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Mathematical Foundations of Computing

February 2022 , Volume 5 , Issue 1

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Convergence of derivative of Szász type operators involving Charlier polynomials
Purshottam N. Agrawal, Thakur Ashok K. Sinha and Avinash Sharma
2022, 5(1): 1-15 doi: 10.3934/mfc.2021016 +[Abstract](643) +[HTML](272) +[PDF](434.39KB)
Abstract:

The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [20]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the \begin{document}$ (2k+2)th $\end{document} order modulus of continuity using Steklov mean.

Sliding mode observer based control for T-S fuzzy descriptor systems
Dongyun Wang
2022, 5(1): 17-32 doi: 10.3934/mfc.2021017 +[Abstract](710) +[HTML](260) +[PDF](424.34KB)
Abstract:

In this paper, the problem of sliding mode observer (SMO) based sliding mode control (SMC) for nonlinear descriptor delay systems is studied. First, based on the T-S fuzzy dynamic modeling technique, the nonlinear descriptor system is transformed into a combination of local linear models. Then, a integral-type sliding surface (ITSS) based SMO is constructed for the error system. In the sequel, sufficient linear matrix inequality (LMI) conditions are established to ensure the admissibility of the sliding motions and obtain the observer gain matrix. Furthermore, two novel SMC laws are developed to ensure the reachability conditions and stabilize the descriptor systems. Finally, simulations are provided to show the effectiveness of the method.

Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors
Yuyan Yao and Gang Wang
2022, 5(1): 33-44 doi: 10.3934/mfc.2021018 +[Abstract](524) +[HTML](274) +[PDF](325.36KB)
Abstract:

\begin{document}$ M $\end{document}-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum \begin{document}$ M $\end{document}-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

Some generalizations of delay integral inequalities of Gronwall-Bellman type with power and their applications
Haoyue Song and Fanwei Meng
2022, 5(1): 45-55 doi: 10.3934/mfc.2021022 +[Abstract](501) +[HTML](195) +[PDF](283.1KB)
Abstract:

Noting the diverse generalizations of the Gronwall-Bellman inequality, this paper investigates some new delay integral inequalities with power, deriving explicit bound on the solution and providing an example. The inequalities given here can act as powerful tools for studying qualitative properties such as existence, uniqueness, boundedness, stability and asymptotics of solutions of differential and integral equations.

Bound state solutions for fractional Schrödinger-Poisson systems
Xinsheng Du, Qi Li, Zengqin Zhao and Gen Li
2022, 5(1): 57-66 doi: 10.3934/mfc.2021023 +[Abstract](553) +[HTML](196) +[PDF](340.93KB)
Abstract:

In this work, we investigate a class of fractional Schrödinger - Poisson systems

where \begin{document}$ s\in(\frac{3}{4}, 1) $\end{document}, \begin{document}$ p\in(3, 5) $\end{document}, \begin{document}$ \lambda $\end{document} is a positive parameter. By the variational method, we show that there exists \begin{document}$ \delta(\lambda)>0 $\end{document} such that for all \begin{document}$ \mu\in[\mu_1, \mu_1+\delta(\lambda)) $\end{document}, the above fractional Schrödinger -Poisson systems possess a nonnegative bound state solutions with positive energy. Here \begin{document}$ \mu_1 $\end{document} is the first eigenvalue of \begin{document}$ (-\triangle)^s +V(x) $\end{document}.

On $ {L}(2,1) $-labelings of some products of oriented cycles
Lucas Colucci and Ervin Győri
2022, 5(1): 67-74 doi: 10.3934/mfc.2021029 +[Abstract](399) +[HTML](173) +[PDF](302.67KB)
Abstract:

We refine two results of Jiang, Shao and Vesel on the \begin{document}$ L(2,1) $\end{document}-labeling number \begin{document}$ \lambda $\end{document} of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of \begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document} for \begin{document}$ m $\end{document}, \begin{document}$ n \geq 40 $\end{document}; in the case of strong product, we either compute the exact value or establish a gap of size one for \begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document} for \begin{document}$ m $\end{document}, \begin{document}$ n \geq 48 $\end{document}.

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