A simple proof of the Gauss-Bonnet-Chern formula for Finsler manifolds

From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey

A Gauss-Bonnet-Chern theorem for Finsler vector bundles

In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any metric-compatible connection is established on Finsler manifolds

A comparison theorem for Finsler submanifolds and its applications

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained

Integral curvature bounds and diameter estimates on Finsler manifolds

In this paper, we study the integral curvatures of Finsler manifolds and prove several Myers type theorems

A Gauss-Bonnet-Chern theorem for complex Finsler manifolds

In this paper, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds

An empirical Bayes testing procedure for detecting variants in analysis of next generation sequencing data

Because of the decreasing cost and high digital resolution, next-generation sequencing (NGS) is expected to replace the traditional hybridization-based microarray technology. For genetics study, the first-step analysis of NGS data is often to identify genomic variants among sequenced samples. Several statistical models and tests have been developed for variant calling in NGS study. The existing approaches, however, are based on either conventional Bayesian or frequentist methods, which are unable to address the multiplicity and testing efficiency issues simultaneously. In this paper, we derive an optimal empirical Bayes testing procedure to detect variants for NGS study. We utilize the empirical Bayes technique to exploit the across-site information among many testing sites in NGS data. We prove that our testing procedure is valid and optimal in the sense of rejecting the maximum number of nonnulls while the Bayesian false discovery rate is controlled at a given nominal level. We show by both simulation studies and real data analysis that our testing efficiency can be greatly enhanced over the existing frequentist approaches that fail to pool and utilize information across the multiple testing sites.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS660 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiresolution topological simplification

Persistent homology has been devised as a promising tool for the topological simplification of complex data. However, it is computationally intractable for large data sets. In this work, we introduce multiresolution persistent homology for tackling large data sets. Our basic idea is to match the resolution with the scale of interest so as to create a topological microscopy for the underlying data. We utilize flexibility-rigidity index (FRI) to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution, we are able to focus the topological lens on a desirable scale. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA and RNA molecules. In particular, the topological persistence of a virus capsid with 240 protein monomers is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks and graphs.Comment: 22 pages and 14 figure

Tensor train-Karhunen-Lo\`eve expansion for continuous-indexed random fields using higher-order cumulant functions

The goals of this work are two-fold: firstly, to propose a new theoretical framework for representing random fields on a large class of multidimensional geometrical domain in the tensor train format; secondly, to develop a new algorithm framework for accurately computing the modes and the second and third-order cumulant tensors within moderate time. The core of the new theoretical framework is the tensor train decomposition of cumulant functions. This decomposition is accurately computed with a novel rank-revealing algorithm. Compared with existing Galerkin-type and collocation-type methods, the proposed computational procedure totally removes the need of selecting the basis functions or collocation points and the quadrature points, which not only greatly enhances adaptivity, but also avoids solving large-scale eigenvalue problems. Moreover, by computing with third-order cumulant functions, the new theoretical and algorithm frameworks show great potential for representing general non-Gaussian non-homogeneous random fields. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the efficiency and accuracy of the proposed algorithm framework

Revised Iterative Solution of Ground State of Double-Well Potential

A revised new iterative method based on Green function defined by quadratures along a single trajectory is developed and applied to solve the ground state of the double-well potential. The result is compared to the one based on the original iterative method. The limitation of the asymptotic expansion is also discussed.Comment: 19 page

Revised Iterative Solution for Groundstate of Schroedinger Equation

A revised iterative method based on Green function defined by quadratures along a single trajectory is proposed to solve the low-lying quantum wave function for Schroedinger equation. Specially a new expression of the perturbed energy is obtained, which is much simpler than the traditional one. The method is applied to solve the unharmonic oscillator potential. The revised iteration procedure gives exactly the same result as those based on the single trajectory quadrature method. A comparison of the revised iteration method to the old one is made using the example of Stark effect. The obtained results are consistent to each other after making power expansion