An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

Recently, Mursaleen et al. (On $(p,q)$ -analogue of Bernstein operators, arXiv:1503.07404 ) introduced and studied the $(p,q)$ -analog of Bernstein operators by using the idea of $(p,q)$ -integers. In this paper, we generalize the q-Bernstein-Schurer operators using $(p,q)$ -integers and obtain a Korovkin type approximation theorem. Furthermore, we obtain the convergence of the operators by using the modulus of continuity and prove some direct theorems.

We study statistical approximation properties of -Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function by operators and give its algorithm.

The objective of the present exploration is to examine the nanoliquid flow amid two horizontal infinite plates. The lower plate is stretchable and permeable. The uniqueness of the flow model is assimilated with the Hall effect, variable thermal conductivity, thermal radiation, and irregular heat source/sink. Transmission of mass is enhanced with the impression of chemical reaction incorporated with activation energy. Appropriate similarity transformation is applied to transform the formulated problem into ordinary differential equations (ODEs). The numerical solution is obtained by employing MATLAB software function bvp4c. The dimensionless parameters are graphically illustrated and discussed for the involved profiles. An increasing behavior is exhibited by the temperature field on escalating the Brownian motion, thermophoresis parameter, variable thermal conductivity, and radiation parameter. For larger values of Schmidt number and chemical reaction parameter, the concentration profile deteriorates, while a reverse trend is seen for activation energy. The rate of heat transfer is strengthened at the lower wall on amplifying the Prandtl number. A comparative analysis of the present investigation with already published work is also added to substantiate the envisioned problem.

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically $C_1^q$-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically $C_1^q$-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.