Ring structure of selected two-dimensional procrystalline lattices

Photonic band structure of two-dimensional atomic lattices

Dimers on two-dimensional lattices

We consider close-packed dimers, or perfect matchings, on twodimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simple-quartic (44), honeycomb (63), triangular (36), kagomé (3 · 6 · 3 · 6), 3-12 (3 · 122) and its dual [3 · 122], and 4-8 (4 · 82) and its dual Uni...

Lattices of local two-dimensional languages

The aimof this paper is to study local two-dimensional languages froman algebraic point of view.We show that local two-dimensional languages over a finite alphabet, with the usual relation of set inclusion, form a lattice. The simplest case Loc1 of local languages defined over the alphabet consisting of one element yields a distributive lattice, which can be easily described. In the general cas...

The L2 Discrepancy of Two-Dimensional Lattices

Let α be an irrational number with bounded partial quotients of the continued fraction ak. It is well known that symmetrizations of the irrational lattice {( μ/N,{μα}N−1 μ=0 have optimal order of L2 discrepancy, √ logN. The same is true for their rational approximations Ln(α) = {( μ/qn,{μ pn/qn} )}qn−1 μ=0 , where pn/qn is the nth convergent of α . However, the question whether and when the sym...

Covering Radius of Two-dimensional Lattices

The covering radius problem in any dimension is not known to be solvable in nondeterministic polynomial time, but when in dimension two, we give a deterministic polynomial time algorithm by computing a reduced basis using Gauss’ algorithm in this paper.