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701
Freeform deformation of solid geometric models
 IN PROC. SIGGRAPH 86
, 1986
"... A technique is presented for deforming solid geometric models in a freeform manner. The technique can be used with any solid modeling system, such as CSG or Brep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, f ..."
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Cited by 701 (1 self)
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, 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
Quantum Circuit Complexity
, 1993
"... We study a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomialsize quantum circuit. This result also enables us to construct a universal quantum compu ..."
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Cited by 320 (1 self)
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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
Some Combinatorial Properties of Schubert Polynomials
, 1993
"... 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 of the permutation ..."
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Cited by 158 (11 self)
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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
Multivariate Bernstein polynomials and convexity
 Comp. Aided Geom. Design
"... 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 polyno ..."
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Cited by 14 (0 self)
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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
Division Algorithms for Bernstein Polynomials
 COMPUTER AIDED GEOMETRIC DESIGN
, 2007
"... 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 µbasis for the ..."
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Cited by 7 (1 self)
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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
Trivariate C r polynomial macroelements
 Constr. Approx
"... Abstract. Trivariate C r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1, 2, these spaces reduce to wellknown macroelement spaces used in data fitting and in the finiteelement method. We determine the dimension of these spaces, and describe st ..."
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Cited by 2 (1 self)
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Abstract. Trivariate C r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1, 2, these spaces reduce to wellknown macroelement spaces used in data fitting and in the finiteelement method. We determine the dimension of these spaces, and describe
Circular BernsteinBézier Polynomials
 Vanderbilt University Press (Nashville
, 1995
"... . We discuss a natural way to define barycentric coordinates associated with circular arcs. This leads to a theory of BernsteinB'ezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials. x1. Introduction BernsteinB'ezier ..."
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Cited by 7 (5 self)
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. We discuss a natural way to define barycentric coordinates associated with circular arcs. This leads to a theory of BernsteinB'ezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials. x1. Introduction Bernstein
Bernstein operators for exponential polynomials
"... 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 { ..."
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Cited by 10 (7 self)
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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].
On the Bernstein Constants of Polynomial Approximation
"... 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 approximation of x  α on [−1, 1] by polynomials of degree ≤ n. Bernstein prove ..."
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Cited by 6 (2 self)
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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 approximation of x  α on [−1, 1] by polynomials of degree ≤ n. Bernstein
Generalized Bernstein Polynomials
, 2014
"... 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 and ..."
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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
Results 1  10
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701