When learning processes depend on samples but not on the order of the information in the sample, then the Bernoulli distribution is relevant and Bernstein polynomials enter into the analysis. We derive estimates of the approximation of the entropy function x log x that are sharper than the bounds from Voronovskaja’s theorem. In this way we get the correct asymptotics for the Kullback-Leibler di...

Stability criteria are proposed for two variable D polynomials having interval parameters in polynomic uncertain ty structures Both the left half plane and unit circle domains are considered Save for a minor condition the criteria reduce robust stability testing of D polynomials to testing positivity of only two polynomials The appealing feature of the new robustness criteria is that positivity...

which are now called Bernstein polynomials, in order to present a short proof of the Weierstrass Approximation Theorem. The subsequent history is well documented, see, e.g., [29] for the period up to 1955, the monograph [18] published in 1953, and the survey article [9] which appeared on the occasion of the hundredth anniversary of the above paper by Bernstein. Since the latter publication prov...

We consider the problem of minimizing a polynomial on the hypercube [0, 1] and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmüdgen [26]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the ...

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti’s finite and infinite representation theorems. We show that the finite representation in terms of count vector...

This is the first part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. The main result of this part is a constructibility result for the analytic Gröbner fan of a parametric ideal in the ring of analytic differential operators. In this part, the main tool is the notion of generic reduced standard basis.

Derivatives and normals of rational Bézier curves and surface patches are discussed. A nonuniformly scaled hodograph of a degree m × n tensor-product rational surface, which provides correct derivative direction but not magnitude, can be written as a degree (2m−2) × 2n or 2m× (2n−2) vector function in polynomial Bézier form. Likewise, the scaled normal direction is degree (3m−2)× (3n−2). Effici...

This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC’2009 conference (Andres, Levandovskyy, and Mart́ın-Morales, 2009). We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element...

Keywords: Integro-differential equations Volterra equations Abel's integral equations Bernstein polynomials Singular volterra integral equations a b s t r a c t In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is pol...