The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size $h$. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.

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We study chiral algebras associated with Argyres-Douglas theories engineered from M5 brane. For the theory engineered using 6d (2,0) type J theory on a sphere with a single irregular singularity (without mass parameter), its chiral algebra is the minimal model of W algebra of J type. For the theory engineered using an irregular singularity and a regular full singularity, its chiral algebra is the affine Kac-Moody algebra of J type. We can obtain the Schur index of these theories by computing the vacua character of the corresponding chiral algebra.

In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [9] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties.