International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 174372, 10 pages
http://dx.doi.org/10.1155/2012/174372
Research Article

Properties of Carry Value Transformation

Suryakanta Pal,1 Sudhakar Sahoo,2 and Birendra Kumar Nayak3

1Department of Basic Science and Humanities, Silicon Institute of Technology, Silicon Hills, Patia, Bhubaneswar 751024, India
2Department of Computer Science, Institute of Mathematics and Applications, Andharua, Bhubaneswar 751003, India
3P.G. Department of Mathematics, Utkal University, Bhubaneswar 751004, India

Received 28 December 2011; Accepted 20 January 2012

Academic Editor: V. R. Khalilov

Copyright © 2012 Suryakanta Pal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Carry Value Transformation (CVT) is a model of discrete deterministic dynamical system. In the present study, it has been proved that (1) the sum of any two nonnegative integers is the same as the sum of their CVT and XOR values. (2) the number of iterations leading to either C V T = 0 or X O R = 0 does not exceed the maximum of the lengths of the two addenda expressed as binary strings. A similar process of addition of modified Carry Value Transformation (MCVT) and XOR requires a maximum of two iterations for MCVT to be zero. (3) an equivalence relation is shown to exist on 𝑍 × 𝑍 which divides the CV table into disjoint equivalence classes.