We nd the shortest non-zero vector in the lattice of all integer multiples of the vector a; b modulo m, for given integers 0 < a ; b < m. W e reduce the problem to the computation of a Minkowski-reduced basis for a planar lattice and thereby show that the problem can be solved in Olog mlog logm 2 bit operations.

We give a method for approximating any n-dimensional lattice with a lattice Λ whose factor group Z/Λ has n− 1 cycles of equal length with arbitrary precision. We also show that a direct consequence of this is that the Shortest Vector Problem and the Closest Vector Problem cannot be easier for this type of lattices than for general lattices.

We demonstrate the formation of fundamental and dipolelike vector solitons in an optically induced two-dimensional photonic lattice. Such vector solitons are realized by mutual trapping of two beams in the lattice. Our theoretical results are in good agreement with experimental observations.

The lattice Sommerfield model, describing a massive vector gauge field coupled to light fermion in 2d, is an ideal candidate verify perturbative conclusions. In contrast with continuum exact solutions, we prove that there no infinite renormalization, implying the reduction of degree ultraviolet divergence, and anomalies are non renormalized. Such features counterpart analogue properties at basi...