This paper presents a new method to generate curves, which allows the designer to produce a curve in combinational way allowing him to get the shape that he had in his imagination with keeping the four control point for the curve design. Basing on the modified BEZIER curve, the upgraded algorithm used de Casteljau algorithm on interval [r, s] as a mathematical base to upgrade the suggested meth...

A method for calculating flow polynomials based on a transfer matrix is described. It is analogous to the method used for chromatic polynomials, although there is as yet no parallel development of the theory. The new method is applied to a family of bracelets, and the limit curves for the flow roots are obtained. There is an unexplained similarity between these calculations and the correspondin...

In this paper we introduce Bezier Lights, an intelligent lighting system that allows users to intuitively establish virtual boundaries in indoor environments. Boundaries can be easily created/modified by arranging the layouts of disk-shaped devices (“stones”) on the building floor. The immediate purpose of the system is to assist users in incorporating location-aware capabilities into private p...

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F -polyno...

ABSTRACT: Knop and Sahi introduced a family of non-homogeneous and nonsymmetric polynomials, Gα(x; q, t), indexed by compositions. An explicit formula for the bivariate Knop-Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macd...

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F -polyno...

Where a 2D problem of optimal profile in variable speed flow is resolved in a class of convex Bezier curves, using symbolic and numerical computations.

The q-Legendre polynomials can be treated as some special ”functions in the quantum double cosets U(1) \ SUq(2)/U(1)”. They form a family (depending on a parameter q) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense ”q-commutative”, the latter one is related t...

in this paper we demonstrate the existence of a set of polynomials pi , 1 i  n , which arepositive semi-definite on an interval [a , b] and satisfy, partially, the conditions of polynomials found in thelagrange interpolation process. in other words, if a  a1  an  b is a given finite sequence of realnumbers, then pi (a j )  ij (ij is the kronecker delta symbol ) ; moreover, the sum of ...