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The run length distribution of progressive mean (PM) control charts is highly skewed when the process is in-control or there is small shift in the process, so the interpretation based on average run length (ARL) may not be suitable. Furthermore, skewness varies at different shifts which cause difficulty in its interpretation on the basis of ARL. In the presence of skewness in run length, median run length (MRL) is the best approach for interpretation and accuracy in monitoring the process. The discussion of run length on the base of MRL is quite easy and readily understood. In this article, performance of PM control chart is evaluated in MRL and it is compared with ARL at the same shifts. This article indicates that MRL is more versatile average for understanding and explaining the distribution of run length because it is quick detector in case of small and moderate shift (highly and moderately skewed run length distribution) and gives good results in case of large shifts (symmetrical run length distribution). Along with standard deviation of run length (SDRL) is computed which is showing higher variability in process when shift is small but in case of moderate and large shifts smaller values of (SDRL). The performance of proposed control is compared with existing optimal EWMA control charts based on MRL and optimal CUSUM control charts based on MRL. The proposed control chart found to be more efficient than competitors.

In this paper, the classical notions of m-convex and m-concave sets are characterized under soft set environment and their important aggregation operations are discussed. Moreover, certain classical approaches (i.e. first and second senses) are employed on m-convex and m-concave soft sets to get more generalized results for uncertain scenarios.

The s-convex functions of first kind and of second kind are well known functions. We would use recently introduced notion of s-convex functions of mixed kind and we would call it (s; r)-convex function. It would generalize the notion of first kind and second kind convexity in the sense that both kinds of convexities would be obtained easily by imposing certain specific conditions on it. We would state generalization of Hermite-Hadamard type inequalities by using our new notion of (s; r)-convex function via Riemann-Lioville fractional integrals.

Home automation is one of the fields working on automating home or household activities. The core objectives of the domotics technology are to save time and effort. Home automation can be implemented in various ways to achieve desire results. Currently, no home automation system can track the location of the person to perform some task at a particular time. We can make home automation more effective and efficient if we can implement a combination of techniques. This research aims to design and implement a hybrid technique based on the Global Position System and hand gestures in the field of home automation. Global Position System works when a person is outside the home, and the hand gesture technique is implemented when a person is present at home. By implementing a hybrid technique, we can overcome the limitations of the Global Position System and hand gesture technique. It is designed to replace sensors used in home automation system with another state-of-the-art technology, the Global Position System. Global Position System is a satellite-based technology that provides location and time information in all weather, anywhere on or near the Earth and hand gestures are used for making it intelligent by setting the trend. Idea is to integrate these two technologies and develop an automatic home automation system using Cellular Global Position System and moves a step forward in the field of automation. This research is useful for the disabled and elderly people for convenience, energy efficiency, and safety benefits and the general population as well.

Through this research article we prove parameterized inequality of fractional Hermite-Hadamard type. So, we acquired many results of Hermite-Hadamard type of inequalities.

This paper includes new versions of generalized Fejer-Hadamard (FH) type fractional integral inequalities (FIIs) for exponentially (p,h)-convex functions. We utilize an extracted generalized fractional integral operator (FIO) involving Mittag-Leffler function (MLF) which contains a monotone increasing function. Some special already published results are deduced from the presented results.

Education is a very important medium for obtaining skills and knowledge. The world is continually changing so it is significant to teach and bring up those peoples who can understand and solve modern social problems. Forecasting techniques are necessary to find out the future trends that are meet the upcoming problems. For the desired purpose, different time series models are applied and selected the best one for better-forecasted values of the gross domestic product (GDP) percentage share on education. The best model is determined based on the lowest value of the Akaike information criterion (AIC) and Bayesian information criterion. Moreover, the quality and predictability power of the selected models are measured based on the minimum value of the mean error, root mean square error, mean percentage error, and mean percentage absolute error. The AIC criteria showed that the best time series model is the autoregressive integrated moving average (ARIMA) (2, 1, 1) to forecast the (GDP) percentage share on the education of Pakistan. Furthermore, the assumptions of the fitted model such as independence, no heteroscedasticity, and normality are also evaluated. The selected model showed that the (GDP) percentage share on education is decreasing slowly, which is not good for the education sector and development of the country.

