Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidentl...

Let Covering be the category of fuzzy coverings, and Partition, partitions. We geometrically construct an isomorphism categories between Partition a full subcategory Covering, which can used to derive bijections partitions coverings with finitely many sets. Also, we establish Covering[n], n sets, whose objects are sets satisfy certain conditions.

Let A be the algebra generated by the power series ∑nn−1qn/n! and ∑ nnqn/n! . We prove that many natural generating functions lie in this algebra: those appearing in graph enumeration problems, in the intersection theory of moduli spaces Mg,n and in the enumeration of ramified coverings of the sphere. We argue that ramified coverings of the sphere with a large number of sheets provide a model o...

Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Bäcklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings ar...

In a study of surface branched coverings, one can ask naturally: In how many different ways can a given surface be a branched covering of another given surface? This problem was studied by many authors in Quart. J. Math. Oxford Ser. 2 46 (1995) 485, Math. Scand. 84 (1999) 23, Discrete Math. 156 (1996) 141, Discrete Math. 183 (1998) 193, Discrete Math. (in press), European J. Combin. 22 (2001) 1...