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