We study the integral Hasse principle for affine varieties of shape a x 2 + y z − = m , using Brauer–Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements. describe these elements explicitly, in some cases, show that there is no obstruction to them.

In this paper, we prove a theorem on local existence and uniqueness of integral solutions to a class of partial neutral functional differential equations with infinite delay. Our method of proof is based on the integrated semigroup theory and the well known Banach fixed point theorem.

Sharp smoothing estimates are proven for magnetic Schrr odinger semigroups in two dimensions under the assumption that the magnetic eld is bounded below by some positive constant B 0. As a consequence the L 1 norm of the associated integral kernel is bounded by the L 1 norm of the Mehler kernel of the Schrr odinger semigroup with the constant magnetic eld B 0 .