Naslov
Torus cube packings

Autori
Itoh, Yoshiaki ; Dutour Sikirić, Mathieu

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, pp prezentacija, znanstveni

Izvornik
Seminar of computational homology and applications, National university of Ireland, Galway / - , 2007

Skup
Seminar of computational homology and applications, National university of Ireland, Galway

Mjesto i datum
Galway, Irska, 17.09.2007

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
cube packing; second moment; holes; enumeration

Sažetak
We consider sequential random packing of cubes z+ [0, 1]^n with z in (1/N)Z^n into the cube [0, 2]^n and the torus R^n/(2\ZZ^n) as N goes to infinity. In the cube case [0, 2]^n as N goes to infinity the random cube packings thus obtained are reduced to a single cube with probability 1-O(1/N). In the torus case the situation is different: for n=1 or 2, sequential random cube packing yields cube tilings, but for n>=3 with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on N depend on some parameters. We use use them to derive an expansion of the packing density in powers of 1/N. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case N goes to infinity of strictly positive probability. We prove the following results for torus combinatorial cube packings: * We give a general Cartesian product construction. * We prove that the number of parameters is at least n(n+1)/2 and we conjecture it to be at most 2^n-1. * We prove that cube packings with at least 2^n-3 cubes are extensible. * We find the minimal number of cubes in non- extensible cube packings for n odd and n<=6.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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