20,305 research outputs found
Collision Times in Multicolor Urn Models and Sequential Graph Coloring With Applications to Discrete Logarithms
Consider an urn model where at each step one of colors is sampled
according to some probability distribution and a ball of that color is placed
in an urn. The distribution of assigning balls to urns may depend on the color
of the ball. Collisions occur when a ball is placed in an urn which already
contains a ball of different color. Equivalently, this can be viewed as
sequentially coloring a complete -partite graph wherein a collision
corresponds to the appearance of a monochromatic edge. Using a Poisson
embedding technique, the limiting distribution of the first collision time is
determined and the possible limits are explicitly described. Joint distribution
of successive collision times and multi-fold collision times are also derived.
The results can be used to obtain the limiting distributions of running times
in various birthday problem based algorithms for solving the discrete logarithm
problem, generalizing previous results which only consider expected running
times. Asymptotic distributions of the time of appearance of a monochromatic
edge are also obtained for other graphs.Comment: Minor revision. 35 pages, 2 figures. To appear in Annals of Applied
Probabilit
Strain induced band gap deformation of H/F passivated graphene and h-BN sheet
Strain induced band gap deformations of hydrogenated/fluorinated graphene and
hexagonal BN sheet have been investigated using first principles density
functional calculations. Within harmonic approximation, the deformation is
found to be higher for hydrogenated systems than for the fluorinated systems.
Interestingly, our calculated band gap deformation for hydrogenated/fluorinated
graphene and BN sheets are positive, while those for pristine graphene and BN
sheet are found to be negative. This is due to the strong overlap between
nearest neighbor {\pi} orbitals in the pristine sheets, that is absent in the
passivated systems. We also estimate the intrinsic strength of these materials
under harmonic uniaxial strain, and find that the in-plane stiffness of
fluorinated and hydrogenated graphene are close, but larger in magnitude as
compared to those of fluorinated and hydrogenated BN sheet.Comment: Submitted to PR
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