We study the inverse scattering for Schrodinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
For the recent issues on discretization of Riemannian manifolds and their spectral properties, see @cite_46 and the references therein. The monograph @cite_5 contains an exposition of the spectral theory due to @cite_43 , over which leans the method of this paper.
@cite_5 @cite_46 @cite_43
2030605635 157436748 2093468829 1628709819
A convenient way of analysing Riemannian manifolds is to embed them in Euclidean spaces, with the embedding typically obtained by flattening the manifold via tangent spaces. This general approach is not free of drawbacks. For example, only distances between points to the tangent pole are equal to true geodesic distances. This is restrictive and may lead to inaccurate modelling. Instead of using tangent spaces, we propose embedding into the Reproducing Kernel Hilbert Space by introducing a Riemannian pseudo kernel. We furthermore propose to recast a locality preserving projection technique from Euclidean spaces to Riemannian manifolds, in order to demonstrate the benefits of the embedding. Experiments on several visual classification tasks (gesture recognition, person re-identification and texture classification) show that in comparison to tangent-based processing and state-of-the-art methods (such as tensor canonical correlation analysis), the proposed approach obtains considerable improvements in discrimination accuracy. The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (Ext-MS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangent spaces are only valid in a small neighborhood). In [3] was proposed an intrinsic mean shift designed to operate on two particular Riemannian manifolds (IntGS-MS), i.e. Grassmann and Stiefel manifolds (using manifold-dedicated density kernels). It is then natural to ask whether mean shift could be intrinsically extended to work on a large class of manifolds. We propose a novel paradigm to intrinsically reformulate the mean shift on general Riemannian manifolds. This is accomplished by embedding the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using a general and mathematically well-founded Riemannian kernel function, i.e. heat kernel [5]. The key issue is that when the data is implicitly mapped to the Hilbert space, the curvature of the manifold is taken into account (i.e. exploits the underlying information of the data). The inherent optimization is then performed on the embedded space. Theoretic analysis and experimental results demonstrate the promise and effectiveness of this novel paradigm. Abstract We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Frechet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Frechet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.