We study Schrdinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by -couplings with a parameter\alpha\in {\bb R}. If the graph is' straight', ie periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever 0. We consider a'bending'deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling and the'bending angle'as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23 (2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The 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 wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method.

In 2017, Ward Beullens et al. submitted Lifted Unbalanced Oil and Vinegar [3], which is a modification to the Unbalanced Oil and Vinegar Scheme by Patarin. Previously, Ding et al. proposed the Subfield Differential Attack [22] which prompted a change of parameters by the authors of LUOV for the second round of the NIST post quantum standardization competition [4]. In this paper we propose a modification to the Subfield Differential Attack called the Nested Subset Differential Attack which fully breaks half of the parameter sets put forward. We also show by experimentation that this attack is practically possible to do in under 210 min for the level I security parameters and not just a theoretical attack. The Nested Subset Differential attack is a large improvement of the Subfield differential attack which can be used in real world circumstances. Moreover, we will only use what is called the “lifted” structure of LUOV, and our attack can be thought as a development of solving “lifted” quadratic systems.

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