We consider a Hamiltonian with N point interactions in {\bb R}^ d,\: d= 2, 3, all with the same coupling constant, placed at vertices of an equilateral polygon {\cal P} _N. It is shown that the ground-state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.

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.

We discuss a modification of Smilansky model in which a singular potential channel is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential channels.

Recently, the Little Dragon Two and Poly-Dragon multivariate based public-key cryptosystems were proposed as efficient and secure schemes. In particular, the inventors of the two schemes claim that Little Dragon Two and Poly-Dragon resist algebraic cryptanalysis. In this paper, we show that MXL2, an algebraic attack method based on the XL algorithm and Ding’s concept of Mutants, is able to break Little Dragon Two with keys of length up to 229 bits and Poly-Dragon with keys of length up to 299. This contradicts the security claim for the proposed schemes and demonstrates the strength of MXL2 and the Mutant concept. This strength is further supported by experiments that show that in attacks on both schemes the MXL2 algorithm outperforms the Magma’s implementation of F4.

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair (7,8) is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Mu¨nzner. This completes the classification of isoparametric hypersurfaces in spheres that ´E. Cartan initiated in the late 1930s.