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No abstract uploaded!

Results are obtained on the scattering theory for the Schrödinger equation $$i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$$ in spaces$L$^{$r$}($R$;$L$^{$q$}($R$^{$d$})) for a certain range of$r, q$, the so-called space-time scattering. In the linear case (i.e.$F$≡)) the relation with usual configuration space scattering is established.

We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces in$R$^{$n$}. The resulting conditions are close to optimal.

We prove a second fundamental theorem in the sense of Nevanlinna's theory of meromorphic functions replacing the constants$a$in $$\bar N$$ ($r, f, a$) by rational functions$R$with$R(∞)=a$. The key argument is Ahlfors's second fundamental theorem from his theory of covering surfaces.