Upward and downward statistical continuities

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0$$ is statistically downward half quasi-Cauchy if $$\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0$$ for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous.Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.069

ON GENERALIZED STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES VIA IDEALS IN INTUITIONISTIC FUZZY NORMED SPACES

In this paper, we introduce the concept of I₂-lacunary statistical convergence and strongly I₂-lacunary convergence with respect to the intuitionistic fuzzy norm (μ,v), investigate their relationship, and make some observations about these classes. We mainly examine the relation between these two new methods and the relation between I₂-statistical convergence in the corresponding intuitionistic fuzzy normed space

Sezgisel fuzzy normlu uzaylarda ℐ-lacunary istatiksel yakınsaklık

In this study, first, we investigate the notions of ℐ-lacunary statistical convergence and strongly ℐlacunary convergence with regards to the intuitionistic fuzzy norm (IFN for short) (μ,ν). Then, we investigate relationships among this new concepts and make important observations about them. Futhermore, we examine the relations among ℐ-lacunary statistical convergence and ℐ-statistical convergence in terms of IFN (μ,ν) in the corresponding intuitionistic fuzzy normed space.Bu çalışmada, ilk olarak (μ,ν) sezgisel normuna göre ℐ-lacunary istatistiksel yakınsaklık ve kuvvetli ℐlacunary yakınsaklık kavramları tanımlandı. Daha sonra bu kavramlar arasındaki ilişkiler incelendi ve bu kavramlar üzerine önemli gözlemler yapıldı. Bununla birlikte, ilgili sezgisel fuzzy normlu uzayda (μ,ν) sezgisel normuna göre ℐ-lacunary istatistiksel yakınsaklık ile ℐ-istatistiksel yakınsaklık arasındaki ilişkiler incelendi

Generalized I of strongly Lacunary of x2 over p-metric spaces defined by Musielak Orlicz function

In this paper, we introduce generalized difference sequence spaces via ideal convergence, lacunary of x2 sequence spaces over p-metric spaces defined by Musielak function, and examine the Musielak-Orlicz function which satisfies uniform Δ2 condition, and we also discuss some topological properties of the resulting spaces of x2 with respect to ideal structures which is solid and monotone. Hence, given an example of the space x2 this is not solid and not monotone. This theory is very useful for statistical convergence and also is applicable to rough convergence