1,619 research outputs found
Integral geometry, hypergroups, and I.M. Gelfand's question
This note is an attempt to give an answer for the following old I.M.
Gelfand's question: why some important problems of integral geometry (e.g., the
Radon transform and others) are related to harmonic analysis on groups, but for
other quite similar problems such relations are not clear? In the note we
examine standard problems of integral geometry generating harmonic analysis
(the Plancherel theorem etc.) on pairs of commutative hypergroups that are in a
duality of Pontryagin's type. As a result new meaningful examples of
hypergroups are constructed.Comment: 10 pages, to be published in Doklady Mathematics, 201
Indirect coupling between spins in semiconductor quantum dots
The optically induced indirect exchange interaction between spins in two
quantum dots is investigated theoretically. We present a microscopic
formulation of the interaction between the localized spin and the itinerant
carriers including the effects of correlation, using a set of canonical
transformations. Correlation effects are found to be of comparable magnitude as
the direct exchange. We give quantitative results for realistic quantum dot
geometries and find the largest couplings for one dimensional systems.Comment: 4 pages, 3 figure
Cyclic projectors and separation theorems in idempotent convex geometry
Semimodules over idempotent semirings like the max-plus or tropical semiring
have much in common with convex cones. This analogy is particularly apparent in
the case of subsemimodules of the n-fold cartesian product of the max-plus
semiring it is known that one can separate a vector from a closed subsemimodule
that does not contain it. We establish here a more general separation theorem,
which applies to any finite collection of closed semimodules with a trivial
intersection. In order to prove this theorem, we investigate the spectral
properties of certain nonlinear operators called here idempotent cyclic
projectors. These are idempotent analogues of the cyclic nearest-point
projections known in convex analysis. The spectrum of idempotent cyclic
projectors is characterized in terms of a suitable extension of Hilbert's
projective metric. We deduce as a corollary of our main results the idempotent
analogue of Helly's theorem.Comment: 20 pages, 1 figur
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