, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved. The scheme is based on trivariate Bernstein polynomials

computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by Bernstein and Vazirani [BV93], thus answering an open question raised in [BV93]. We also develop a theory of quantum communication complexity, and use it as a tool to prove

Schubert polynomials were introduced by Bernstein Gelfand Gelfand and De- mazure, and were extensively developed by Lascoux, Schiitzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polyno- mial in terms of the reduced decompositions

It is well known that in two or more variables Bernstein polynomials do not preserve convexity. Here we introduce two variations, one stronger than the classical notion, the other one weaker, which are preserved. Moreover, a weaker sufficient condition for the monotony of subsequent Bernstein

Three division algorithms are presented for univariate Bernstein polynomials: an algorithm for finding the quotient and remainder of two univariate polynomials, an algorithm for calculating the GCD of an arbitrary collection of univariate polynomials, and an algorithm for computing a µ

Abstract. Trivariate C r macro-elements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1, 2, these spaces reduce to well-known macro-element spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe

. We discuss a natural way to define barycentric coordinates associated with circular arcs. This leads to a theory of Bernstein-B'ezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials. x1. Introduction Bernstein

Abstract. Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ0,..., λn. Assume that the set Un of all solutions of the equation Lf = 0 is closed under complex conjugation. If the length of the interval [a, b] is smaller than π/Mn, where Mn: = max {|Imλj | : j = 0,..., n}, then there exists a basis pn,k, k = 0,...n, of the space Un with the property that each pn,k has a zero of order k at a and a zero of order n − k at b, and each pn,k is positive on the open interval (a, b). Under the additional assumption that λ0 and λ1 are real and distinct, our first main result states that there exist points a = t0 < t1 <... < tn = b and positive numbers α0,.., αn, such that the operator n∑ Bnf: = αkf (tk)pn,k (x) k=0 satisfies Bne λjx = e λjx, for j = 0, 1. The second main result gives a sufficient condition guaranteeing the uniform convergence of Bnf to f for each f ∈ C [a, b].

Let α> 0 not be an integer. In papers published in 1913 and 1938, S. N. Bernstein established the limit Λ ∗ ∞,α = lim n→ ∞ nα En [|x | α; L ∞ [−1, 1]]. Here En [|x | α; L ∞ [−1, 1]] denotes the error in best uniform approx-imation of |x | α on [−1, 1] by polynomials of degree ≤ n. Bernstein

This thesis consisting of three chapters is concerned with Bernstein polynomials. In the first chapter, an introduction to Bernstein polynomials is given. Then, basic properties of Bernstein polynomials are studied in the second chapter. Last chapter studies the generalized Bernstein polynomials