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Distant perturbations of the Laplacian in a multi-dimensional space
We consider the Laplacian in perturbed by a finite number of
distant perturbations those are abstract localized operators. We study the
asymptotic behaviour of the discrete spectrum as the distances between
perturbations tend to infinity. The main results are the convergence theorem
and the asymptotics expansions for the eigenelements. Some examples of the
possible distant perturbations are given; they are potential, second order
differential operator, magnetic Schrodinger operator, integral operator, and
\d-potential
Ultrafilter extensions of linear orders
It was recently shown that arbitrary first-order models canonically extend to
models (of the same language) consisting of ultrafilters. The main precursor of
this construction was the extension of semigroups to semigroups of
ultrafilters, a technique allowing to obtain significant results in algebra and
dynamics. Here we consider another particular case where the models are
linearly ordered sets. We explicitly calculate the extensions of a given linear
order and the corresponding operations of minimum and maximum on a set. We show
that the extended relation is not more an order however is close to the natural
linear ordering of nonempty half-cuts of the set and that the two extended
operations define a skew lattice structure on the set of ultrafilters
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