On the resonances and eigenvalues for a 1D half-crystal with localised impurity

We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, for any potential $q$ and for any sequences (\s_n)_{1}^\iy, \s_n\in \{0,1\}, and (\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0, there exists a potential $p$ such that \vk_n is the length of the $n$-th gap, $n\in\N$, and $H$ has exactly \s_n eigenvalues and 1-\s_n antibound state in each high-energy gap. Moreover, we show that between any two eigenvalues in a gap, there is an odd number of antibound states, and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit.Comment: 25 page

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

In this work we consider Runge-Kutta discontinuous Galerkin methods (RKDG) for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics

Computation of volume potentials over bounded domains via approximate approximations

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.Comment: 18 page

Approximate Approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.Comment: 29 pages, 17 figure

Mission Drift of large MFIs?

Since the Mexican Microfinance Institution (MFI) Compartamos went public in 2007 – whereby promoting NGOs and private investors earned about USD 425 million – leading journals and magazines have repeatedly run rather sceptical articles about microfinance. They are mostly inspired by antagonists of MFIs growing into market driven enterprises. This antagonism has been blended with contemplation about assumed “subprime issues” of microfinance. However, the sector showed a steady performance, different from most other segments of the financial sector. The unholy blend of these two lines of thought risks to create an unwarranted image of microfinance.Microfinance; Mission Drift; Subprime; Sustainability

Compression for Smooth Shape Analysis

Most 3D shape analysis methods use triangular meshes to discretize both the shape and functions on it as piecewise linear functions. With this representation, shape analysis requires fine meshes to represent smooth shapes and geometric operators like normals, curvatures, or Laplace-Beltrami eigenfunctions at large computational and memory costs. We avoid this bottleneck with a compression technique that represents a smooth shape as subdivision surfaces and exploits the subdivision scheme to parametrize smooth functions on that shape with a few control parameters. This compression does not affect the accuracy of the Laplace-Beltrami operator and its eigenfunctions and allow us to compute shape descriptors and shape matchings at an accuracy comparable to triangular meshes but a fraction of the computational cost. Our framework can also compress surfaces represented by point clouds to do shape analysis of 3D scanning data