Vertex Operator Algebra Analogue of Embedding D8D_8 into E8E_8

Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{{\mathbf g}}_{D_{8}}$ and $\widehat{{\mathbf g}}_{E_8}$ with level 1 respectively. In the 1980s by I. Frenkel, Lepowsky and Meurman as one of the many important preliminary steps toward their construction of the moonshine module vertex operator algebra, they use roots lattice showing that $L_{D_8}(1, 0)$ can embed into $L_{E_8}(1, 0)$ as a vertex operator subalgebra(\cite{5, 6, 8}). Their construct is a base of vertex operator theory. But the embedding they gave using the fact $L_{\mathbf g}(1,0)$ is isomorphic to its root lattice vertex operator algebra $V_L$. In this paper, we give an explicitly construction of the embedding and show that as an $L_{D_8}(1, 0)$-module, $L_{E_8}(1, 0)$ is isomorphic to the extension of $L_{D_8}(1, 0)$ by its simple module $L_{D_8}(1, \overline{\omega}_8)$. It may be convenient to be used for conformal field theory.Comment: The abstrat and section 1 are modified. We give more informations about the embeddin

Cluster algebra structure on the finite dimensional representations of Uq(A3^)U_q(\widehat{A_{3}}) for ll=2

In this paper, we prove one case of the conjecture given by Hernandez and Leclerc\cite{HL0}. Specifically, we give a cluster algebra structure on the Grothendieck ring of a full subcategory of the finite dimensional representations of a simply-laced quantum affine algebra U_q(\widehat{\g}). In the procedure, we also give a specific description of compatible subsets of type $E_{6}$. As a conclusion, for every exchange relation of cluster algebra there exists a exact sequence of the full subcategory corresponding to it.Comment: 58 pages, 2 figures. Correct some typoes and add a new example. arXiv admin note: text overlap with arXiv:0903.1452 by other author

Embedded connectivity of recursive networks

Let $G_n$ be an $n$-dimensional recursive network. The $h$-embedded connectivity $\zeta_h(G_n)$ (resp. edge-connectivity $\eta_h(G_n)$) of $G_n$ is the minimum number of vertices (resp. edges) whose removal results in disconnected and each vertex is contained in an $h$-dimensional subnetwork $G_h$. This paper determines $\zeta_h$ and $\eta_h$ for the hypercube $Q_n$ and the star graph $S_n$, and $\eta_3$ for the bubble-sort network $B_n$

End-to-End Neural Ranking for eCommerce Product Search: an application of task models and textual embeddings

We consider the problem of retrieving and ranking items in an eCommerce catalog, often called SKUs, in order of relevance to a user-issued query. The input data for the ranking are the texts of the queries and textual fields of the SKUs indexed in the catalog. We review the ways in which this problem both resembles and differs from the problems of IR in the context of web search. The differences between the product-search problem and the IR problem of web search necessitate a different approach in terms of both models and datasets. We first review the recent state-of-the-art models for web search IR, distinguishing between two distinct types of model which we call the distributed type and the local-interaction type. The different types of relevance models developed for IR have complementary advantages and disadvantages when applied to eCommerce product search. Further, we explain why the conventional methods for dataset construction employed in the IR literature fail to produce data which suffices for training or evaluation of models for eCommerce product search. We explain how our own approach, applying task modeling techniques to the click-through logs of an eCommerce site, enables the construction of a large-scale dataset for training and robust benchmarking of relevance models. Our experiments consist of applying several of the models from the IR literature to our own dataset. Empirically, we have established that, when applied to our dataset, certain models of local-interaction type reduce ranking errors by one-third compared to the baseline tf-idf. Applied to our dataset, the distributed models fail to outperform the baseline. As a basis for a deployed system, the distributed models have several advantages, computationally, over the local-interaction models. This motivates an ongoing program of work, which we outline at the conclusion of the paper.Comment: Accepted to appear at the SIGIR 2018 workshop on eCommerc

Quantum cluster algebra structure on the finite dimensional representations of Uq(sl2^)U_q(\widehat{sl_{2}})

In this paper, we give a quantum cluster algebra structure on the deformed Grothendieck ring of \CC_{n}, where \CC_{n} is a full subcategory of finite dimensional representations of $U_q(\widehat{sl_{2}})$ defined in section II.Comment: 11pages, no figure

On small set of one-way LOCC indistinguishability of maximally entangled states

In this paper, we study the one-way local operations and classical communication (LOCC) problem. In $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d\geq4$, we construct a set of $3\lceil\sqrt{d}\rceil-1$ one-way LOCC indistinguishable maximally entangled states which are generalized Bell states. Moreover, we show that there are four maximally entangled states which cannot be perfectly distinguished by one-way LOCC measurements for any dimension $d\geq 4$.Comment: 10 pages.Very recently, we became aware of related work \cite{Zhang2} in which the same $\lceil\frac{d}{2}\rceil+2$ states in $\mathbb{C}^d\otimes\mathbb{C}^d$ is proved to be one-way LOCC indistinguishable. arXiv admin note: text overlap with arXiv:1310.4220 by other autho

A Vertex Algebra Commutant for the βγ\beta\gamma-System and Howe pairs

Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation $V$ of Lie algebra sl(2,\C), we describe a commutant of $\beta\gamma$- System $S(V)$ by giving its generators, moreover, we get a new Howe pair of vertex algebras.Comment: 15 pages, No figures. This is our first paper of series on this topi

The radio environment of the 21 Centimeter Array: RFI detection and mitigation

Detection and mitigation of radio frequency interference (RFI) is the first and also the key step for data processing in radio observations, especially for ongoing low frequency radio experiments towards the detection of the cosmic dawn and epoch of reionization (EoR). In this paper we demonstrate the technique and efficiency of RFI identification and mitigation for the 21 Centimeter Array (21CMA), a radio interferometer dedicated to the statistical measurement of EoR. For terrestrial, man-made RFI, we concentrate mainly on a statistical approach by identifying and then excising non-Gaussian signatures, in the sense that the extremely weak cosmic signal is actually buried under thermal and therefore Gaussian noise. We also introduce the so-called visibility correlation coefficient instead of conventional visibility, which allows a further suppression of rapidly time-varying RFI. Finally, we briefly discuss removals of the sky RFI, the leakage of sidelobes from off-field strong radio sources with time-invariant power and a featureless spectrum. It turns out that state of the art technique should allow us to detect and mitigate RFI to a satisfactory level in present low frequency interferometer observations such as those acquired with the 21CMA, and the accuracy and efficiency can be greatly improved with the employment of low-cost, high-speed computing facilities for data acquisition and processing.Comment: 18 pages, 14 figure

Constructing unextendible product bases from the old ones

We studied the construction problem of the unextendible product basis (UPB). We mainly give a method to construct a UPB of a quantum system through the UPBs of its subsystem. Using this method and the UPBs which are known for us, we construct different kinds of UPBs in general bipartite quantum system. Then we use these UPBs to construct a family of UPBs in multipartite quantum system. The UPBs can be used to construct the bound entangled states with different ranks.Comment: 6 pages, 1 figure

The local distinguishability of any three generalized Bell states

We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable.Comment: 11 pages, 1 figure, accepted by Quantum Information Processin