We prove the existence of complete conformally flat metrics of constant positive scalar curvature on the complement in ${\mathbb S}^n$ of a finite number of $(n-2)/2$-dimensional smooth submanifolds, provided $n \geq 4$ is even.

We present some sufficient conditions on the existence of a complete noncompact convex hypersurface whose Gauss curvature is equal to a prescribed function on the unit sphere. By the Legendre transform this problem is reduced to the solvability of the Monge-Ampère equation subject to certain boundary conditions.

Multiplicity results for both periodic and anti-periodic forced oscillations for the pendulum equation are given on the background of a detailed exposition of the results related to the so-called problem of Moser and to the Mawhin conjecture.