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In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic NavierStokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.

Image matching is a fundamental problem in computer vision. One of the well-known techniques is SIFT (scale-invariant feature transform). SIFT searches for and extracts robust features in hierarchical image scale spaces for object identification. However it often lacks efficiency as it identifies many insignificant features such as tree leaves and grass tips in a natural building image. We introduce a content adaptive image matching approach by preprocessing the image with a color-entropy based segmentation and harmonic inpainting. Natural structures such as tree leaves have both high entropy and distinguished color, and so such a combined measure can be both discriminative and robust. The harmonic inpainting smoothly interpolates the image functions over the tree regions and so blurrs and reduces the features and their unnecessary matching there. Numerical experiments on building images show

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Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.