We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators....

Generalized strictly diagonally dominant matrices have wide applications in science and engineering, but it is very difficult to determine whether a given matrix is a generalized strictly diagonally dominant matrix or not in practice. In this paper, we give several practical conditions for generalized strictly diagonally dominant matrices by constructing different positive diagonal matrix and a...

It is shown that for every undirected graph G over a nite set N and for every nonempty T N there exists an undirected graph G T over T, called the marginal graph of G for T, such that the class of marginal distributions for T of (discrete) G-Markovian distributions coincides with the class of G T-Markovian distributions. An example shows that this is not true within the framework of strictly po...

For the sake of completeness, we first present the Sinkhorn’s theorem and explain how it was used to derive the solution of the regularized optimal transport. Theorem ((Sinkhorn & Knopp, 1967)). If A is an n × n matrix with strictly positive elements, then there exist diagonal matrices D1 and D2 with strictly positive diagonal elements such that D1 ∗ A ∗ D2 is doubly stochastic. The matrices D1...

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the strictly positive discrete probability distributions with the prescribed sample space that factorize according to a chain graph G with dimension d have positive Lebesgue measure wrt R, whereas those that factorize according to G but are not faithful to it have zero Lebesgue measur...

We consider functions f v = min A∈K Av and gv = max A∈K Av, where K is a finite set of nonnegative matrices and by " min " and " max " we mean coordinate-wise minimum and maximum. We transfer known results about properties of g to f. In particular we show existence of nonnegative generalized eigenvectors for f , give necessary and sufficient conditions for existence of strictly positive eigenve...

We observe that the decision problem for the ∃ theory of real closed fields (RCF) is simply reducible to the decision problem for RCF over a connective-free ∀ language in which the only relation symbol is a strict inequality. In particular, every ∃ RCF sentence φ can be settled by deciding a proposition of the form “polynomial p (which is a sum of squares) takes on strictly positive values over...

Various types of stability are introduced in DEFINITION 2. A one-parameter family of discretizations, {B(k)}, is said to be (1) stable if and only if for any T>0 there exist positive constants C and ko such that ||S(fe)|| g C for all 0<k<k0 and all positive integers n such that nk S T, (2) strictly stable if and only if there exists positive constants k0 and C such that ||S(fe)|| g C for all 0 ...