Blossoms are polar forms

Ramshaw, Lyle

doi:10.1016/0167-8396(89)90032-0

Cited by 243 publications

(97 citation statements)

References 4 publications

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“…We can thus replace (43) by (44) in the system (33)- (37). Now, taking account of the first and last inequalities in (40), (44) proves the positivity of the product α is not a problem because none of these two coefficients is involved in the connection conditions at any knot t j , j k − 1.…”

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confidence: 99%

Constructing totally positive piecewise Chebyshevian B-spline bases

Mazure

2018

Journal of Computational and Applied Mathematics

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confidence: 99%

Constructing totally positive piecewise Chebyshevian B-spline bases

Mazure

2018

Journal of Computational and Applied Mathematics

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“…Hence, our construction reduces to the familiar univariate construction of B-splines. In the univariate setting, formula (4.2) also has a counterpart; see [13]. In this case identity (4.2) becomes…”

Section: Reproduction Of Polynomialsmentioning

confidence: 99%

“…Polar forms, while well known in an algebraic context for quite some time, have been introduced into the spline theory only relatively recently by de Casteljau and independently by Ramshaw (see [13], for a comprehensive introduction). Given an s-variate polynomial p of total degree at most n, i.e., p ∈ Π n , the polar form P of p is defined as the unique function of n vector variables x 1 , .…”

Section: Reproduction Of Polynomialsmentioning

confidence: 99%

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

Neamtu

2007

Trans. Amer. Math. Soc.

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Abstract. In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

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“…This fact is also well-known [25], and we present its proof in Appendix, which can facilitate understanding subsequent development.…”

Section: Theorem 2 (Blossoming Principle) For Any Polynomialmentioning

confidence: 99%

“…We shall use a more efficient approach based on the blossoming principle, which is summarized in the following theorem. A thorough description of this principle and its various applications can be found in [25,27].…”

Section: Computing the Bézier Control Netmentioning

confidence: 99%

Approximate Reachability Computation for Polynomial Systems

Dang

2006

Hybrid Systems: Computation and Control

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Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

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Blossoms are polar forms

Cited by 243 publications

References 4 publications

Constructing totally positive piecewise Chebyshevian B-spline bases

Constructing totally positive piecewise Chebyshevian B-spline bases

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

Approximate Reachability Computation for Polynomial Systems

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