The family of rings the form
 \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \rangle}
 is investigated which contains generalized Hamilton quaternions over $\Z_4$. These are local order 256. This has 256 contained in 88 distinct isomorphism classes. Of non-isomorphic rings, 10 minimal reversible nonsymmetric and 21 abelian reflexive nonsem...

The two properties of the radial mass distribution a gravitational lens that are well-constrained by Einstein rings radius R_E and xi2 = alpha''(R_E)/(1-kappa_E), where alpha''(R_E) kappa_E second derivative deflection profile convergence at R_E. However, if there is tight mathematical relationship between angular structure, as true ellipsoids, an ring can appear to strongly distinguish distrib...