bernstein type
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2021 ◽  
pp. 4903-4915
Ali Jassim Muhammad ◽  
Asma Jaber

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter  and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions ,  is a non-negative integer

Robotica ◽  
2021 ◽  
pp. 1-25
Alireza Izadbakhsh

Abstract Thisarticle presents an observer-based output tracking control method for electrically actuated cooperative multiple manipulators using Bernstein-type operators as a universal approximator. This efficient mathematical tool represents lumped uncertainty, including external perturbations and unmodeled dynamics. Then, adaptive laws are derived through the stability analysis to tune the polynomial coefficients. It is confirmed that all the position and force tracking errors are uniformly ultimately bounded using the Lyapunov stability theorem. The theoretical achievements are validated by applying the proposed observer-based controller to a cooperative robotic system comprised of two manipulators transporting a rigid object. The outcomes of the introduced method are also compared to RBFNN, which is a powerful state-of-the-art approximator. The results demonstrate the efficacy of the introduced adaptive control approach in controlling the system even in the presence of disturbances and uncertainties.

2021 ◽  
Vol 45 (4) ◽  
pp. 615-622

In this paper, we shall use a parameter β and obtain some Bernstein-type inequalities for rational functions with prescribed poles which generalize the results of Qasim and Liman and Li, Mohapatra and Rodriguez and others.

2021 ◽  
Vol 149 ◽  
pp. 16-22
Joshua Erde ◽  
J. Pascal Gollin ◽  
Atilla Joó ◽  
Paul Knappe ◽  
Max Pitz

Ulrich Dierkes ◽  
Nico Groh

AbstractWe classify all rotational symmetric solutions of the singular minimal surface equation in both cases $$\alpha <0$$ α < 0 and $$\alpha >0$$ α > 0 . In addition, we discuss further geometric and analytic properties of the solutions, in particular stability, minimizing properties and Bernstein-type results.

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Ning Zhang

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

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