Critical power of collapsing vortices

We calculate the critical power for collapse of linearly-polarized phase vortices, and show that this expression is more accurate than previous results. Unlike the non-vortex case, deviations from radial symmetry do not increase the critical power for collapse, but rather lead to disintegration into collapsing non-vortex filaments. The cases of circular, radial and azimuthal polarizations are also considered

All-Pay Auctions with Weakly Risk-Averse Buyers

We use perturbation analysis to study independent private-value all-pay auctions with weakly risk-averse buyers. We show that under weak risk aversion: 1) Buyers with low values bid lower and buyers with high values bid higher than they would bid in the risk neutral case. 2) Buyers with low values bid lower and buyers with high values bid higher than they would bid in a first-price auction. 3) Buyers' expected utilities in an all-pay auction are lower than in a first-price auction. 4) The seller's expected payoff in an all-pay auction may be either higher or lower than in the risk neutral case. 5) The seller's expected payoff in an all-pay auction may be either higher or lower than in a first-price auction.Private-value auctions, Risk aversion, Perturbation analysis

Funnel Theorems for Spreading on Networks

We derive novel analytic tools for the discrete Bass model, which models the diffusion of new products on networks. We prove that the probability that any two nodes adopt by time t, is greater than or equal to the product of the probabilities that each of the two nodes adopts by time t. We introduce the notion of an "influential node", and use it to determine whether the above inequality is strict or an equality. We then use the above inequality to prove the "funnel inequality", which relates the adoption probability of a node to the product of its adoption probability on two sub-networks. We introduce the notion of a "funnel node", and use it to determine whether the funnel inequality is strict or an equality. The above analytic tools can be exptended to epidemiological models on networks. We then use the funnel theorems to derive a new inequality for diffusion on circles and a new explicit expression for the adoption probabilities of nodes on two-sided line, and to prove that the adoption level on one-sided lines is strictly slower than on anisotropic two-sided lines, and that the adoption level on multi-dimensional Cartesian networks is bounded from below by that on one-dimensional networks

Multiple filamentation induced by input-beam ellipticity

The standard explanation for multiple filamentation (MF) of intense laser beams has been that it is initiated by input beam noise (modulational instability). In this study we provide the first experimental evidence that MF can also be induced by input beam ellipticity. Unlike noise-induced beam breakup, the MF pattern induced by ellipticity is reproducible shot to shot. Moreover, our experiments show that ellipticity can dominate the effect of noise, thus providing the first experimental methodology for controlling the MF pattern of noisy beams. The results are explained using a theoretical model and simulations