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We investigate whether a Stein manifold$M$which allows proper holomorphic embedding into ℂ^{$n$}can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of$M$is smaller than$n$/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.

The validity of using the tight-binding approximation for the nonlinear Schrödinger equations with a two-dimensional optical lattice is considered. This work provides a rigorous foundation for a technique based on “orbital” functions that is central to solid-state physics and nonlinear optics. Simple and honeycomb lattices are addressed, and it is therefore shown that the use of tight-binding approximations is justified in complicated situations.

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