Bernstein polynomials have been recently used for the solution of some linear and non-linear differential equations, both partial and ordinary, by Bhatta and Bhatti [1] and Bhatti and Bracken [2]. Also these have been used to solve some classes of inegral equations of both first and second kinds, by Mandal and Bhattacharya [3]. These were further used to solve a Cauchy singular integro-differen...

This paper uses the symmetry properties of circles and Bernstein polynomials to establish a series of interesting barycentric properties of rational biquadratic Bézier patches. A robust algorithm is presented, based on these properties, for the conversion of Dupin cyclide patches into Bézier form. A set of conversion examples illustrates the use of this algorithm.

An approximation method based on operational matrices of Bernstein polynomials used for the solution of Hammerstein integral equations. The operational matrices of these functions are utilized to reduce a nonlinear Hammerstein and Volterra Hammerstein integral equation to a system of nonlinear algebraic equations. The method is computationally very simple and attractive, and applications are de...

We study the finitely dimensional approximations of the elliptic problem (Lu)(x, y) + φ(λ, (x, y), u(x, y)) = 0 for (x, y) ∈ Ω u(x, y) = 0 for (x, y) ∈ ∂Ω, defined for a smooth bounded domain Ω on a plane. The approximations are derived from Bernstein polynomials on a triangle or on a rectangle containing Ω. We deal with approximations of global bifurcation branches of nontrivial solutions as w...

The Bernstein operator Bn for a simplex in Rd is naturally defined via the Bernstein basis obtained from the barycentric coordinates given by its vertices. Here we consider a generalisation of this basis and the Bernstein operator, which is obtained from generalised barycentric coordinates that are given for more general configurations of points in Rd . We call the associated polynomials a Bern...

A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inc...

The lowest two-sided cell of the extended affine Weyl group We is the set {w ∈ We : w = x · w0 · z, for some x, z ∈ We}, denoted W(ν). We prove that for any w ∈ W(ν), the canonical basis element C w can be expressed as 1 [n]!χλ(Y )C ′ v1w0 C w0v2 , where χλ(Y ) is the character of the irreducible representation of highest weight λ in the Bernstein generators, and v1 and v −1 2 are what we call ...

We consider the category of modules over the affine Kac-Moody algebra ĝ of critical level with regular central character. In our previous paper [FG2] we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove...

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H⊗QSymk is defined, where QSymk denotes the Hopf algebra of quasisymmetric functions. This homomorphism generalizes the “internal comultiplication” on QSymk, and extends what Hazewinkel (in §18.24 of his “Witt vectors”) calls the Bernstein homomorphism. We construct...