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In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf.[4]). The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.

In this article we define a new information theoretical quantity for any bipartite mixed state ρ_{AB} . We call it the balanced partial entanglement (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show that in holographic CFT_{2} the BPE equals to the area of the entanglement wedge cross section (EWCS) divided by 4G. More interestingly, when we consider the canonical purification the BPE is just half of the reflected entropy, which also directly relate to the EWCS. The BPE can be considered as an generalization of the reflected entropy for a generic purification of the mixed state ρ_{AB}. We interpret the correspondence between the BPE and EWCS using the holographic picture of the entanglement contour.

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in S3 is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-polynomial of a knot in the 3-sphere.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap when the potential is not necessarily convex.