We consider the Dirichlet Laplacian for astrip in \mathbb{R}^2 with one straight boundary and a width \mathbb{R}^2 , where \mathbb{R}^2 is a smooth function of acompact support with a length 2<i>b</i>. We show that in the criticalcase, \mathbb{R}^2 , the operator has nobound statesfor small \mathbb{R}^2 .On the otherhand, a weakly bound state existsprovided \mathbb{R}^2 . In thatcase, there are positive <i>c</i> ₁,<i>c</i> ₂ suchthat the corresponding eigenvalue satisfies \mathbb{R}^2 for all \mathbb{R}^2 sufficiently small.

We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modelled by a delta potential. We show that if the coupling strength of the latter is modified locally, ie, it reaches the same asymptotic value in both directions along the line, there is always a bound state below the bottom of the essential spectrum provided the effective coupling function is attractive in the mean. The eigenvalues and eigenfunctions, as well as the scattering matrix for energies above the threshold, are found numerically by the mode-matching technique. In particular, we discuss the rate at which the ground state energy emerges from the continuum and properties of the nodal lines. Finally, we investigate a system with a modified geometry: an infinite cylindrical surface threaded by a homogeneous magnetic field parallel to the

We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called StefanovUhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space. Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate the new method.

MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008 and improved in PQC 2008. This article gives an overview over the MutantXL algorithm. It also presents experimental results comparing the behavior of the MutantXL algorithm to the $F_4$ algorithm on HFE and randomly generated multivariate systems. In both cases MutantXL is faster and uses less memory than the Magma's implementation of $F_4$.

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