Let R be a unital ring with involution. The pseudo core inverses of differences and products two projections are investigated in R. Some equivalent conditions obtained. As applications, the invertibility commutator pq - qp anti-commutator + characterized, where p; q

Let F2 be the free group generated by x and y . In this article, we prove that the commutator of xm and yn is a product of two squares if and only if mn is even. We also show using topological methods that there are infinitely many obstructions for an element in F2 to be a product of two squares. AMS Classification 20F12; 57M07

Let T be a bounded linear operator on X = (Σ `q)p with 1 ≤ q < ∞ and 1 < p <∞. Then T is a commutator if and only if for all non zero λ ∈ C, the operator T − λI is not X-strictly singular.

The density functional theory (DFT) is a remarkably successful of electronic structure matter. At the foundation this lies Kohn-Sham (KS) equation. In paper, we describe long-time behaviour time-dependent KS Assuming weak self-interactions, prove global existence and scattering in (almost) full "short-range" regime. This achieved with new simple techniques, naturally compatible DFT involving co...