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Stochastic Completeness of Graphs We analyze the stochastic completeness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal follows by studying a specific family of trees.… (More)
We study an example of an index problem for a Dirac-like operator subject to Atiyah-Patodi-Singer boundary conditions on a noncommu-tative manifold with boundary, namely the quantum unit disk.
We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold M split along a hypersurface Σ (M = X ∪Σ Y). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula takes the form SF (D) = SF (D |X , BX) + SF (D |Y… (More)
We review the work of the authors and their collaborators on the decomposition of the ζ-determinant of the Dirac operator into the contributions coming from different parts of a manifold.
We survey geometric properties which imply the stochastic incom-pleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of… (More)
Anilines react with 5-nitroindoles in the presence of t-BuOK in N,N-dimethylformamide (DMF) to form 5-nitroso-4-arylaminoindoles that in turn when treated with N,O-bis(trimethylsilyl)acetamide cyclize to pyrrolo[3,2-a]phenazines. In an alternative approach pyr-rolo[3,2-a]phenazines are formed from aminoindoles and nitroarenes.
Alkylation of 5-nitroindol-4-ylacetonitriles with ethyl chloroacetate, α-halomethyl ketones, and chloroacetonitrile followed by a treatment of the products with chlorotrimethylsilane in the presence of DBU gives 1-cyanopyrrolo[3,2-e]indoles substituted in position 2 with electron-withdrawing groups.