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In this paper, we present a new algorithm to solve algebraically the following lattice-related problems: 1) the small integer solution (SIS) problem under the condition: if the solution is bounded by an integer β in l_∞ norm, which we call a bounded SIS (BSIS) problem, and if the difference between the row dimension n and the column dimension m of the corresponding basis matrix is relatively small with respect the row dimension m; 2) the learning with errors (LWE) problems under the condition: if the errors are bounded - the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D < q), which we call a learning with bounded errors (LWBE) problem. We will show that we can solve these problems with polynomial complexity.

No abstract uploaded!

No abstract uploaded!

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or-type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small