# American Institute of Mathematical Sciences

eISSN:
2577-8838

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## Mathematical Foundations of Computing

November 2022 , Volume 5 , Issue 4

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2022, 5(4): 269-293 doi: 10.3934/mfc.2022003 +[Abstract](518) +[HTML](171) +[PDF](581.45KB)
Abstract:

The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B\begin{document}$\ddot{o}$\end{document}gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.

2022, 5(4): 295-314 doi: 10.3934/mfc.2022006 +[Abstract](588) +[HTML](194) +[PDF](1701.76KB)
Abstract:

This paper focuses on the dynamical analysis of the permanent magnet asynchronous motor with the aim of subsequently designing effective robust control laws for the indirect field-oriented control (IFOC) devices. We first perform some tasks which demonstrate the existence of chaos phenomenon in the IFOC using relevant indicators such as phase portraits, bifurcations diagrams and Lyapunov exponents. Chaotic signature and some striking transitions are revealed such as period-doubling, torus, period-adding and chaos when an accessible parameter of the IFOC motor is changed. More interestingly, a certain range of the parameter space corresponds to the transient chaos. This behavior was not reported previously and can be considered as an enriching contribution. Secondly, due to the great interest to reduce the upper bound of uncertainties and interference, conventional sliding mode control (SMC) has been abundantly investigated for fault-tolerant control (FTC) systems. However, this approach presents several drawbacks in terms of overshoot, less robustness, transient state error, large chattering and speed of convergence that limit its use for industrial applications. For these reasons, the integral sliding mode control (ISMC) and the fuzzy sliding mode control (FISMC) are proposed to keep the IFOC motor in the regular operation zone. The optimal feedback gains and a sufficient condition are proposed for the stability of the overall IFOC system is drawn based on the linear quadratic regulator (LQR) method. To highlight the effectiveness and applicability of the proposed control scheme, numerical simulation results are presented. This analysis allows us a great knowledge of engineers for interpreting the operation of the IFOC motor. To highlight the effectiveness and the applicability of the proposed control scheme, numerical simulations results are presented and clearly demonstrated the feasibility of these techniques.

2022, 5(4): 315-330 doi: 10.3934/mfc.2022007 +[Abstract](538) +[HTML](154) +[PDF](332.11KB)
Abstract:

The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz\begin{document}$\acute{a}$\end{document}sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.

2022, 5(4): 331-342 doi: 10.3934/mfc.2022008 +[Abstract](572) +[HTML](183) +[PDF](294.98KB)
Abstract:

In the present paper, we introduce a new type of Bernstein-Durrmeyer operators preserving linear functions in movable interval. The approximation rate of the new operators for continuous functions and Voronovskaja's asymptotic estimate are obtained.

2022, 5(4): 343-350 doi: 10.3934/mfc.2022009 +[Abstract](503) +[HTML](128) +[PDF](264.9KB)
Abstract:

In this paper, we propose and analyze iterative method based on projection techniques to solve a non-singular linear system \begin{document}$Ax = b$\end{document}. In particular, for a given positive integer \begin{document}$m$\end{document}, \begin{document}$m$\end{document}-dimensional successive projection method (\begin{document}$m$\end{document}D-SPM) for symmetric positive definite matrix \begin{document}$A$\end{document}, is generalized for non-singular matrix \begin{document}$A$\end{document}. Moreover, it is proved that \begin{document}$m$\end{document}D-SPM gives better result for large values of \begin{document}$m$\end{document}. Numerical experiments are carried out to demonstrate the superiority of the proposed method in comparison with other schemes in the scientific literature.

2022, 5(4): 351-362 doi: 10.3934/mfc.2022021 +[Abstract](298) +[HTML](97) +[PDF](835.03KB)
Abstract:

Readability of Chinese texts considered in this paper is a multi-class classification problem with \begin{document}$12$\end{document} grade classes corresponding to \begin{document}$6$\end{document} grades in primary schools, \begin{document}$3$\end{document} grades in middle schools, and \begin{document}$3$\end{document} grades in high schools. A special property of this problem is the strong ambiguity in determining the grades. To overcome the difficulty, a measurement of readability assessment methods used empirically in practice is adjacent accuracy in addition to exact accuracy. In this paper we give mathematical definitions of these concepts in a learning theory framework and compare these two quantities in terms of the ambiguity level of texts. A deep learning algorithm is proposed for readability of Chinese texts, based on convolutional neural networks and a pre-trained BERT model for vector representations of Chinese characters. The proposed CNN model can extract sentence and text features by convolutions of sentence representations with filters and is efficient for readability assessment, which is demonstrated with some numerical experiments.

2021 CiteScore: 0.2