56 research outputs found
A minimax inequality and its applications
A minimax inequality of Ky Fan for mapping with non-compact domain is given. Moreover, some minimax inequalities are obtained.Publisher's Versio
Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems
This is an updated and expanded version of our earlier survey article
\cite{Gut5}. Section introduces the subject matter. Sections expose the basic material following the paradigm of elliptic, hyperbolic and
parabolic billiard dynamics. In section we report on the recent work
pertaining to the problems and conjectures exposed in the survey \cite{Gut5}.
Besides, in section we formulate a few additional problems and
conjectures. The bibliography has been updated and considerably expanded
On compression of Bruhat-Tits buildings
We obtain an analog of the compression of angles theorem in symmetric spaces
for Bruhat--Tits buildings of the type .
More precisely, consider a -adic linear space and the set of
all lattices in . The complex distance in is a complete system of
invariants of a pair of points of under the action of the complete
linear group. An element of a Nazarov semigroup is a lattice in the duplicated
linear space . We investigate behavior of the complex distance under
the action of the Nazarov semigroup on the set .Comment: 6 page
Minimax inequality equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz Theorem
The purpose of this note is to give further generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional and to show that our minimax inequalities are equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and a modified FKKM theorem given in this note
Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in
We investigate a class of generalized Schr\"{o}dinger operators in
with a singular interaction supported by a smooth curve
. We find a strong-coupling asymptotic expansion of the discrete
spectrum in case when is a loop or an infinite bent curve which is
asymptotically straight. It is given in terms of an auxiliary one-dimensional
Schr\"{o}dinger operator with a potential determined by the curvature of
. In the same way we obtain an asymptotics of spectral bands for a
periodic curve. In particular, the spectrum is shown to have open gaps in this
case if is not a straight line and the singular interaction is strong
enough.Comment: LaTeX 2e, 30 pages; minor improvements, to appear in Rev. Math. Phy
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