This paper has two distinguishable, but interwoven themes: (i) maximising the use of the context information available from a given task, by the application of a systems engineering approach to the packing problem, and (ii) the development of an adaptive packing strategy for random shapes, using morphological and heuristic techniques. The packing techniques and applications discussed in previou...

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the eeectiveness of these paradigms for two optimization problems on matroids: nding an optimum matroid basis and packing disjoint matroid bases. Applications of th...

For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and > 0, the random graph Gn,p, with n a multiple of t, with high probability contains a family of edge-disjoint T -factors covering all but an -f...

We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We prove that for k/2 < ` ≤ k, with high probability almost all edges of the ran...

Structures for the ramdam distributions of disks in one dimension are used ta calculate trie packing density of two dimensional random heaps of disks. Trie calculated values are compared with those obtained from computer simulations. Studies bave been done on trie random

The Percus-Yevick theory for monodisperse hard spheres gives very good results for the pressure and structure factor of the system in a whole range of densities that lie within the gas and liquid phases. However, the equation seems to lead to a very unacceptable result beyond that region. Namely, the Percus-Yevick theory predicts a smooth behavior of the pressure that diverges only when the vol...

Best-t is the best known algorithm for on-line bin-packing, in the sense that no algorithm is known to behave better both in the worst case (when Best-t has performance ratio 1.7) and in the average uniform case, with items drawn uniformly in the interval 0; 1] (then Best-t has expected wasted space O(n 1=2 (log n) 3=4)). In practical applications, Best-t appears to perform within a few percent...

Shortly after the determination of the first protein x-ray crystal structures, researchers analyzed their cores and reported packing fractions ϕ ≈ 0.75, a value that is similar to close packing of equal-sized spheres. A limitation of these analyses was the use of extended atom models, rather than the more physically accurate explicit hydrogen model. The validity of the explicit hydrogen model w...

We propose an interpretation of the random close packing of granular materials as a phase transition, and discuss the possibility of experimental verification.

We review results about the density of typical lattices in R. This density is of the order of 2−n. We then obtain similar results for random (non-lattice) sphere packings in R: after suitably taking a fraction ν of centers of spheres in a typical random packing σ, the resulting packing τ has density C(ν)2−n with a reasonable C(ν). We obtain estimates of C(ν). 2000 Math. Subj. Class. 82B05.