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Normal bivariate Birkhoff interpolation schemes and Pell equation

Marius Crainic, Nicolae Crainic (2009)

Commentationes Mathematicae Universitatis Carolinae

Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular “shape” often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of “shapes”. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...

Note on universal algorithms for learning theory

Karol Dziedziul, Barbara Wolnik (2007)

Applicationes Mathematicae

We study the universal estimator for the regression problem in learning theory considered by Binev et al. This new approach allows us to improve their results.

Numerical curves and their applications to algebraic curves

H. Gevorgian, H. Hakopian, A. Sahakian (1996)

Studia Mathematica

Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme N = n 1 , . . . , n s ; n , i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization...

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