We prove equivalence of two definitions of representability of matroids: representation by vector configurations and representation by retraction of buildings of type An. Proofs are given in a more general context of representation of matroids in semimodular lattices and Coxeter matroids in chamber systems with group metric.

The bond-percolation process is studied on periodic planar random lattices and their duals. The thresholds and critical exponents of the percolation transition are determined. The scaling functions of the percolating probability, the existence probability of the appearance of percolating clusters, and the mean cluster size are also calculated. The simulation result of the percolation threshold ...

Mutual stochastic independences among-algebras and mutual algebraic inde-pendences among elements of semimodular lattices are observed to have a very similar behaviour. We suggest abstract independence structures called I-relations describing it. Presented examination of I-relations resembles a theory of abstract connectedness: a dual characterization of I-relations by families of connected set...

This review presents the potential that lattice ~or spring network! models hold for micromechanics applications. The models have their origin in the atomistic representations of matter on one hand, and in the truss-type systems in engineering on the other. The paper evolves by first giving a rather detailed presentation of one-dimensional and planar lattice models for classical continua. This i...