The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes a1, . . . , an in (0, 1]. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized O( n)(log log n) ∑n i=1 ai +( 1 2 ) 1 2 ) time (1+2)approximation scheme for the bin packing problem. We show that every randomized algorithm with unifo...

Advancements in the synthesis of faceted nanoparticles and colloids have spurred interest in the phase behavior of polyhedral shapes. Regular tetrahedra have attracted particular attention because they prefer local symmetries that are incompatible with periodicity. Two dense phases of regular tetrahedra have been reported recently. The densest known tetrahedron packing is achieved in a crystal ...

For the last 50 years, the flow of a granular material through an aperture has been intensely studied in gravity-driven vertical systems (e.g., silos and hoppers). Nevertheless, in many industrial applications, grains are horizontally transported at constant velocity, lying on conveyor belts or floating on the surface of flowing liquids. Unlike fluid flows, that are controlled by the pressure, ...

We present experimental and theoretical results for geometrical properties of 2D packings of disks. We were mainly interested in the study of mixtures with disk size distribution which are of more practical interest than equal disks. Average geometrical properties, such as packing fraction or coordination number do not depend on the composition of the mixture, contrary to what would be expected...

Approximately 40% of the mitochondria in the sphincter muscle of the crayfish vas deferens have prismatic-type cristae. In cross section, the angular cristae have either triangular or rhomboid profiles which are surrounded by a hexagonal array of electron-dense dots. In longitudinal section, these mitochondria exhibit both thick and thin parallel lines, which represent cristae and filaments, r...

This is a slightly expanded version of a talk given at the János Bolyai Conference on Hyperbolic Geometry, held in Budapest in July, 2002. The general subject of the talk was the densest packings of simple bodies, for instance spheres or polyhedra, in Euclidean or hyperbolic spaces, and describes recent joint work with Lewis Bowen. One of the main points was to report on our solution of the old...

‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...