An edge irregular total h -labeling of a simple, undirected and connected graph G(V,E) is a labeling defined by H : V ∪E → {1,2,3,...,h} so that for any two distinct edges pq,rs their weights are different; i.e. W_H (pq) 6 \ne W_H (rs) where W_H (pq) = H(p)+H(q)+H(pq). A total edge irregular strength [TEIS] of a graph G, denoted by tes(G), is a labelling with the minimum h. In this paper, we have calculated the exact value of TEIS of triangle related graphs namely half gear graph, double fan graph and triangular snake graph.

In this study, wave equations with initial boundary conditions have been studied. The general form of the wave equation has been derived. The first order and second order difference schemes were established for the presented IBVP. The stability of the difference schemes has been guaranteed. The approximation solution of the problem was achieved by using finite difference methods. Two different examples are provided. A comparison between the exact and approximation solution has been carried out. Absolute errors of the problem have been presented by using MATLAB software. Moreover, the comparison shows that the second order difference scheme is a more accurate result than the first order. It is shown that the results of the comparison guaranty the reliability and accuracy of the presented method.

We define three topologies induced by a raw binary operation which is weaker than the concept of binary operation. In addition, we examine some effects of a raw binary operation on the characteristics of each induced topology.

In this paper, on the basis of the proved identities, we obtain generalized integral inequalities of the Hermite-Hadamard type for exponentially (m 1 ,m 2 )−convex functions in terms of the Riemann-Liouville fractional integration operators. Some results existing in the literature are some special cases of our results.

New general linear (integral and discrete) identities and inequalities are given for convex functions of order n via extension of Montgomery identity using new Green functions. We also state positivity conditions for these inequalities. We also study n-convexity at a point for our proposed inequalities. Bounds for reminders for proposed results are also given by using Grüss- and Ostrowski-types inequalities. We would also state mean value results of Cauchy type and Lagrange type.

Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

The aim of this study attempted to shed light on the issues such as forecasting of life expectancy rate at birth in Pakistan. Data on life expectancy rate at birth are collected over 37 years from period 1980 to 2017. A variety of time series models are applied to find the most appropriate model for forecasting the life expectancy rate at birth (LEB) of Pakistan. In the study, the autoregressive integrated moving average ARIMA (3, 2, 2) is found to be the most adequate model for forecasting the LEB of Pakistan. The best model is selected based on the various model selection’s tools. Furthermore, different test of runs and Jarque-Bera (JB) tests are used to justify the assumption of the randomness and normality of residuals. Based on the best ARIMA (3, 2, 2) model the forecasted value of the LEB in 2020,2022, and 2025 are 67.05,67.70, and 68.97 respectively.

Topological index is an important numerical magnitude that can reflect the whole structure of a graph. Degree-based indices are mathematical descriptors worked in chemical graph theory to compute the connectivity and underlying descriptions of molecules. These indices, such as the Wiener index, Randi´ c index, and Zagreb indices, are obtained from the quantities of vertices in molecular graphs. The Wiener index correspond to the calculation of gaps between all pairs of nodes in a graph, presenting information about molecular dimension and separating. Zagreb indices, including the first and second Zagreb indices, summarize the measure allocation and node connectivity within a molecular graph. Degree-based indices portray a critical position in molecular modeling, QSAR investigations, and medication layout, requiring perceptions into molecular regional anatomy and forecasting various physicochemical assets. In this article, we compute topological descriptors for complete p-ary trees. Interesting comparison of these indices are is also shown in tabular and graphical format. Moreover, expressions for multiple Zagreb indices and polynomials for these important classes are found.