Norm estimates and asymptotic faithfulness of the quantum SU(n)SU(n) representations of the mapping class groups

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in [10] using skein theory.Comment: Geometriae Dedicata (online), 10 pages, minor change

A Bayesian approach to parameter estimation for kernel density estimation via transformations

In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibit a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel choice. In the current literature, there have been some developments in the area of estimating densities based on transformed data, but bandwidth selection depends on pre-determined transformation parameters. Moreover, in the bivariate situation, each dimension is considered separately and the correlation between the two dimensions is largely ignored. We extend the Bayesian sampling algorithm proposed by Zhang, King and Hyndman (2006) and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is well captured through the bivariate density estimator based on transformed data.Bandwidth parameter; kernel density estimator; Markov chain Monte Carlo; Metropolis-Hastings algorithm; power transformation; transformation parameter.

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Let π: X→ M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over X. We obtain the asymptotic of the curvature of L -metric and Qullien metric on the direct image bundle π (L ⊗ K ) up to the lower order terms than k , for large k. As an application we prove that the analytic torsion τ (∂\uaf) satisfies ∂∂\uaflog(τk(∂\uaf))2=o(kn-1), where n is the dimension of fibers