5,202 research outputs found
The structure of decomposition of a triconnected graph
We describe the structure of triconnected graph with the help of its
decomposition by 3-cutsets. We divide all 3-cutsets of a triconnected graph
into rather small groups with a simple structure, named complexes. The detailed
description of all complexes is presented. Moreover, we prove that the
structure of a hypertree could be introduced on the set of all complexes. This
structure gives us a complete description of the relative disposition of the
complexes.
Keywords: connectivity, triconneted graphs.Comment: 49 pages, 8 figures. Russian version published in Zap. Nauchn. Sem.
POMI v.391 (2011), http://www.pdmi.ras.ru/znsl/2011/v391/abs090.htm
Three-block exceptional collections over Del Pezzo surfaces
We study complete exceptional collections of coherent sheaves over Del Pezzo
surfaces, which consist of three blocks such that inside each block all Ext
groups between the sheaves are zero. We show that the ranks of all sheaves in
such a block are the same and the three ranks corresponding to a complete
3-block exceptional collection satisfy a Markov-type Diophantine equation that
is quadratic in each variable. For each Del Pezzo surface, there is a finite
number of these equations; the complete list is given. The 3-string braid group
acts by mutations on the set of complete 3-block exceptional collections. We
describe this action. In particular, any orbit contains a 3-block collection
with the sum of ranks that is minimal for the solutions of the corresponding
Markov-type equation, and the orbits can be obtained from each other via
tensoring by an invertible sheaf and with the action of the Weyl group. This
allows us to compute the number of orbits up to twisting.Comment: LaTex v2.09, 32 pages with 1 figure. To appear in Izvestiya Mat
Polariton condensation in photonic crystals with high molecular orientation
We study Frenkel exciton-polariton Bose-Einstein condensation in a
two-dimensional defect-free triangular photonic crystal with an organic
semiconductor active medium containing bound excitons with dipole moments
oriented perpendicular to the layers. We find photonic Bloch modes of the
structure and consider their strong coupling regime with the excitonic
component. Using the Gross- Pitaevskii equation for exciton polaritons and the
Boltzmann equation for the external exciton reservoir, we demonstrate the
formation of condensate at the points in reciprocal space where photon group
velocity equals zero. Further, we demonstrate condensation at non-zero momentum
states for TM-polarized photons in the case of a system with incoherent
pumping, and show that the condensation threshold varies for different points
in the reciprocal space, controlled by detuning.Comment: 6 pages, 4 figure
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