We compare the convergence behavior of best polynomial approximations and Legendre Chebyshev projections derive optimal rates for analytic differentiable functions in maximum norm. For functions, we show that approximation degree n is better than projection same by a factor $$n^{1/2}$$ . such as piecewise fractional smoothness, however, only some constant factors. Our results provide new insig...

We determine the infinite sequences (ak) of integers that can be generated by polynomials with integral coefficients, in the sense that for each finite initial segment of length n there is an integral polynomial fn(x) of degree < n such that ak = fn(k) for k = 0, 1, . . . , n − 1. Let P be the set of such sequences and Π the additive group of all infinite sequences of integers. Then P is a subg...

This work presents the eigenfrequencies of a study on fluid-structure interaction for pipe with circular section, under pressure constraints in laminar, incompressible and irrotational flow, followed by comparison between stiffness coupling bar different geometric ratio Radius/Length; Radius/Thickness, as frequency identity industry. The realization this is to have better coupling, where we use...

Abstract We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits DG approximation constructed from polynomial basis functions and integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. theoretical discussion re-contextualizes stable results finite difference into setting. Numerical tests verify...

In this paper, we extend the operational matrix method to solve tempered fractional differential equation, via shifted Legendre polynomial. Although is widely used in solving various calculus problems, it yet apply equations defined derivatives. We first derive analytical expression for derivative x p</...

For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. Ther...

We present a high-order method employing Jacobi polynomial-based shape functions, as an alternative to the typical Legendre polynomial-based shape functions in solid mechanics, for solving dynamic threedimensional geometrically nonlinear elasticity problems. We demonstrate that the method has an exponential convergence rate spatially and a second-order accuracy temporally for the four classes o...

This paper is concerned with the stability analysis of distributed delay systems using complete-Lyapunov functionals. Numerous articles aim at approximating their parameters thanks to a discretization method or polynomial modeling. The interest of such approximations is the design of tractable sufficient stability conditions. In the present article, we provide an alternative method based on pol...

Let φ be a polynomial over K (a field of characteristic 0) such that the Hessian of φ is a nonzero constant. Let φ̄ be the formal Legendre Transform of φ. Then φ̄ is well-defined as a formal power series over K. The Hessian Conjecture introduced here claims that φ̄ is actually a polynomial. This conjecture is shown to be true when K = R and the Hessian matrix of φ is either positive or negative de...