In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which incl...

We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier-Tchebycheff coefficients given by Michhelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a func...

Abstract Finite elements with polynomial basis functions on the simplex a symmetric distribution of nodes should have unique representation. Unisolvence not only requires that number equals independent polynomials spanning space given degree, but also Vandermonde matrix controlling their mapping to Lagrange interpolating can be inverted. Here, necessary condition for unisolvence is presented sp...

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...

A form of Gauss-Quadrature rule over [0,1] has been investigated that involves the derivative of the integrand at the pre-assigned left or right end node. This situation arises when the underlying polynomials are orthogonal with respect to the weight function (): 1 x x ω = − over [0,1]. Along the lines of Golub's work, the nodes and weights of the quadrature rule are computed from a Jacobi-type...

In this paper, we revisit Grid-Based Key Pre-Distribution Scheme in Wireless Sensor Network to investigate improving the connectivity of the network and maintain both the security level and communication overhead. Both of the original work and our modification are based on using symmetric bivariate polynomials for generating cryptographic keys. In addition, their work relies on the usage of mul...

Lagrange interpolation is a classical method for approximating a continuous function by a polynomial that agrees with the function at a number of chosen points (the “nodes”). However, the accuracy of the approximation is greatly influenced by the location of these nodes. Now, a useful way to measure a given set of nodes to determine whether its Lagrange polynomials are likely to provide good ap...

The infrastructure-less property of wireless ad hoc network makes the traditional central server based security management schemes unsuitable and requires the use of a distributed key management mechanism. In this paper, we propose a distributed pairwise key establishment scheme based on the concept of bivariate polynomials. In our method, any mobile node in an ad hoc network can securely commu...