Under the assumption that a residually finite-dimensional Hopf algebra H has an Artinian ring of fractions, it is proved flat module over any right coideal subalgebra satisfying polynomial identity and faithfully subalgebra. As consequence we find large class algebras which are all subalgebras subalgebras.

In the present study, the frontal and facial features of 112 populations from around the world are compared in terms of frontal and facial flatness measurements. Univariate analyses and canonical correlation analysis were applied to six indices representing flatness of frontal and facial bones. The deep infraglabellar notch, marked prognathism, and flat frontal bone show distinctive Australian/...

The evolution of a universe with Brans-Dicke gravity and nonzero curvature is investigated here. We present the equations of motion and their solutions during the radiation dominated era. In a Friedman-Robertson-Walker cosmology we show explicitly that the three possible values of curvature κ = +1, 0,−1 divide the evolution of the Brans-Dicke universe into dynamically distinct classes just as f...