Measuring the transition to homogeneity with photometric redshift surveys

We study the possibility of detecting the transition to homogeneity using photometric redshift catalogs. Our method is based on measuring the fractality of the projected galaxy distribution, using angular distances, and relies only on observable quantites. It thus provides a way to test the Cosmological Principle in a model-independent unbiased way. We have tested our method on different synthetic inhomogeneous catalogs, and shown that it is capable of discriminating some fractal models with relatively large fractal dimensions, in spite of the loss of information due to the radial projection. We have also studied the influence of the redshift bin width, photometric redshift errors, bias, non-linear clustering, and surveyed area, on the angular homogeneity index H2 ({\theta}) in a {\Lambda}CDM cosmology. The level to which an upcoming galaxy survey will be able to constrain the transition to homogeneity will depend mainly on the total surveyed area and the compactness of the surveyed region. In particular, a Dark Energy Survey (DES)-like survey should be able to easily discriminate certain fractal models with fractal dimensions as large as D2 = 2.95. We believe that this method will have relevant applications for upcoming large photometric redshift surveys, such as DES or the Large Synoptic Survey Telescope (LSST).Comment: 14 pages, 14 figure

On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{th}$ Feng-Rao number is obtained.Comment: 23 pages, 6 figure

Full two-photon downconversion of just a single photon

We demonstrate, both numerically and analytically, that it is possible to generate two photons from one and only one photon. We characterize the output two photon field and make our calculations close to reality by including losses. Our proposal relies on real or artificial three-level atoms with a cyclic transition strongly coupled to a one-dimensional waveguide. We show that close to perfect downconversion with efficiency over 99% is reachable using state-of-the-art Waveguide QED architectures such as photonic crystals or superconducting circuits. In particular, we sketch an implementation in circuit QED, where the three level atom is a transmon

Isocausal spacetimes may have different causal boundaries

We construct an example which shows that two isocausal spacetimes, in the sense introduced by Garc\'ia-Parrado and Senovilla, may have c-boundaries which are not equal (more precisely, not equivalent, as no bijection between the completions can preserve all the binary relations induced by causality). This example also suggests that isocausality can be useful for the understanding and computation of the c-boundary.Comment: Minor modifications, including the title, which matches now with the published version. 12 pages, 3 figure