In this article, we give explicit minimal generators of the first syzygy Hibi ring for a planar distributive lattice in terms sublattices. We also characterization when it is linearly related and derive an exact formula Betti number lattice.

We report a novel form of planar magnetic colloidal crystals formed by coated magnetic microspheres floating on a liquid meniscus. Under an external magnetic field, the balance between the repulsive magnetic interaction and the "attractive" interaction, due to the weight of the particles projected along the surface tangent, yields not only the triangular lattice with a variable lattice constant...

Energies of four-quark systems have been extracted in a static quenched SU(2) lattice Monte Carlo calculation for six diierent geometries, both planar and non-planar, with = 2:4 and lattice size 16 3 32. In all cases, it is found that the binding energy is greatly enhanced when the four quarks can be partitioned in two ways with comparable energies. Also it is shown that the energies of the fou...

A Monte Carlo dynamics study of the motions of hydrocarbon chains containing cis double bonds is presented. The simulations utilize the high-coordination {2 1 0) lattice for the simultaneous representation of the tetrahedrally bonded carbon atoms and the planar unsaturated segment. Results on single chains undergoing free motion in space and tethered to an impenetrable planar interface are repo...

Phase transitions of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on several decorated planar lattices consisting of interconnected diamonds are investigated within the framework of the generalized decoration-iteration transformation. The main attention is paid to the systematic study of the finite-temperature phase diagrams in dependence on the lattice topology. The critical behaviour ...

We conjecture that the classical geometric 2-designs PGd(n, q), where 2 ≤ d ≤ n− 1, are characterized among all designs with the same parameters as those having line size q + 1. The conjecture is known to hold for the case d = n − 1 (the Dembowski-Wagner theorem) and also for d = 2 (a recent result established by Tonchev and the present author). Here we extend this result to the cases d = 3 and...

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w)× Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of thi...