Let P be a poset. A subset A of P is a k-family ii A contains no (k+1)-element chain. For i 1, let A i be the set of elements of A at depth i?1 in A. The k-families of P can be ordered by deening A B ii for all i, A i is included in the order ideal generated by B i. This paper examines minimal r-element k-families, deened as k-families A such that A = r and for every B < A, B < r. Minimal k-fam...

The feasibility of ion channeling in quasicrystalline structures is investigated in a computer experiment. For this purpose, a quasilattice is constructed by the grid method. As this model crystal shows, also in a quasicrystal the lattice points are arranged in strings and planes and form pronounced axial and planar channels. Their occupation by lattice points is, however, not constant, but exh...

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on ∆-bonds with inva...

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on ∆-bonds with inva...

We explore the relationship between the Ising model with inverse temperature β, the q = 2 random-cluster model with edge-parameter p = 1− e−2β , and the random even subgraph with edge-parameter 12p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual of G forms a random even subgraph of G. A coupling of the random even subgraph of G and the q = 2 ran...

In this article, Temperley’s bijection between spanning trees of a subgraph of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of an associated subgraph of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. W...

A good dimensional stability is a crucial property of any base-platform structure for the attachment high precision optical or mechanical devices, such as imaging equipment, satellite antennas, thermal sensors, etc., where surrounding temperature may fluctuate substantially. The present study demonstrates how in form dual-composite, planar-lattice can be designed and rapidly, conveniently, manu...