We study finite dimensional representations over some Noetherian algebras a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category locally to be closed under taking injective hulls extend results known group rings enveloping Ore extensions, Hopf crossed products, affine low Gelfand-Kirillov dimension.

In this paper, we consider two typical problems on a locally finite connected graph. The first one is to study the Bochner formula for Laplacian operator other obtain global nontrivial nonnegative solution porous-media equation via use of Aronson-Benilan argument. We curvature dimension condition give characterization point also criterion about stochastic completeness There not much work in dir...