No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds X1 and X2 of dimension 2n􀀀 1, is there a non-constant CR morphism between them? In this paper, we use Kohn{Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specically, if dimHp;q KR(X1) < dimHp;q KR(X2) for any (p; q) with 1  q  n 􀀀 2, then there is no non-constant CR morphism from X1 to X2.

The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a potential and a vector potential coupled to the loose edges by a coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense.