In this paper, we prove that the degree of regularity of the family of Square systems, an HFE type of systems, over a prime finite field of odd characteristics q is exactly q, and therefore prove that \begin{itemize} \item inverting Square systems algebraically is exponential, when q=O(n), where n is the number of variables of the system. \end{itemize}

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We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called ST-form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of boundary conditions, called the P Q R S-form, also gives a natural scheme to design generalized low and high pass quantum filters.

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.

We establish a generic vanishing theorem for surfaces in characteristic p that lift to p and use it for classification of surfaces of general type with Euler characteristic p and large Albanese dimension.