Denote by Ln the projection operator obtained by applying the Lagrange interpolation method, weighted by (1 − x), at the zeros of the Chebyshev polynomial of the second kind of degree n + 1. The norm ‖Ln‖ = max ‖f‖∞≤1 ‖Lnf‖∞, where ‖ · ‖∞ denotes the supremum norm on [−1, 1], is known to be asymptotically the same as the minimum possible norm over all choices of interpolation nodes for unweight...

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈ n/2 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpo...

According to a result of S. N. Bernstein, «-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 or n ä 10. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We...

Straight beam ®nite elements with greater than two nodes are used for edge sti€ening in plane stress analyses and elsewhere. It is often necessary to match the number of nodes on the edge sti€ener to the number on a whole plane stress element side. Beam elements employ shape functions which are recognised to be level one Hermitian polynomials. An alternative to the commonly adopted method for d...

The trajectory planning, which is known as a movement from starting to end point by satisfying the constraints along the path is an essential part of robot motion planning. A common way to create trajectories is to deal with polynomials which have independent coefficients. This paper presents a trajectory formulation as well as a procedure to arrange the suitable trajectories for applications. ...

The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

in this paper two nodes to model isotropic and anisotropic media in transmission line matrix method (tlm) are presented. by using these two nodes, the permeability and permittivity of an anisotropic media can be modeled simultaneously. another application of these nodes is in modeling media with permittivity and permeability less than one. in the conventional 2 dimensional tlm, the stubs repres...

designing wireless sensor networks should meet appropriate parameters such as quality of service (qos) defined by different users. the variable physical conditions of the environment, processing and transmission power limitations and limited communication capabilities are the most important obstacles that influence qos parameters such as throughput, delay, reliability and network lifetime. the ...

pollution has become a very serious threat to our environment. monitoring pollution is the rst step toward planning to save the environment. the use of dierential equations, monitoring pollution has become possible. in this paper, a ritz-collocation method is introduced to solve non-linear oscillatory systems such as modelling the pollution of a system of lakes. the method is based upon bernoul...

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits which polynomial computed at every node has a bound on individual degree r ≥ 1 with respect to all its variables (referred as multi- -ic circuits). The goal study is make progress towards proving superpolynomial lower bounds for general polynomials, by better value i...