We introduce a notion of complexity for systems linear forms called sequential Cauchy–Schwarz complexity, which is parametrized by two positive integers k,ℓ and refines the introduced Green Tao. prove that if system has at most (k,ℓ) then any average 1-bounded functions over this controlled 21−ℓ-th power Gowers Uk+1-norms functions. For ℓ=1 agrees with but ℓ>1 there are have whereas their great...

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist forms are not sums of squares via a nonconstructive argument. We provide an explicit example form degree four in 272 variables sum squares. The related to the Cauchy-Schwarz inequality over octonions. proof uses symmetry reduction together with fact (due Blekherman) even degree, near-constant on ...

The goal of this project is to investigate constant properties (called the Liouville-type Problem) for a p-stable map as a local or global minimum of a p-energy functional where the domain is a Euclidean space and the target space is a closed half-ellipsoid. The First and Second Variation Formulas for a p-energy functional has been applied in the Calculus Variation Method as computation techniq...

Some inequalities of Cauchy-Bunyakovsky-Schwarz type for sequences of bounded linear operators in Hilbert spaces and some applications are given.

The results of the recent published paper by Masjed-Jamei et al. 2009 are extended to a larger class and some of subclasses are studied in the sequel. In other words, we generalize the well known Cauchy-Schwarz and Cauchy-Bunyakovsky inequalities having eight free parameters and then introduce some of their interesting subclasses.