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Ideal statistically quasi Cauchy sequences
(AMER INST PHYSICS, 2016) Savas, Ekrem; Cakalli, Huseyin; Ashyralyev, A; Lukashov, A
An ideal I is a family of subsets of N, the set of positive integers which is closed under taking finite unions and subsets of its elements. A sequence (x(k)) of real numbers is said to be S(I)-statistically convergent to a real number L, if for each epsilon > 0 and for each delta > 0 the set {n is an element of N: 1/n {k <= n: vertical bar x(k) - L vertical bar >= epsilon}vertical bar >= delta} belongs to I. We introduce S(I)-statistically ward compactness of a subset of R, the set of real numbers, and S(I)-statistically ward continuity of a real function in the senses that a subset E of R is S(I)-statistically ward compact if any sequence of points in E has an S(I)-statistically quasi Cauchy subsequence, and a real function is S(I)-statistically ward continuous if it preserves S(I)-statistically quasi-Cauchy sequences where a sequence (x(k)) is called to be S(I)-statistically quasi-Cauchy when (Delta x(k)) is S(I)-statistically convergent to 0. We obtain results related to S(I)-statistically ward continuity, S(I)-statistically ward compactness, N-theta-ward continuity, and slowly oscillating continuity.