Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(\Lambda_0)$ of ${\mathscr D}(\Delta_n)$ of the cyclic quiver $\Delta_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $\Delta_n$, we obtain a decomposition of the basic representation $L(\Lambda_0)$ which induces the Kashiwara--Miwa--Stern decomposition of $\bigwedge^\infty$ and a construction of the canonical basis of $\bigwedge^\infty$ defined by Leclerc and Thibon in terms of certain monomial basis elements in ${\mathscr D}(\Delta_n)$.

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We study Legendrian embeddings of a compact Legendrian submanifold L sitting in a closed contact manifold (M,ξ) whose contact structure is supported by a (contact) open book OB on M. We prove that if OB has Weinstein pages, then there exist a contact structure ξ′ on M, isotopic to ξ and supported by OB, and a contactomorphism f:(M,ξ)→(M,ξ′) such that the image f(L) of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of OB.

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrdinger equation. Traditional Gaussian beam type methods for the Schrdinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al.[S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrdinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in$B$-regular domain. This theorem implies that the two classes of Jensen measures coincide in$B$-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.