The task of obtaining the exact values best approxima-tions by trigonometric polynomials continuous or summable functions originates from works P. L. Chebyshev, who, back in 1950s, posed problem finding polynomial, that de-viates least a given function. Subsequently this direction theory approximation got further development thanks to K. Weierstrass, D. Jackson, S. N. Bernstein, Valle-Poussin a...

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in C outside some singular set Θ. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea we have initiated in a pr...

Kiefer and Wolfowitz [14] showed that if F is a strictly curved concave distribution function (corresponding to a strictly monotone density f), then the Maximum Likelihood Estimator F̂n, which is, in fact, the least concave majorant of the empirical distribution function Fn, differs from the empirical distribution function in the uniform norm by no more than a constant times (n logn)2/3 almost s...

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary value problems in [11] wit...

Two probability distributions with common support are said to exhibit density ratio ordering when they admit a nonincreasing density ratio. Existing statistical tests of the null hypothesis of density ratio ordering are known to be conservative, with null limiting rejection rates below the nominal significance level whenever the two distributions are unequal. We show how a bootstrap procedure c...

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesguemea...

Criteria containing a barrier function i.e., an unbounded function at the boundary of the feasible solution domain are frequently encountered in the optimization framework. When an iterative descent method is used, a search along the line supported by the descent direction through the minimization of the underlying scalar function has to be performed at each iteration. Usual line search strateg...

The geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic xed point are analyzed in the framework of resonant normal forms. The interpolating ow is not obtained from a map tangent to the identity, but from the normal form of the given map and a time independent interpolating hamiltonian H is introduced. On this hamiltonian the local stability properties...

This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundary-value problems. Along with the detailed discussion of the most crucial properties of such estimates, we present the algorithm of a reliable solution approximation together with the scheme of efficient a posteriori error bound generation that is b...