applications of the Lefschetz number to digital images, Bulletin of the Belgian Mathematical Society, Simon Stevin 21, no. 5 (2014), 823-839. https://doi.org/10.36045/bbms/1420071856

S.-E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018

S.-E. Han, Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences 177, no. 16 (2007), 3314-3326. https://doi.org/10.1016/j.ins.2006.12.013

G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993),381-396. https://doi.org/10.1006/cgip.1993.1029

I. Karaca and O. Ege, Some results on simplicial homology groups of 2D digital images, International Journal of Information and Computer Science 1, no. 8 (2012), 198-203.

Proceedings IEEE Intl. Conf. on Systems, Man, and Cybernetics, pp. 227-234, 1987.

T. Y. Kong, A digital fundamental group, Computers and Graphics 13 (1989), 159-166. https://doi.org/10.1016/0097-8493(89)90058-7

A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6

E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. https://doi.org/10.1007/978-1-4684-9322-1_5

Show more Show less