The minimum linear arrangement problem (MinLA) was first stated by Harper in [5]. His aim was to design error-correcting codes with minimal average absolute errors on certain classes of graphs. Latter, in the 1970’s it was used as an abstract model of the placement phase in VLSI layout, where nodes of the graph represented modules and edges represented interconnections. In this case, the cost o...

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work wit...

In this paper Kolomogorov entropy of a simulated cavity quantum electrodynamics in a multi-partite system consisting of eight quantum dots in interaction with one cavity mode has been estimated. It has been shown that the Kolmogorov Entropy monotonically increases with the increasing coupling strength, which is a sufficient condition for chaotic behavior under ultrastrong coupling regime. The a...

We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewise-linear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions...

To model discriminative, i.e. competition induced, self-thinning in even-aged forest stands a concept has been explored that discriminative mortality alters spatial arrangement of trees which in turn alters the mortality. Function of density was suggested to be a balance of initial density minus mortality that was dependent on initial spatial arrangement and mean horizontal size in the forest s...

We prove tight and near-tight combinatorial complexity bounds for vertical decompositions of arrangements of hyperplanes and 3-simplices in four dimensions. In particular, we prove a tight upper bound of Θ(n) for the vertical decomposition of an arrangement of n hyperplanes in four dimensions, improving the best previously known bound [8] by a logarithmic factor. We also show that the complexit...

Let D n;k denote the subspace arrangement formed by all linear subspaces in R n given by equations of the form k. Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In previous work on a larger class of subspace arrangements by Bjj orner & Sagan BS] the topology of the intersection lattice L(D n;k) = n;k turned out to be a pa...

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that ce...