Research Article Asymptotic I -Equivalence of Two Number Sequences and Asymptotic I -Regular Matrices Hafize Gumus, 1 Jeff Connor, 2 and Fatih Nuray 3 1 Faculty of Eregli Education, Necmettin Erbakan University, Eregli, Konya, Turkey 2 Department of Mathematics, Ohio University, Athens, OH, USA 3 Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkey Correspondence should be addressed to Fatih Nuray; fnuray@aku.edu.tr Received 8 November 2013; Accepted 5 January 2014; Published 24 February 2014 Academic Editors: W. Xiao and J. Zhang Copyright © 2014 Hafze Gumus et al. Tis 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. We study I -equivalence of the two nonnegative sequences  = (  ) and  = (  ). Also we defne asymptotic I -regular matrices and obtain conditions for a matrix  = (  ) to be asymptotic I -regular. 1. Introduction Te notion of -convergence was introduced by Kostyrko et al. for real sequences (see [1]) and then extended to metric spaces by Kostyrko et al. (see [2]). Fast [3] introduced statis- tical convergence and -convergence, which is based on using ideals of N to defne sets of density 0; is a natural extension of Fast’s defnition. Defnition 1. A family of sets ⊆2 N is called an ideal if and only if (i) 0∈; (ii) for each ,∈ we have ∪∈; (iii) for each ∈ and each ⊆ we have ∈. An ideal is called nontrivial if N ∉ and a nontrivial ideal is called admissible if {} ∈  for each ∈ N (see [2]). Defnition 2. A family of sets ⊂2 N is a flter in N if and only if (i) 0∉; (ii) for each ,∈ we have ∩∈; (iii) for each ∈ and each ⊇ we have ∈ (see [2]). Proposition 3.  is a nontrivial ideal in N if and only if  =  () = { = N \:∈} (1) is a flter in N (see [2]). Defnition 4. A real sequence  = (  ) is said to be - convergent to ∈ R if and only if for each >0 the set   = { ∈ N :       −     ≥ } (2) belongs to . Te number  is called the -limit of the sequence  (see [2]). Let  = (  ) and  = (  ) be real sequences. Pobyvanets introduced asymptotic equivalence for  and  as follows: if lim →∞     = 1, (3) then  and  are called asymptotic equivalent; this is denoted by ∼ (see [4]). Pobyvanets also introduced asymptotic regular matrices which preserve the asymptotic equivalence of two nonnegative number sequences; that is, for the nonnegative matrix  = (  ) if ∼ then  ∼  (see [5]). Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2014, Article ID 805857, 5 pages http://dx.doi.org/10.1155/2014/805857