In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the HausdorffYoung inequality, Young's inequality, the HirschmanBeckner uncertainty principle, the DonohoStark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardy's uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's <i></i>-lattices, modular tensor categories, etc.

In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

With the rapid development of internet economy, transparent logistics is stepping into a prosperity period with massive transportation data generated and collected every day. In this paper, we focus on the segmentation of GPS trajectory data generated in logistics transportation to analyze the vehicle behaviors and extract business affair information according to the vehicle behavior characteristics, which is challenging due to the complexity of trajectory data and unavailability of road information. We extract the stopping points from the trajectory data sequence based on the duration of nonmovement, and construct business time window and electronic fence by analyzing the driving habits of vehicles. Furthermore, we propose a probabilistic logic based data segmentation method (PLDSM) which not only helps finding all the business points but also assists in inferring the business affair categories. An efficient

In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. In Cheng etal. (2013) a dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within the framework of generalized multiscale finite element method (GMsFEM) for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical