Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian

Hyperbolicity, irrationality exponents and the eta invariant

We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a logarithmic improvement of the remainder for Anosov Reeb flows, while for certain elliptic flows the improvement is in terms of irrationality measures of corresponding Floquet exponents

Use of data mining techniques for the analysis of consumer’s electricity consumption over a year in particular region

Data mining techniques are used to discover electricity consumption pattern at regional level in a city and used to extract knowledge concerning to the electricity consumption with respect to atmospheric temperature and physical distance from geographic features like river, farm, ground and highway. The demand for electricity keeps on increasing almost every year and almost in every region. Installing and developing such a new electricity generation plants is practically impossible due to environmental preservation awareness and pollution control policies of government. To deal with such situation, one needs to find the optional ways for handling and management electricity loads in future with current capacity of electricity generation

Geometric quantization results for semi-positive line bundles on a Riemann surface

In earlier work the authors proved the Bergman kernel expansion for semipositive line bundles over a Riemann surface whose curvature vanishes to atmost finite order at each point. Here we explore the related results and consequences of the expansion in the semipositive case including: Tian's approximation theorem for induced Fubini-Study metrics, leading order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections and the asymptotics of holomorphic torsion.Comment: arXiv admin note: substantial text overlap with arXiv:1811.0099

K\"ahler-Einstein Bergman metrics on pseudoconvex domains of dimension two

We prove that a two dimensional pseudoconvex domain of finite type with a K\"ahler-Einstein Bergman metric is biholomorphic to the unit ball. This answers an old question of Yau for such domains. The proof relies on asymptotics of derivatives of the Bergman kernel along critically tangent paths approaching the boundary, where the order of tangency equals the type of the boundary point being approached