Grothendieck's constant and local models for noisy entangled quantum states

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states \rho^W_p=p \proj{\psi^-}+(1-p){\one}/{4}, we show that there is a local model for projective measurements if and only if $p \le 1/K_G(3)$, where $K_G(3)$ is Grothendieck's constant of order 3. Known bounds on $K_G(3)$ prove the existence of this model at least for $p \lesssim 0.66$, quite close to the current region of Bell violation, $p \sim 0.71$. We generalize this result to arbitrary quantum states.Comment: 6 pages, 1 figur

The Panchromatic Hubble Andromeda Treasury. VI. The reliability of far-ultraviolet flux as a star formation tracer on sub-kpc scales

We have used optical observations of resolved stars from the Panchromatic Hubble Andromeda Treasury (PHAT) to measure the recent (< 500 Myr) star formation histories (SFHs) of 33 FUV-bright regions in M31. The region areas ranged from ~$10^4$ to $10^6$ pc$^2$, which allowed us to test the reliability of FUV flux as a tracer of recent star formation on sub-kpc scales. The star formation rates (SFRs) derived from the extinction-corrected observed FUV fluxes were, on average, consistent with the 100-Myr mean SFRs of the SFHs to within the 1$\sigma$ scatter. Overall, the scatter was larger than the uncertainties in the SFRs and particularly evident among the smallest regions. The scatter was consistent with an even combination of discrete sampling of the initial mass function and high variability in the SFHs. This result demonstrates the importance of satisfying both the full-IMF and the constant-SFR assumptions for obtaining precise SFR estimates from FUV flux. Assuming a robust FUV extinction correction, we estimate that a factor of 2.5 uncertainty can be expected in FUV-based SFRs for regions smaller than $10^5$ pc$^2$, or a few hundred pc. We also examined ages and masses derived from UV flux under the common assumption that the regions are simple stellar populations (SSPs). The SFHs showed that most of the regions are not SSPs, and the age and mass estimates were correspondingly discrepant from the SFHs. For those regions with SSP-like SFHs, we found mean discrepancies of 10 Myr in age and a factor of 3 to 4 in mass. It was not possible to distinguish the SSP-like regions from the others based on integrated FUV flux.Comment: Accepted for publication in The Astrophysical Journa

Snow Cover and Precipitation Impacts on Dry Season Streamflow in the Lower Mekong Basin

Climate change impacts on dry season streamflow in the Mekong River are relatively understudied, despite the fact that water availability during this time is critically important for agricultural and ecological systems. Analyses of two gauging stations (Vientiane and Kratie) in the Lower Mekong Basin (LMB) show significant positive correlations between dry season (March through May, MAM) discharge and upper basin snow cover and local precipitation. Using snow cover, precipitation, and upstream discharge as predictors, we develop skillful regression models for MAM streamflow at Vientiane and Kratie, and force these models with output from a suite of general circulation model (GCM) experiments for the twentieth and twenty-first centuries. The GCM simulations predict divergent trends in snow cover (decreasing) and precipitation (increasing) over the twenty-first century, driving overall negligible long-term trends in dry season streamflow. Our study demonstrates how future changes in dry season streamflow in the LMB will depend on changes in snow cover and precipitation, factors that will need to be considered when assessing the full basin response to other climatic and non-climatic drivers

Triangulations, order polytopes, and generalized snake posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every regular triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of filters comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Comment: 39 pages, 26 figures, comments welcomed