Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

MAX-DOAS measurements of bromine explosion events in McMurdo Sound, Antarctica

Reactive halogen species (RHS) are responsible for ozone depletion and oxidation of gaseous elemental mercury and dimethyl sulphide in the polar boundary layer, but the sources and mechanisms controlling their catalytic reaction cycles are still not completely understood. To further investigate these processes, ground– based Multi–Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) observations of boundary layer BrO and IO were made from a portable instrument platform in McMurdo Sound during the Antarctic spring of 2006 and 2007. Measurements of surface ozone, temperature, pressure, humidity, and wind speed and direction were also made, along with fourteen tethersonde soundings and the collection of snow samples for mercury analysis. A spherical multiple scattering Monte Carlo radiative transfer model (RTM) was developed for the simulation of box-air-mass-factors (box-AMFs), which are used to determine the weighting functions and forward model differential slant column densities (DSCDs) required for optimal estimation. The RTM employed the backward adjoint simulation technique for the fast calculation of box-AMFs for specific solar zenith angles (SZA) and MAX-DOAS measurement geometries. Rayleigh and Henyey-Greenstein scattering, ground topography and reflection, refraction, and molecular absorption by multiple species were included. Radiance and box-AMF simulations for MAX-DOAS measurements were compared with nine other RTMs and showed good agreement. A maximum a posteriori (MAP) optimal estimation algorithm was developed to retrieve trace gas concentration profiles from the DSCDs derived from the DOAS analysis of the measured absorption spectra. The retrieval algorithm was validated by performing an inversion of artificial DSCDs, simulated from known NO2 profiles. Profiles with a maximum concentration near the ground were generally well reproduced, but the retrieval of elevated layers was less accurate. Retrieved partial vertical column densities (VCDs) were similar to the known values, and investigation of the averaging kernels indicated that these were the most reliable retrieval product. NO₂ profiles were also retrieved from measurements made at an NO₂ measurement and profiling intercomparison campaign in Cabauw, Netherlands in July 2009. Boundary layer BrO was observed on several days throughout both measurement periods in McMurdo Sound, with a maximum retrieved surface mixing ratio of 14.4±0.3 ppt. The median partial VCDs up to 3km were 9.7±0.07 x 10¹² molec cm ⁻ in 2007, with a maximum of 2.3±0.07 x 10¹³ molec cm⁻², and 7.4±0.06 x 10¹² molec cm⁻² in 2006, with a maximum of 1.05 ± 0.07 x 1013 molec cm⁻². The median mixing ratio of 7.5±0.5 ppt for 2007 was significantly higher than the median of 5.2±0.5 ppt observed in 2006, which may be related to the more extensive first year sea ice in 2007. These values are consistent with, though lower than estimated boundary layer BrO concentrations at other polar coastal sites. Four out of five observed partial ozone depletion events (ODEs) occurred during strong winds and blowing snow, while BrO was present in the boundary layer in both stormy and calm conditions, consistent with the activation of RHS in these two weather extremes. Air mass back trajectories, modelled using the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) model, indicated that the events were locally produced rather than transported from other sea ice zones. Boundary layer IO mixing ratios of 0.5–2.5±0.2 ppt were observed on several days. These values are low compared to measurements at Halley and Neumayer Stations, as well as mid-latitudes. Significantly higher total mercury concentrations observed in 2007 may be related to the higher boundary layer BrO concentrations, but further measurements are required to verify this

Cured meat, vegetables, and bean-curd foods in relation to childhood acute leukemia risk: A population based case-control study

<p>Abstract</p> <p>Background</p> <p>Consumption of cured/smoked meat and fish leads to the formation of carcinogenic <it>N-</it>nitroso compounds in the acidic stomach. This study investigated whether consumed cured/smoked meat and fish, the major dietary resource for exposure to nitrites and nitrosamines, is associated with childhood acute leukemia.</p> <p>Methods</p> <p>A population-based case-control study of Han Chinese between 2 and 20 years old was conducted in southern Taiwan. 145 acute leukemia cases and 370 age- and sex-matched controls were recruited between 1997 and 2005. Dietary data were obtained from a questionnaire. Multiple logistic regression models were used in data analyses.</p> <p>Results</p> <p>Consumption of cured/smoked meat and fish more than once a week was associated with an increased risk of acute leukemia (OR = 1.74; 95% CI: 1.15–2.64). Conversely, higher intake of vegetables (OR = 0.55; 95% CI: 0.37–0.83) and bean-curd (OR = 0.55; 95% CI: 0.34–0.89) was associated with a reduced risk. No statistically significant association was observed between leukemia risk and the consumption of pickled vegetables, fruits, and tea.</p> <p>Conclusion</p> <p>Dietary exposure to cured/smoked meat and fish may be associated with leukemia risk through their contents of nitrites and nitrosamines among children and adolescents, and intake of vegetables and soy-bean curd may be protective.</p