476,164 research outputs found

    On quantum vertex algebras and their modules

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    We give a survey on the developments in a certain theory of quantum vertex algebras, including a conceptual construction of quantum vertex algebras and their modules and a connection of double Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras.Comment: 18 pages; contribution to the proceedings of the conference in honor of Professor Geoffrey Maso

    Modules-at-infinity for quantum vertex algebras

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    This is a sequel to \cite{li-qva1} and \cite{li-qva2} in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian DY(sl2)DY_{\hbar}(sl_{2}), denoted by DYq(sl2)DY_{q}(sl_{2}) and DYq(sl2)DY_{q}^{\infty}(sl_{2}) with qq a nonzero complex number. For each nonzero complex number qq, we construct a quantum vertex algebra VqV_{q} and prove that every DYq(sl2)DY_{q}(sl_{2})-module is naturally a VqV_{q}-module. We also show that DYq(sl2)DY_{q}^{\infty}(sl_{2})-modules are what we call VqV_{q}-modules-at-infinity. To achieve this goal, we study what we call §\S-local subsets and quasi-local subsets of \Hom (W,W((x^{-1}))) for any vector space WW, and we prove that any §\S-local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with WW as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.Comment: Latex, 48 page

    Empirical risk minimization as parameter choice rule for general linear regularization methods.

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    We consider the statistical inverse problem to recover f from noisy measurements Y = Tf + sigma xi where xi is Gaussian white noise and T a compact operator between Hilbert spaces. Considering general reconstruction methods of the form (f) over cap (alpha) = q(alpha) (T*T)T*Y with an ordered filter q(alpha), we investigate the choice of the regularization parameter alpha by minimizing an unbiased estiate of the predictive risk E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. The corresponding parameter alpha(pred) and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk E[parallel to f - (f) over cap (alpha pred)parallel to(2)] with the oracle prediction risk inf(alpha>0) E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense. Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations

    Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model

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    Owing to its conceptual simplicity and computational efficiency, the pseudopotential multiphase lattice Boltzmann (LB) model has attracted significant attention since its emergence. In this work, we aim to extend the pseudopotential LB model to simulate multiphase flows at large density ratio and relatively high Reynolds number. First, based on our recent work [Li et al., Phys. Rev. E. 86, 016709 (2012)], an improved forcing scheme is proposed for the multiple-relaxation-time pseudopotential LB model in order to achieve thermodynamic consistency and large density ratio in the model. Next, through investigating the effects of the parameter a in the Carnahan-Starling equation of state, we find that the interface thickness is approximately proportional to 1/sqrt(a). Using a smaller a will lead to a wider interface thickness, which can reduce the spurious currents and enhance the numerical stability of the pseudopotential model at large density ratio. Furthermore, it is found that a lower liquid viscosity can be gained in the pseudopotential model by increasing the kinematic viscosity ratio between the vapor and liquid phases. The improved pseudopotential LB model is numerically validated via the simulations of stationary droplet and droplet oscillation. Using the improved model as well as the above treatments, numerical simulations of droplet splashing on a thin liquid film are conducted at a density ratio in excess of 500 with Reynolds numbers ranging from 40 to 1000. The dynamics of droplet splashing is correctly reproduced and the predicted spread radius is found to obey the power law reported in the literature.Comment: 9 figures, 2 tables, accepted by Physical Review E (in press

    Incubators vs Zombies: Fault-Tolerant, Short, Thin and Lanky Spanners for Doubling Metrics

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    Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed by Arya et al. in a seminal STOC 1995 paper. However, Elkin and Solomon's spanner construction is extremely complicated; we offer a simple alternative construction that is very intuitive and is based on the standard technique of net tree with cross edges. Indeed, our approach can be readily applied to our previous construction of k-fault tolerant spanners (ICALP 2012) to achieve k-fault tolerance, maximum degree O(k^2), hop-diameter O(log n) and lightness O(k^3 log n)