Abstract. AMotzkin path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (1, 0), up-steps (1, 1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment ) of a Motzkin path is a maximum sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics...

Let W : R → (0, 1] be continuous. Bernstein’s approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → ‖fW‖L∞(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950’s. Quantitative forms of the problem were actively investigated starting from the 1960’s. We survey old and recent aspects of this t...

We propose an alternative method for computing effectively the solution of non-linear, fixed-terminal-time, optimal control problems when they are given in Lagrange, Bolza or Mayer forms. This method works well when the nonlinearities in the control variable can be expressed as polynomials. The essential of this proposal is the transformation of a nonlinear, non-convex optimal control problem i...

A discrete Fourier analysis on the fundamental domain Ωd of the d-dimensional lattice of type Ad is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas o...

The performance of a single or the collection microswimmers strongly depends on hydrodynamic coupling among their constituents and themselves. We present numerical study for pair based lattice Boltzmann method (LBM) simulations. Our algorithm consists two separable parts. Lagrange polynomials provide discretization captures dynamics surrounding fluid. components couple via an immersed boundary ...

For n 1, let fxjngnj=1 be n distinct points in a compact set K R and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let v be a suitably restricted function on K. What conditions on the array fxjng1 j n; n 1 ensure the existence of p > 0 such that lim n!1 k (f Ln[f ]) v kLp(K)= 0 for every continuous f :: K ! R ? We show that it is necessary and su cient that there exists r ...

Article introduces an extension of the approximating functions method, a particular case finite element method (FEM) with interpolating in form Lagrange polynomials special form, to solve electrodynamics problems planar waveguide constant polarization spatial-temporal domain using Volterra integral equation method. The main goal article is expand area ​​applicability this three-dimensional pola...

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861–877], the authors have shown t...

For n 1, let fxjngnj=1 be n distinct points and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let W : R ! [0;1). What conditions on the array fxjng1 j n; n 1 ensure the existence of p > 0 such that lim n!1 k (f Ln[f ])W b kLp(R)= 0 for every continuous f : R ! Rwith suitably restricted growth, and some “weighting factor” ? We obtain a necessary and su¢ cient condition for ...