We discuss quantum graphs consisting of a compact part and semi-infinite leads. Such a system may have embedded eigenvalues if some edge lengths in the compact part are rationally related. If such a relation is perturbed, these eigenvalues may turn into resonances; we analyze this effect both generally and in simple examples.

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space-the natural extension of hyperbolic space by the de Sitter space-except for the degenerate case where all simplices are Euclidean in a generalized sense.

In this paper, we show the claims in the original Kipnis-Shamir attack on the HFE cryptosystems and the improved attack by Courtois that the complexity of the attacks is polynomial in terms of the number of variables are invalid. We present computer experiments and a theoretical argument using basic algebraic geometry to explain why it is so. Furthermore we show that even with the help of the powerful new Gröbner basis algorithm like F_4, the Kipnis-Shamir attack still should be exponential but not polynomial. This again is supported by our theoretical argument.

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For a one-parameter family (V, \Omega_{i}) with I=1,...,p_g of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.