Extensions of simple modules over Leavitt path algebras

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\infty$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[c^\infty]}$. Further, when $E$ is row-finite, for each irrational infinite path $p$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[p]}$. For Chen simple modules $S,T$ we describe ${\rm Ext}_{L_K(E)}^1(S,T)$ by presenting an explicit $K$-basis. For any graph $E$ containing at least one cycle, this description guarantees the existence of indecomposable left $L_K(E)$-modules of any prescribed finite length.Comment: updated: dedication to Alberto Facchini on the occasion of his 60th Birthday added in front matte

How Much Does Money Matter in a Direct Democracy?

The fine-structure splitting of quantum confined InxGa1-x Nexcitons is investigated using polarization-sensitive photoluminescence spectroscopy. The majority of the studied emission lines exhibits mutually orthogonal fine-structure components split by 100-340 mu eV, as measured from the cleaved edge of the sample. The exciton and the biexciton reveal identical magnitudes but reversed sign of the energy splitting.Original Publication:Supaluck Amloy, Y T Chen, K F Karlsson, K H Chen, H C Hsu, C L Hsiao, L C Chen and Per-Olof Holtz, Polarization-resolved fine-structure splitting of zero-dimensional InxGa1-xN excitons, 2011, PHYSICAL REVIEW B, (83), 20, 201307.http://dx.doi.org/10.1103/PhysRevB.83.201307Copyright: American Physical Societyhttp://www.aps.org

On a theorem of K T Chen

Proving normal form of mappings of real line into itself by contracting mapping principl

Analysis of Quark Mixing Using Binary Tetrahedral Flavor Symmetry

Using the binary tetrahedral group $T^{'}$, the three angles and phase of the quark CKM mixing matrix are pursued by symmetry-breaking which involves $T^{'}$-doublet VEVs and the Chen-Mahanthappa CP-violation mechanism. The NMRT$^{'}$M, Next-to-Minimal-Renormalizable -T$^{'}$-Model is described, and its one parameter comparison to experimental data is explored.Comment: 14 pages latex. Two .eps figures include