Busemann functions and barrier functions

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they are locally semi-concave with linear modulus. We also analysis the structure of singularity sets of Busemann functions. Moreover we study barrier functions, which are analogues to Mather's barrier functions in Mather theory, and provide some fundamental properties. Based on barrier functions, we could define some relations on the set of lines and thus classify them. We also discuss some initial relations with the ideal boundary of the Riemannian manifold.Comment: comments are welcome

Testing the Universal Structured Jet Models of Gamma-Ray Bursts by BATSE Observations

Assuming that the observed gamma-ray burst (GRB) rate as a function of redshift is proportional to a corrected star formation rate, we derive the empirical distribution of the viewing angles of long BATSE GRBs, $P^{\rm em}(\theta)$, and the distribution of these bursts in the plane of $\theta$ against redshift, $P^{\rm em}(\theta, z)$, by using a tight correlation between $E_{\gamma}$) and $E_{\rm p}^{'}$). Our results show that $P^{\rm em}(\theta)$ is well fitted by a log-normal distribution centering at $\log \theta/{\rm rad}=-0.76$ with a width of $\sigma_{\log \theta}=0.57$. We test different universal structured jet models by comparing model predictions with our empirical results. To make the comparisons reasonable, an "effective" threshold, which corresponds to the sample selection criteria of the long GRB sample, is used. We find that the predictions of a two-Gaussian jet model are roughly consistent with our empirical results. A brief discussion shows that cosmological effect on the $E_{\gamma}-E_{\rm p}^{'}$ relation does not significantly affect our results, but sample selection effects on this relationship might significantly influence our results.Comment: 5 pages, 6 figures, accepted for publication in A

Illusion Media: Generating Virtual Objects Using Realizable Metamaterials

We propose a class of optical transformation media, illusion media, which render the enclosed object invisible and generate one or more virtual objects as desired. We apply the proposed media to design a microwave device, which transforms an actual object into two virtual objects. Such an illusion device exhibits unusual electromagnetic behavior as verified by full-wave simulations. Different from the published illusion devices which are composed of left-handed materials with simultaneously negative permittivity and permeability, the proposed illusion media have finite and positive permittivity and permeability. Hence the designed device could be realizable using artificial metamaterials.Comment: 9 pages, 4 figures, published in Appl. Phys. Lett