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.

Let L be a distributive lattice and R(L) the associated Hibi ring. We compute regR(L) when L is a planar lattice and give bounds for regR(L) when L is nonplanar, in terms of the combinatorial data of L. As a consequence, we characterize the distributive lattices L for which the associated Hibi ring has a linear resolution.

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a low-dimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces.

We consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k. We find that besides the standard power-law factor the leading term also has a logarithmic multiplier.

This paper is a brief mathematical excursion which starts from quantum electrodynamics and leads to the Möbius function of the Tamari lattice of planar binary trees, within the framework of groups of tree-expanded series. First we recall Brouder’s expansion of the photon and the electron Green’s functions on planar binary trees, before and after the renormalization. Then we recall the structure...

A subset X of a lattice L with 0 is called CDW-independent if (1) it is CDindependent, i.e., for any x, y ∈ X , either x ≤ y or y ≤ x or x ∧ y = 0 and (2) it is weakly independent, i.e., for any n ∈ N and x, y1, . . . , yn ∈ X the inequalityx ≤ y1∨· · ·∨yn implies x ≤ yi for some i. A maximal CDW-independent subset is called a CDW-basis. With combinatorial examples and motivations in the backgr...