The classical error analysis for the Raviart-Thomas interpolation on triangular elements requires the so-called regularity of the elements, or equivalently, the minimum angle condition. However, in the lowest order case, optimal order error estimates have been obtained in [1] replacing the regularity hypothesis by the maximum angle condition, which was known to be sufficient to prove estimates ...

We discuss a few of Erdös’results on Lagrange interpolation, and then focus on some of the rami…cations of the Erdös-Turan Theorem on Mean Convergence of Lagrange interpolation.

Let Ln [f ] denote the Lagrange interpolation polynomial to a function f at the zeros of a polynomial Pn with distinct real zeros. We show that f − Ln [f ] = −PnHe [ H [f ] Pn ] , where H denotes the Hilbert transform, and He is an extension of it. We use this to prove convergence of Lagrange interpolation for certain functions analytic in (−1, 1) that are not assumed analytic in any ellipse wi...

Let {pm(wα)}m be the sequence of the polynomials orthonormal w.r.t. the Sonin-Markov weight wα(x) = e−x 2 |x|. The authors study extended Lagrange interpolation processes essentially based on the zeros of pm(wα)pm+1(wα), determining the conditions under which the Lebesgue constants, in some weighted uniform spaces, are optimal.

A t-out-of-n secret-sharing scheme allows an honest dealer D to distribute a secret s among n players, such that any subset of t players has no information about s, but every set of t + 1 players can collaboratively reconstruct the secret. The most famous secret-sharing scheme is Shamir’s Sharing Scheme [Sha79] (cf. Section 8.1.2). It uses polynomials to obtain the desired properties. Before pr...

Following recent work by Lambiase and Nesterenko we study in detail the interquark potential for a Nambu-Goto string with point masses attached to its ends. We obtain exact solutions to the gap equations for the Lagrange multipliers and metric components and determine the potential without simplifying assumptions. We also discuss Lüscher term and argue that it remains universal. 12.38.Aw,12.39....

We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell’s problem. We use polynomials of degree k + 1, k,...

-In this paper, we investigate some properties of Michel Penot subdifferentials of locally Lipschitz functions and establish Lagrange multiplier rule in terms of Michel-Penot subdifferentials for nonsmooth mathematical programming problem. Key-Words: Nonsmooth optimization; approximate subdifferentials; generalized gradient; Michel Penot subdifferential; Banach space.