We present a characterization of the approximation errors of the Post-Widder and the Gamma operators in Lp[0, ∞), 1 ≤ p ≤ ∞, with a weight x γ for any real γ. Two types of characteristics are used – weighted K-functionals of the approximated function itself and the classical fixed step moduli of smoothness taken on a simple modification of it.

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also anal...

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1 − 1/e ≈ .63, derived in Syrgkanis and Tardos (2013) using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the wor...

We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number μ > τ 2, where τ > m − 1 is the exponent in the small divisors condition and m is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible ma...

In this paper we will try to introduce a good smoothness notion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should has.