3,399 research outputs found
The hyperbolic geometry of random transpositions
Turn the set of permutations of objects into a graph by connecting
two permutations that differ by one transposition, and let be the
simple random walk on this graph. In a previous paper, Berestycki and Durrett
[In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of
the distance from the identity at time has a phase transition at .
Here we investigate some consequences of this result for the geometry of .
Our first result can be interpreted as a breakdown for the Gromov hyperbolicity
of the graph as seen by the random walk, which occurs at a critical radius
equal to . Let be a triangle formed by the origin and two points
sampled independently from the hitting distribution on the sphere of radius
for a constant . Then when , if the geodesics are suitably
chosen, with high probability is -thin for some , whereas
it is always O(n)-thick when . We also show that the hitting
distribution of the sphere of radius is asymptotically singular with
respect to the uniform distribution. Finally, we prove that the critical
behavior of this Gromov-like hyperbolicity constant persists if the two
endpoints are sampled from the uniform measure on the sphere of radius .
However, in this case, the critical radius is .Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Library-Based Synthesis Methodology for Reversible Logic
In this paper, a library-based synthesis methodology for reversible circuits
is proposed where a reversible specification is considered as a permutation
comprising a set of cycles. To this end, a pre-synthesis optimization step is
introduced to construct a reversible specification from an irreversible
function. In addition, a cycle-based representation model is presented to be
used as an intermediate format in the proposed synthesis methodology. The
selected intermediate format serves as a focal point for all potential
representation models. In order to synthesize a given function, a library
containing seven building blocks is used where each building block is a cycle
of length less than 6. To synthesize large cycles, we also propose a
decomposition algorithm which produces all possible minimal and inequivalent
factorizations for a given cycle of length greater than 5. All decompositions
contain the maximum number of disjoint cycles. The generated decompositions are
used in conjunction with a novel cycle assignment algorithm which is proposed
based on the graph matching problem to select the best possible cycle pairs.
Then, each pair is synthesized by using the available components of the
library. The decomposition algorithm together with the cycle assignment method
are considered as a binding method which selects a building block from the
library for each cycle. Finally, a post-synthesis optimization step is
introduced to optimize the synthesis results in terms of different costs.Comment: 24 pages, 8 figures, Microelectronics Journal, Elsevie
Frobenius-Schur indicators for some fusion categories associated to symmetric and alternating groups
We calculate Frobenius-Schur indicator values for some fusion categories
obtained from inclusions of finite groups , where more concretely
is symmetric or alternating, and is a symmetric, alternating or cyclic
group. Our work is strongly related to earlier results by
Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product
Hopf algebras obtained from exact factorizations of groups. We can generalize
some of their results, settle some open questions and offer shorter proofs;
this already pertains to the Hopf algebra case, while our results also cover
fusion categories not associated to Hopf algebras.Comment: 15 page
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