Heisenberg double of the generalized quantum euclidean group and its representations

The generalized quantum Euclidean group \oq(\frak{b}_{m,n}) is a natural generalization of the quantum Euclidean group \oq(\frak{b}_{1,1}). The Heisenberg double \od(\frak{b}_{m,n}) of \oq(\frak{b}_{m,n}) is the smash product of \oq(\frak{b}_{m,n}) with its Hopf dual \ou(\frak{b}_{m,n}). In this paper, we study the weight modules, the prime spectrum and the automorphism group of the Heisenberg double \od(\frak{b}_{m,n}).Comment: 11pages. comments are welcom

BL-MNE: Emerging Heterogeneous Social Network Embedding through Broad Learning with Aligned Autoencoder

Network embedding aims at projecting the network data into a low-dimensional feature space, where the nodes are represented as a unique feature vector and network structure can be effectively preserved. In recent years, more and more online application service sites can be represented as massive and complex networks, which are extremely challenging for traditional machine learning algorithms to deal with. Effective embedding of the complex network data into low-dimension feature representation can both save data storage space and enable traditional machine learning algorithms applicable to handle the network data. Network embedding performance will degrade greatly if the networks are of a sparse structure, like the emerging networks with few connections. In this paper, we propose to learn the embedding representation for a target emerging network based on the broad learning setting, where the emerging network is aligned with other external mature networks at the same time. To solve the problem, a new embedding framework, namely "Deep alIgned autoencoder based eMbEdding" (DIME), is introduced in this paper. DIME handles the diverse link and attribute in a unified analytic based on broad learning, and introduces the multiple aligned attributed heterogeneous social network concept to model the network structure. A set of meta paths are introduced in the paper, which define various kinds of connections among users via the heterogeneous link and attribute information. The closeness among users in the networks are defined as the meta proximity scores, which will be fed into DIME to learn the embedding vectors of users in the emerging network. Extensive experiments have been done on real-world aligned social networks, which have demonstrated the effectiveness of DIME in learning the emerging network embedding vectors.Comment: 10 pages, 9 figures, 4 tables. Full paper is accepted by ICDM 2017, In: Proceedings of the 2017 IEEE International Conference on Data Mining

Simple smooth modules over the superconformal current algebra

In this paper, we classify simple smooth modules over the superconformal current algebra $\frak g$. More precisely, we first classify simple smooth modules over the Heisenberg-Clifford algebra, and then prove that any simple smooth $\frak g$-module is a tensor product of such modules for the super Virasoro algebra and the Heisenberg-Clifford algebra, or an induced module from a simple module over some finite-dimensional solvable Lie superalgebras. As a byproduct, we provide characterizations for both simple highest weight $\frak g$-modules and simple Whittaker $\frak g$-modules. Additionally, we present several examples of simple smooth $\frak g$-modules that are not tensor product of modules over the super Virasoro algebra and the Heisenberg-Clifford algebra.Comment: Latex, 30pages, comments are welcome

An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method