Hyperbolic periodic points for chain hyperbolic homoclinic classes

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.Comment: 15 pages, 1 figure

Multi-stage Suture Detection for Robot Assisted Anastomosis based on Deep Learning

In robotic surgery, task automation and learning from demonstration combined with human supervision is an emerging trend for many new surgical robot platforms. One such task is automated anastomosis, which requires bimanual needle handling and suture detection. Due to the complexity of the surgical environment and varying patient anatomies, reliable suture detection is difficult, which is further complicated by occlusion and thread topologies. In this paper, we propose a multi-stage framework for suture thread detection based on deep learning. Fully convolutional neural networks are used to obtain the initial detection and the overlapping status of suture thread, which are later fused with the original image to learn a gradient road map of the thread. Based on the gradient road map, multiple segments of the thread are extracted and linked to form the whole thread using a curvilinear structure detector. Experiments on two different types of sutures demonstrate the accuracy of the proposed framework.Comment: Submitted to ICRA 201

Bayesian Conditional Tensor Factorizations for High-Dimensional Classification

In many application areas, data are collected on a categorical response and high-dimensional categorical predictors, with the goals being to build a parsimonious model for classification while doing inferences on the important predictors. In settings such as genomics, there can be complex interactions among the predictors. By using a carefully-structured Tucker factorization, we define a model that can characterize any conditional probability, while facilitating variable selection and modeling of higher-order interactions. Following a Bayesian approach, we propose a Markov chain Monte Carlo algorithm for posterior computation accommodating uncertainty in the predictors to be included. Under near sparsity assumptions, the posterior distribution for the conditional probability is shown to achieve close to the parametric rate of contraction even in ultra high-dimensional settings. The methods are illustrated using simulation examples and biomedical applications

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry

We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.Comment: v5; 49 pages, version to appear in Advances in High Energy Physic