Hall algebras of cyclic quivers and qq-deformed Fock spaces

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 ${\mathscr D}(\Delta_n)$

Meson Decays in an Extended Nambu--Jona-Lasinio model with Heavy Quark Flavors

In a previous work, we proposed an extended Nambu--Jona-Lasinio (NJL) model including heavy quark flavors. In this work, we will calculate strong and radiative decays of vector mesons in this extended NJL model, including light $\rho$, $\omega$, $K^*$, $\phi$ and heavy $D^*$, $D^*_s$, $B^*$, $B^*_s$

Automorphism Groups of the Pancake Graphs

It is well-known that the pancake graphs are widely used as models for interconnection networks \cite{Akers}. In this paper, some properties of the pancake graphs are investigated. We first prove that the pancake graph, denoted by $P_n (n\geq 4),$ is super-connected and hyper-connected. Further, we study the symmetry of $P_n$ and completely determine its full automorphism group,which shows that $P_n (n\geq 5)$ is a graphical regular representation of $S_n.$Comment: 9 pages, 21 reference

A fifth-order shock capturing scheme with BVD algorithm

A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P4-THINC-BVD (4th degree polynomial and THINC reconstruction based on BVD algorithm), is formulated as a two-stage cascade BVD (Boundary Variation Diminishing) algorithm following the BVD principle that minimizes the jumps of reconstructed values at cell boundaries. In the P4-THINC-BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness are used as the candidate reconstruction functions. The final reconstruction function is selected from the candidate functions by a two-stage cascade BVD algorithm so as to effectively control numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P4-THINC-BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it faithfully maintains the free-mode solutions in long term computation. P4-THINC-BVD, as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales

Constructing high-order discontinuity-capturing schemes with linear-weight polynomials and boundary variation diminishing algorithm

In this study, a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called $\mathrm{P}_{n}\mathrm{T}_{m}-\mathrm{BVD}$ (polynomial of $n$-degree and THINC function of $m$-level reconstruction based on BVD algorithm), are designed by employing high-order linear-weight polynomials and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using linear-weight polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order linear-weight polynomials for smooth solution over all wave numbers and realizes $n+1$ order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.Comment: arXiv admin note: text overlap with arXiv:1811.0137

Some practical versions of boundary variation diminishing (BVD) algorithm

This short note presents some variant schemes of boundary variation diminishing (BVD) algorithm in one dimension with the results of numerical tests for linear advection equation to facilitate practical use. In spite of being presented in 1D fashion, all the schemes are simple and easy to implement in multi-dimensions on structured and unstructured grids for nonlinear and system equations

Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras

Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\rm{coh}\mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $\mathbb{X}$. By further dealing with the Ringel--Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits' automorphisms of the Kac--Moody algebra $\frak{g}_Q$ associated with $Q$, as well as for Lusztig's symmetries of the quantum enveloping algebra of ${\frak g}_Q$.Comment: 40 page

Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schr\"odinger equations

A bilinearisation-reduction approach is described for finding solutions for nonlocal integrable systems and is illustrated with nonlocal discrete nonlinear Schr\"odinger equations. In this approach we first bilinearise the coupled system before reduction and derive its double Casoratian solutions, then we impose reduction on double Casoratians so that they coincide with the nonlocal reduction on potentials. Double Caosratian solutions of the classical and nonlocal (reverse space, reverse time and reverse space-time) discrete nonlinear Schr\"odinger equations are presented.Comment: 8 page

Direct numerical study of speed of sound in dispersed air-water two-phase flow

Speed of sound is a key parameter for the compressibility effects in multiphase flow. We present a new approach to do direct numerical simulations on the speed of sound in compressible two-phase flow, based on the stratified multiphase flow model (Chang & Liou, JCP 2007). In this method, each face is divided into gas-gas, gas-liquid, and liquid-liquid parts via reconstruction of volume fraction, and the corresponding fluxes are calculated by Riemann solvers. Viscosity and heat transfer models are included. The effects of frequency (below the natural frequency of bubbles), volume fraction, viscosity and heat transfer are investigated. With frequency 1 kHz, under viscous and isothermal conditions, the simulation results satisfy the experimental ones very well. The simulation results show that the speed of sound in air-water bubbly two-phase flow is larger when the frequency is higher. At lower frequency, for the phasic velocities, the homogeneous condition is better satisfied. Considering the phasic temperatures, during the wave propagation an isothermal bubble behavior is observed. Finally, the dispersion relation of acoustics in two-phase flow is compared with analytical results below the natural frequency. This work for the first time presents an approach to the direct numerical simulations of speed of sound and other compressibility effects in multiphase flow, which can be applied to study more complex situations, especially when it is hard to do experimental study

A parallel space-time domain decomposition method for unsteady source inversion problems

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknowns. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications