It is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct. Thus the corresponding interpolation matrix M is nonsingular. ...Schumaker stated the conjecture that all principal submatrices of M are nonsingular too. Furthermore, all of the corresponding determinants (the principal minors) are conjectured to be positive. This result would solve the constrained interpolation problem. In this paper, the conjecture on minors for polynomial degree ⩽17 and conjecture for some particular configurations of domain points are confirmed.
Artificial neural networks (ANNs) have a large appeal to many researchers due to their feature to simulate and solve different kinds of problems that do not have algorithmic solutions. The outlined ...in this paper is an efficient and robust collocation method based on the ANNs and Bernstein polynomials intended for the fuzzy Abel integral equation problem. To do this, first truncated Bernstein-series polynomial of the solution function is substituted in parametric form of the given fuzzy problem. Then an architecture of ANNs namely the feed-back neural nets is designed to determine values for the unknown coefficients. Eventually, the proposed method is implemented on some numerical examples, and also is compared with an usual and classical technique.
This paper derives an approximation algorithm for multi-degree reduction of a degree
n triangular Bézier surface with corners continuity in the norm
L
2
. The new idea is to use orthonormality of ...triangular Jacobi polynomials and the transformation relationship between bivariate Jacobi and Bernstein polynomials. This algorithm has a very simple and explicit expression in matrix form, i.e., the reduced matrix depends only on the degrees of the surfaces before and after degree reduction. And the approximation error of this degree-reduced surface is minimum and can get a precise expression before processing of degree reduction. Combined with surface subdivision, the piecewise degree-reduced patches possess global
C
0
continuity. Finally several numerical examples are presented to validate the effectiveness of this algorithm.
A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the ...Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are compared through the analysis of data sets famous in the equating literature. Also, the classical percentile-rank, linear, and mean equating models are each proven to be a special case of a Bayesian model under a highly-informative choice of prior distribution.
This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to ...approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.
Aiming at the drawback that the shape of Bézier triangles is fixed with respect to the control points, some blending functions with parameter and with similar properties to the bivariate Bernstein ...polynomials are presented. Few literatures introduce how do the blending functions are derived. This paper aims at providing the general construction method of adjustable Bézier triangles in polynomial space. With the help of degree elevation technique and based on the idea that the adjustable surfaces are defined by the adjustable control points, the shape adjustable Bézier triangles are defined. The construction process of the blending functions is demonstrated in detail.
In this paper we present a new result on the saturation of sequences of linear operators in a multivariate and simultaneous setting. Specifically, a small
o saturation result is obtained for the ...partial derivatives of the classical Bernstein bivariate operators on the unit simplex. Solutions of boundary value problems for certain partial differential equations of elliptic type play an important role.
This paper deals with a modification of the classical Bernstein polynomials defined on the unit simplex. It introduces a new sequence of non-polynomial linear operators which hold fixed the ...polynomials
x
2
+
α
x
and
y
2
+
β
y
with
α
,
β
∈
0
,
+
∞
)
. We study the convergence properties of the new approximation process and certain shape properties that are preserved. Finally, we compare it with Bernstein polynomials and show an improvement of the error of convergence in certain subsets of the simplex.
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