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Page 5752 of Mathematical Reviews Vol. , Issue 94j
[page]
1994
Mathematical Reviews
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Manfred Sommer (Eichstatt)
94j:41008 41A15 Wu, Dong Bing (PRC-BJ; Beijing) Dual bases of a Bernstein polynomial basis on simplices. (English summary) Comput. Aided Geom. ...
The author constructs dual bases for the space spanned by the classical Bernstein basis polynomials of degree n on a simplex in R”+!. ...
Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases
1998
Journal of Approximation Theory
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We describe, in particular, a new change of basis algorithm from a bivariate Lagrange L-basis to a bivariate Bernstein basis with computational complexity O(n 3 ). Academic Press ...
It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of both L-bases and B-bases. ...
ACKNOWLEDGMENTS We would like to thank Phil Barry of the University of Minnesota for discussing some of the topics presented here, and for helping us to improve our presentation. ...
doi:10.1006/jath.1997.3162
fatcat:bo4prnb6hjbq3nk3l4mjielfgq
Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials
2014
The Scientific World Journal
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We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. ...
In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. ...
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. ...
doi:10.1155/2014/257484
pmid:25485293
pmcid:PMC4251784
fatcat:ei53selkpfbdhobnkclwnmlotm
New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials
2015
Baghdad Science Journal
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Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved. ...
Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method. ...
Theorem: [9]
The first derivatives of nth degree
generalized
Bernstein
basis
polynomials can be written as a linear
combination of the generalized
Bernstein basis polynomials of degree
n
( ) ...
doi:10.21123/bsj.12.4.846-853
fatcat:aumki4xb25dedouj55h4ovyw3y
Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma * *This work is supported by the Danish Council for Independent Research under grant number DFF - 4005-00452 in the project CodeMe
2017
IFAC-PapersOnLine
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In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices. ...
Abstract: In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices. ...
Let a polynomial p of degree d be defined in the Bernstein basis on a simplexσ, and let b(p, d,σ) ≥ 0. ...
doi:10.1016/j.ifacol.2017.08.339
fatcat:25l34y2zmnedzizy2x3yceioku
Pyramid Algorithms for Bernstein--Bézier Finite Elements of High, Nonuniform Order in Any Dimension
2014
SIAM Journal on Scientific Computing
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Pyramid algorithms replace an operation on single high order polynomial by a recursive sequence of self-similar affine combinations, and are ubiquitous in CAGD for computations involving high order curves ...
A new, non-uniform order, variant of the de Casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm ...
How are the Bernstein polynomials defined on a face F related to the Bernstein polynomials defined on the original simplex T ? ...
doi:10.1137/130914048
fatcat:pyrgyigq6zdyhnq55j7axldjxy
Differential-recurrence properties of dual Bernstein polynomials
[article]
2018
arXiv
pre-print
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New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. ...
Using these results, a fourth-order differential equation satisfied by dual Bernstein polynomials has been constructed. ...
Bernstein basis polynomials B n i are given by B n i (x) := n i x i (1 − x) n−i (i = 0, 1, . . . , n; n ∈ N). (2.8) One can easily check that polynomials B n 0 , B n 1 , . . . , B n n form a basis of the ...
arXiv:1803.01735v2
fatcat:j3mswjyicjeebceizqjgrkn4be
Page 1467 of Mathematical Reviews Vol. , Issue 91C
[page]
1991
Mathematical Reviews
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The authors introduce a B-net representation of polynomials on simplices and splines over triangulations. ...
Again the author takes the B-net (Bernstein-Bézier-net) approach in his construction of a local basis for S. ...
The apolar bilinear form in geometric modeling
1999
Mathematics of Computation
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These results are applied to study so-called lineal polynomial bases and their dual bases. The B-patch basis is introduced in [6] as the dual of a special lineal basis. ...
A similar binary form on the space of univariate polynomials of a fixed degree has been studied by Goldman [12] . ...
Acknowledgments I wish to thank Lyle Ramshaw for his interest in this paper, and the anonymous referee for suggesting a useful addition to the introductory section. ...
doi:10.1090/s0025-5718-99-01144-8
fatcat:j4wpj3hhwzftldljbzt6k4ht6m
Legendre–Bernstein basis transformations
2000
Journal of Computational and Applied Mathematics
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We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on [0; 1] into each other, and examine the stability of this linear map. ...
The Bernstein form of a polynomial o ers valuable insight into its geometrical behavior, and has thus won widespread acceptance as the basis for BÃ ezier curves and surfaces. ...
One approach [18] to direct least-squares approximation by polynomials in Bernstein form relies on construction of the basis d n 0 (u); : : : ; d n n (u) that is "dual" to the Bernstein basis J uttler ...
doi:10.1016/s0377-0427(00)00376-9
fatcat:5n2xq6wvjnf67d5qldourigrly
Bézier form of dual bivariate Bernstein polynomials
[article]
2016
arXiv
pre-print
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Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L^2-solution of the problem of best polynomial approximation of Bézier curve or surface. ...
In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. ...
Remark that the striking simplicity of Algorithm 2.1 was obtained thanks to using properties of dual bivariate Bernstein polynomials, investigated in Appendix A. ...
arXiv:1510.08246v4
fatcat:vrnj6yy2brdmlh7wfme6rjtosy
Connections between two-variable Bernstein and Jacobi polynomials on the triangle
2006
Journal of Computational and Applied Mathematics
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Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given. ...
Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials. ...
Appendix A. ...
doi:10.1016/j.cam.2005.11.013
fatcat:ftdj5htq5bcbvei5qfvr6dhmay
Efficient merging of multiple segments of Bézier curves
[article]
2015
arXiv
pre-print
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We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (P. Wo\'zny, S. Lewanowicz, Comput. Aided Geom. ...
Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging ...
We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis [12] to compute the control points r i . ...
arXiv:1409.2671v3
fatcat:4ogl3oklmzcglhqvbeuyikqmwy
Fast simplicial quadrature-based finite element operators using Bernstein polynomials
2011
Numerische Mathematik
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We derive low-complexity matrix-free finite element algorithms for simplicial Bernstein polynomials on simplices. ...
Our techniques, based on a sparse representation of differentiation and special block structure in the matrices evaluating B-form polynomials at warped Gauss points, apply to variable coefficient problems ...
The complete set of Bernstein polynoimals, {B n α } |α|=n , form a basis of polynomials of complete degree n on S d . They form a nonnegative partition of unity on S d . ...
doi:10.1007/s00211-011-0431-y
fatcat:uiub7xmxsnawxmioy3c2dhjacq
Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation
2017
Computational and Applied Mathematics
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2016 pre-print arXiv:1605.06744v2 fatcat:mudsi3phnbbgbeulcsc3rrmfouOther Versions
A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones. ...
We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix based approach for the spatial discretization. ...
So it is of interest to explore some new aspects of this basis in order to facilitate the numerical methods for differential equations that are based on Bernstein polynomials and to present a method for ...
doi:10.1007/s40314-017-0455-8
fatcat:6n5drszugnh4zen3rdobhwqxqu
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