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The purpose if this paper is to present a fixed point result constructed by finite sequences. Using iterated function systems and related fractal operators, a mixed patterns generated by the a finite sequence patterns construct the sets of patterns built by black and white squares. A complete metric space related to a mixed pattern sequence is defined using the distance based on difference of the black squares' area. The main result of the paper highlights that these fractal operators has unique fixed points for the sets generated by the mixed patterns. Moreover, the main theorem is also applied for Vicsek fractals such that results also hold for mixed Vicsek patterns. Motivated by various studies on growing graph sequences and related large structures, this paper underlines a new connection between fixed point theory and network science. Using circle patterns, the paper also interprets the main result on sets mixed patterns based on touching circles. Thus, the paper focuses a fixed point theorem on the sets mixed patterns built by iterated function systems and the distances calculated between the areas of these geometric shapes. [ABSTRACT FROM AUTHOR]

The statistical framework to systematically detect mean stationarity in the context of continuous manufacturing is described in this article. The methods presented in this article use econometric and financial time‐series analysis concepts in the form of unit‐root and stationarity hypothesis tests. The tests under discussion are the augmented Dickey‐Fuller, Philips‐Perron, Leybourne‐McCabe, and Kwiatkowski‐Phillips‐Schmidt‐Shin. These hypothesis tests are evaluated on data generated by a focused‐beam reflectance measurement sensor implemented on‐line in a continuous plug‐flow crystallizer. This contribution has shown that the hypothesis tests can be used to detect steady‐state conditions on‐line in a plug‐flow crystallizer. Furthermore, this econometric framework can be used as a mean stationarity “certificate” of collected samples to document that the process was mean stationary during the sampling. The statistical framework described in this article can be applied to any continuously operated unit operation or sensor measurement. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2426–2437, 2018 [ABSTRACT FROM AUTHOR]

Our paper focuses on a new approach to the construction of graph sequences driven using iterated function systems (briey IFSs) and iterated multifunction systems (briey IMSs). We build self-similar graphs sequences generated initally based on an arbitrary graph. We analyze the graphs sequences gotten within the iterations of IFSs and IMSs. We specially analyze the density properties between those graphs which are in the same sequence and we also focus on graph sequences driven by Erdős{ Rényi graphs. The main aim of our paper is to analyze the limit sets of the graph-driven IFSs and IMSs. We characterize them using the Sierpinski carpet and the [0; 1]2 set which we interpret as limits of the constructed graph sequences. [ABSTRACT FROM AUTHOR]