We propose some refinements for the second inequality in $$ \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, where A ∈ B(H). In particular, if is hyponormal, then, by refining Young with Kantorovich constant K K(⋅, ⋅), we show that \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \righ...

Franklin's identity generalizes Euler's and states that the number of partitions n with j different parts divisible by r equals repeated at least times. In this article, we give a refinement when j=1. We prove j=1, r=2 for fixed perimeter, i.e., largest hook. also derive Beck-type perimeter: excess in all into odd perimeter M over distinct whose set even is singleton. provide analytic combinato...

Great algorithmic difficulty of hp-adaptive algorithms is one of the main obstacles preventing adaptive hp-FEM from being employed widely in realistic engineering computations. In order to reduce their complexity, we present a new technique of arbitrary-level hanging nodes that eliminates forced refinements. By forced refinements we mean refinements which are not based on a large value of an er...