Elements with finite Coxeter part in an affine Weyl group

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold for twisted conjugacy classes.Comment: 9 page

On affine Deligne-Lusztig varieties for Sp_4(L)

In this paper, we study the emptiness/nonemptiness and the dimension formulas of affine Deligne-Lusztig varieties for $Sp_4(L)$. We mainly calculate the degree of class polynomials for the Iwahori-Hecke algebra of type $\widetilde{C}_2$. Then, give an explicit description on the emptiness/nonemptiness and dimension formulas of affine Deligne-Lusztig varieties for the group $Sp_4(L)$.Comment: 30 page

DMIS: Dynamic Mesh-based Importance Sampling for Training Physics-Informed Neural Networks

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schr\"odinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed. Code are available at https://github.com/MatrixBrain/DMIS.Comment: Accepted to AAAl-2

Effects of Coronal Density and Magnetic Field Distributions on a Global Solar EUV Wave

We investigate a global extreme-ultraviolet (EUV) wave associated with a coronal mass ejection (CME)-driven shock on 2017 September 10. The EUV wave is transmitted by north- and south-polar coronal holes (CHs), which is observed by the Solar Dynamics Observatory (SDO) and Solar Terrestrial Relations Observatory A (STEREO-A) from opposite sides of the Sun. We obtain key findings on how the EUV wave interacts with multiple coronal structures, and on its connection with the CME-driven shock: (1) the transmitted EUV wave is still connected with the shock that is incurvated to the Sun, after the shock has reached the opposite side of the eruption; (2) the south CH transmitted EUV wave is accelerated inside an on-disk, low-density region with closed magnetic fields, which implies that an EUV wave can be accelerated in both open and closed magnetic field regions; (3) part of the primary EUV wavefront turns around a bright point (BP) with a bipolar magnetic structure when it approaches a dim, low-density filament channel near the BP; (4) the primary EUV wave is diffused and apparently halted near the boundaries of remote active regions (ARs) that are far from the eruption, and no obvious AR related secondary waves are detected; (5) the EUV wave extends to an unprecedented scale of ~360{\deg} in latitudes, which is attributed to the polar CH transmission. These results provide insights into the effects of coronal density and magnetic field distributions on the evolution of an EUV wave, and into the connection between the EUV wave and the associated CME-driven shock.Comment: 16 pages, 8 figures, and 3 animations available at http://doi.org/10.13140/RG.2.2.12408.29442 , http://doi.org/10.13140/RG.2.2.25830.06723 , and http://doi.org/10.13140/RG.2.2.19119.18088 ; published in Ap