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In this note we prove that the essential spectrum of a Schrdinger operator with potential supported on a finite number of compact Lipschitz hypersurfaces is given by [0, +). We emphasize that the union of a family of Lipschitz hypersurfaces is in general not Lipschitz. ( 2013 WileyVCH Verlag GmbH & Co. KGaA, Weinheim)

Photonic graphene, a photonic crystal with honeycomb structures, has been intensively studied in both theoretical and applied fields. Similar to graphene which admits Dirac Fermions and topological edge states, photonic graphene supports novel and subtle propagating modes (edge modes) of electromagnetic waves. These modes have wide applications in many optical systems. In this paper, we propose a novel gradient recovery method based on Bloch theory for the computation of topological edge modes in the honeycomb structure. Compared to standard finite element methods, this method provides higher order accuracy with the help of gradient recovery technique. This high order accuracy is highly desired for constructing the propagating electromagnetic modes in applications. We analyze the accuracy and prove the superconvergence of this method. Numerical examples are presented to show the efficiency by computing the edge mode for the P-symmetry and C-symmetry breaking cases in honeycomb structures.

We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

In this paper, we present an efficient attack to the multivariate Quadratic Quasigroups (MQQ) cryptosystem. Our cryptanalysis breaks MQQ cryptosystems by solving systems of multivariate quadratic polynomial equations using a modified version of the MutantXL algorithm. We present experimental results comparing the behavior of our implementation of MutantXL to Magma's implementation of F_4 on MQQ systems (\ge 135 bit). Based on our results we show that the MutantXL implementation solves with much less memory than Magma's implementation of F_4 algorithm.