  As a significant strengthening of properties of earlier algebras of
generalized functions, here are presented classes of such algebras which can
deal with dense singularities. In fact, the cardinal of the set of singular
points can be larger than that of the nonsingular points. This large class of
singularities allows the solution of large classes of smooth nonlinear PDEs.
This in certain ways overcomes the celebrated 1957 H. Lewy impossibility
result.
