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 $q$ 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 $q$-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