In Hilbert space L2(Rn), we prove the equivalence between the mod-ulus of smoothness and the K-functionals constructed by the Sobolev space cor-responding to the Fourier transform. For this purpose, Using a spherical meanoperator.

in hilbert space l2(rn), we prove the equivalence between the mod-ulus of smoothness and the k-functionals constructed by the sobolev space cor-responding to the fourier transform. for this purpose, using a spherical meanoperator.

For n + 1 points in IR, in general position, the Kergin polynomial interpolant of C functions may be extended to an interpolant of C functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k...

We discuss conditions under which a convex cone K ⊂ R admits a probability m such that supk∈K m(k) ≤ 0. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.

We develop the first-order gradient correction to the exchange-correlation free energy of the homogeneous electron gas for use in finite-temperature density functional calculations. Based on this, we propose and implement a simple temperature-dependent extension for functionals beyond the local density approximation. These finitetemperature functionals show improvement over zero-temperature fun...

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem 2.1. 1. Functionals Wek from rescaled Ricci flow point of view This is the research notes when the author wrote [Li07]. In the first section, we discuss the relation between functionals Wek(g, f, τ) and rescaled Ricci flow. In Theorem 4.2 [Li07] , we have defined funct...