Edge states attract more and more research interests owing to the novel topologically protected properties. In this work, we studied edge modes and traveling edge states via the linear Dirac equation with so-called edge-admissible masses. The unidirectional edge state provides a heuristic approach to more general traveling edge states through the localized behavior along slowly varying edges. We show the dominated asymptotic solutions of two typical edge states that follow circular and curved edges with small curvature by the analytic and quantitative arguments.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

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For an essentially normal operator$T$, it is shown that there exists a unilateral shift of multiplicity$m$in$C$^{$*$}$(T)$if and only if γ($T$)≠0 and γ$(T)/m$. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as$C$^{$*$}-algebras. Finally, we construct a natural$C$^{$*$}-algebra ε + ε_{*}on the Bergman space$L$_{$a$}^{$2$}($B$_{$n$}), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.