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nthispaperwewillpresentthelocalstabilityanalysisandlocalerrorestimatefor the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of O(h log(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge–Kutta time-marching. Here h is the maximum mesh length. The numerical experiments are given also.

By the integral method, we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\mathbb{R}^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in [9] and [1] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [1].

We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s $-conjugacy classes. As an application, we verify the Axioms in\cite {HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s $-conjugacy class in a residually split group, the set of connected components is" controlled" by the set of straight elements associated to the $\s $-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with\cite {Zhou}, this allows one to verify in the residually split case, the description of the mod- p isogeny classes on Shimura varieties conjectured by Langland and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian of\cite {BS} and\cite {Zhu} which is of independent interest.

In this paper, we formulate a problem of simultaneous redundant via insertion and line end extension for via yield optimization. Our problem is more general than previous works in the sense that more than one type of line end extension is considered and the objective function to be optimized directly accounts for via yield. We present a zero-one integer linear program based approach, that is equipped with two speedup techniques, to solve the addressed problem optimally. In addition, we describe how to modify our approach to exactly solve a previous work. Extensive experimental results are shown to demonstrate the effectiveness and efficiency of our approaches.