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The fundamental concept of statistical convergence first was put forward by Steinhaus and at the same time but also by Fast \cite{Fast} independently both for complex and real sequences. In fact, the convergence in terms of statistical manner can be seen as a generalized form of the common convergence notion that is in the parallel of the theory of usual convergence. Measuring how large a subset of the set of natural number can be possible by means of asymptotic density. It is intuitively known that positive integers are in fact far beyond the fact that they are perfect squares. This is due to the fact that each perfect square is positive and besides at the same time there are many other positive integers. But it is also known that the set consisting of integers which are positive is not larger than that of those which are perfect squares: both of those sets are countable and infinite and therefore can be considered in terms of $1$-to-$1$ correspondence. However, when the natural numbers are scanned for increasing order, then the squares are seen increasingly scarcity. It is at this point that the concept of natural density comes into out help and this intuition becomes more precise. In this study, the above mentioned statistical convergence and asymptotic density concepts are examined in a new space and an attempt is made to fill a gap in the literature as follows. Stancu type extension of the widely known Chlodowsky type \linebreak$\left( \lambda,q\right) $-operators is going to be introduced. Moreover, the description of the novel rough statistical convergence having Pascal Fibonacci binomial matrix is going to be presented and several general characteristics of rough statistical convergence are taken into consideration. In the second place, the approximation theory is investigated as the rate of the rough statistical convergence of Chlodowsky type $\left(\lambda,q\right)$-operators.

The aim of this paper is to‎ ‎introduce and study a new concept of convergence almost surely (a.s.)‎, ‎convergence in probability‎, ‎convergence in mean‎, ‎and convergence in‎ ‎distribution are four important convergence concepts of random sequence and‎ ‎also discusses some convergence concepts of the fuzzy sequence‎: ‎convergence‎ ‎almost surely‎, ‎convergence in credibility‎, ‎convergence in mean‎, ‎and‎ ‎convergence in distribution‎.

An eco-friendly, low-cost smart attire was made of metal-free graphitic carbon nitride (GCN) as a semiconductor material in a biodegradable synthetic polymer (polyvinyl alcohol) and cotton material. Various concentrations such as 0.04, 0.08, 0.12, 0.16, and 0.2 weight percentage of GCN is entrenched in PVA. UV absorption spectra displayed two peaks, one matching PVA (317 nm) and the other excitonic peak of GCN (390 nm).0.2% GCN in PVA showed a low bandgap (2.84 eV) harnessing maximum solar light. Due to the charge transfer mechanism, enhanced blue emission at 450 nm was observed for the higher concentration of graphitic carbon nitride. Due to more defect centers of higher weight percentage of GCN, higher electrical conductivity (7.6462 × 10−3 S/cm) and optical conductivity (0.113 S) were noticed. Due to the higher conductivity of 5GCN, it was embedded with cotton fabric. The fabricated smart apparel can be used to manufacture flexible, lightweight, eco-friendly optoelectronic devices. Further, according to literature, GCN possesses high antibacterial activity, hence it can serve as clothing for the military and medical community.

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator A of fractional order, where a is a proper fraction and y = (Ymnk) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order n of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order n. [ABSTRACT FROM AUTHOR]

Let 3 denote the space of all triple gai sequences and  the space of all triple analytic sequences. This paper is devoted to the general properties of 3.

In this paper we introduce the strongly generalized difference sequence spaces of modulus function and is a non-negative four dimensional matrix of complex numbers and (pi(mn)) is a sequence of positive real numbers. We also give natural relationship between strongly generalized difference summable sequences with respect of modulus. We examine some topological properties of the above spaces and investigate some inclusion relations between these spaces.

In this paper we characterize v-invariance of the sequence spaces [[DELTA].sup.2]([[LAMBDA].sup.2]), and [[DELTA].sup.2]([chi square]). Keywords Gai sequence, analytic sequence, modulus function, double sequences. 2010 Mathematics Subject Classification: 40A05, 40C05, 40D05., [section]1. Introduction Throughout w, x and [LAMBDA] denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write [w.sup.2] for the set of all complex sequences [...]

We define the concept of rough limit set of a triple sequence space of Chlodowsky type (λ, q)-Bernstein Stancu operator of Borel summability of fuzzy numbers and obtain the relation between the set of rough limit and the extreme limit points of a triple sequence space of Chlodowsky type (λ, q)-BernsteinStancu operator of Borel summability method of fuzzy numbers. Finally, we investigate some properties of the rough limit set of Chlodowsky type (λ, q)-Bernstein Stancu operator under which Borel summable sequence of fuzzy numbers are convergent. Also, we give the results for Borel summability method of series of fuzzy numbers. [ABSTRACT FROM AUTHOR]

We introduce and study some basic properties of arithmetic rough summable of I− convergent for triple sequence and also study the set of all arithmetic rough summable of I− limits for a triple sequence and relation between analytic ness and arithmetic rough summable of I− statistical convergence for a triple sequences. [ABSTRACT FROM AUTHOR]

This paper aims to improve further on the work of Phu (2001), Aytar (2008), and Ghosal (2013). We propose a new apporach to extend the application area of rough statistical convergence usually used in triple sequence of the Bernstein operator of real numbers to the theory of probability distributions. The introduction of this concept in the probability of Bernstein polynomials of rough statistical convergence, Bernstein polynomials of rough strong Cesàro summable, Bernstein polynomials of rough lacunary statistical convergence, Bernstein polynomials of rough Nθ - convergence, Bernstein polynomials of rough λ - statistical convergence, and Bernstein polynomials of rough strong (V,λ) - summable to generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct Bernstein polynomials of rough statistical convergence: (1) Bernstein polynomials of rough lacunary statistical convergence, (2) Bernstein polynomials of rough λ - statistical convergence where Bernstein polynomials of rough strong Cesàro summable is equivalent to Bernstein polynomials of rough statistical convergence, and (3) Bernstein polynomials of rough Nθ convergence which is equivalent to Bernstein polynomials of rough lacunary statistical convergence. Bernstein polynomials of rough strong (V, λ) - summable is equivalent to Bernstein polynomials of rough λ - statistical convergence. Basic properties and interrelations of these three distinct convergences are investigated and some observations were made in these classes and in this way we demonstrated that rough statistical convergence in probability is the more generalized concept than the usual Bernstein polynomials of rough statistical convergence. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of λmnk - Γ³ and Λ³ sequences. Further, we introduce the spaces ... and ..., which are of nonabsolute type and we prove that these spaces are linearly isomorphic to the spaces Γ3 and Λ3; respectively. Moreover, we establish some inclusion relations between these spaces. [ABSTRACT FROM AUTHOR]

We introduce and study some basic properties of rough Iλ-convergence of weight g, where g: N3 ! [0;1) is a function statisying g (n) ! 1 and g (n) 6! 0 as n ! 1 of a triple sequence of Bernstein Stancu Cheney and Sharma operators and also investigate certain properties of rough I-convergence of weight g. [ABSTRACT FROM AUTHOR]

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In this paper we characterize ν-invariance of the sequence spaces Δ²(Λ²), and Δ² (Χ²). [ABSTRACT FROM AUTHOR]

In this paper, we introduce two general double sequence spaces χ²M (p) and _2 M (p) using Orlicz functions. We establish some inclusion relations, topological results and we characterize the duals of these double sequence spaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce generalized semi normed difference of double gai sequence spaces defined by a modulus function. We study their different properties and obtain some inclusion relations involving these semi normed difference double gai sequence spaces. [ABSTRACT FROM AUTHOR]