17 research outputs found
Top to random shuffles on colored permutations
A deck of cards are shuffled by repeatedly taking off the top card,
flipping it with probability , and inserting it back into the deck at a
random position. This process can be considered as a Markov chain on the group
of signed permutations. We show that the eigenvalues of the transition
probability matrix are and the multiplicity of the
eigenvalue is equal to the number of the {\em signed} permutation having
exactly fixed points. We show the similar results also for the colored
permutations. Further, we show that the mixing time of this Markov chain is
, same as the ordinary 'top-to-random' shuffles without flipping the
cards. The cut-off is also analyzed by using the asymptotic behavior of the
Stirling numbers of the second kind.Comment: Corrected versio
Broadcastings and digit tilings on three-dimensional torus networks
AbstractA tiling in a finite abelian group H is a pair (T,L) of subsets of H such that any h∈H can be uniquely represented as t+l where t∈T and l∈L. This paper studies a finite analogue of self-affine tilings in Euclidean spaces and applies it to a problem of broadcasting on circuit switched networks. We extend the tiling argument of Peters and Syska [Joseph G. Peters, Michel Syska, Circuit switched broadcasting in torus networks, IEEE Trans. Parallel Distrib. Syst., 7 (1996) 246–255] to 3-dimensional torus networks