Diffuse PeV neutrinos from gamma-ray bursts

The IceCube collaboration recently reported the potential detection of two cascade neutrino events in the energy range 1-10 PeV. We study the possibility that these PeV neutrinos are produced by gamma-ray bursts (GRBs), paying special attention to the contribution by untriggered GRBs that elude detection due to their low photon flux. Based on the luminosity function, rate distribution with redshift and spectral properties of GRBs, we generate, using Monte-Carlo simulation, a GRB sample that reproduce the observed fluence distribution of Fermi/GBM GRBs and an accompanying sample of untriggered GRBs simultaneously. The neutrino flux of every individual GRBs is calculated in the standard internal shock scenario, so that the accumulative flux of the whole samples can be obtained. We find that the neutrino flux in PeV energies produced by untriggered GRBs is about 2 times higher than that produced by the triggered ones. Considering the existing IceCube limit on the neutrino flux of triggered GRBs, we find that the total flux of triggered and untriggered GRBs can reach at most a level of ~10^-9 GeV cm^-2 s^-1 sr^-1, which is insufficient to account for the reported two PeV neutrinos. Possible contributions to diffuse neutrinos by low-luminosity GRBs and the earliest population of GRBs are also discussed.Comment: Accepted by ApJ, one more figure added to show the contribution to the diffuse neutrino flux by untriggered GRBs with different luminosity, results and conclusions unchange

Large-N scaling behavior of the quantum fisher information in the Dicke model

Quantum Fisher information (QFI) of the reduced two-atom state is employed to capture the quantum criticality of the superradiant phase transition in the Dicke model in the infinite size and finite-$N$ systems respectively. The analytical expression of the QFI of its ground state is evaluated explicitly. And finite-size scaling analysis is performed with the large accessible system size due to the effective bosonic coherent-state technique. We also investigate the large-size scaling behavior of the scaled QFI of the reduced $N$-atom state and show the accurate exponent.Comment: 6pages,2figure

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form as well as in the usual sense using acceptable portfolios.Comment: Final version. To appear in Mathematics and Financial Economics. Keywords: Proportional Transaction Costs, Unbounded Random Endowments, Acceptable Portfolios, Super-hedging Theorem, Utility Maximization, Shadow Prices, Convex Dualit