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We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive interaction supported by a smooth surface in R 3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrdinger type operator with an effective potential expressed in terms of the interaction support curvatures.

We investigate a periodic quantum graph in the form of a square lattice with a general self-adjoint coupling at the vertices. We analyse the spectrum's high-energy behaviour. Depending on the coupling type, bands and gaps have different asymptotics. Bands may be flat even if the edges are coupled, and non-flat band widths may behave like\mathcal O (n^ j),\, j= 1, 0,-1,-2,-3, as the band index n. The gaps may be of asymptotically constant width or linearly growing with the latter case being generic.

Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.