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Network flow theory has been used across a number of disciplines, including theoretical computer science, operations research, and discrete math, to model not only problems in the transportation of goods and information, but also a wide range of applications from image segmentation problems in computer vision to deciding when a baseball team has been eliminated from contention. This graduate text and reference presents a succinct, unified view of a wide variety of efficient combinatorial algorithms for network flow problems, including many results not found in other books. It covers maximum flows, minimum-cost flows, generalized flows, multicommodity flows, and global minimum cuts and also presents recent work on computing electrical flows along with recent applications of these flows to classical problems in network flow theory.
Evolution of thermally stratified open channel flow after removal of a volumetric heat source is investigated using direct numerical simulation. The heat source models radiative heating from above and varies with height due to progressive absorption. After removal of the heat source the initial stable stratification breaks down and the channel approaches a fully mixed isothermal state. The initial state consists of three distinct regions: a near-wall region where stratification plays only a minor role, a central region where stratification has a significant effect on flow dynamics and a near-surface region where buoyancy effects dominate. We find that a state of local energetic equilibrium observed in the central region of the channel in the initial state persists until the late stages of the destratification process. In this region local turbulence parameters such as eddy diffusivity
and flux Richardson number
are found to be functions only of the Prandtl number
and a mixed parameter
, which is equal to the ratio of the local buoyancy Reynolds number
and the friction Reynolds number
. Close to the top and bottom boundaries turbulence is also affected by
and vertical position
. In the initial heated equilibrium state the laminar surface layer is stabilised by the heat source, which acts as a potential energy sink. Removal of the heat source allows Kelvin–Helmholtz-like shear instabilities to form that lead to a rapid transition to turbulence and significantly enhance the mixing process. The destratifying flow is found to be governed by bulk parameters
and the friction Richardson number
. The overall destratification rate
is found to be a function of
Characterising the three-dimensional (3D) distribution of hydraulic conductivity and its variability in the shallow subsurface is fundamental to understanding groundwater behaviour and to developing conceptual and numerical groundwater models to manage the subsurface. However, directly measuring in situ hydraulic conductivity can be difficult and expensive and is rarely carried out with sufficient density in urban environments. In this study we model hydraulic conductivity for 603 sites in the unconsolidated Quaternary deposits underlying Glasgow using particle size distribution and density description widely available from geotechnical investigations. Six different models were applied and the MacDonald formula was found to be most applicable in this heterogeneous environment, comparing well with the few available in situ hydraulic conductivity data. The range of the calculated hydraulic conductivity values between the 5th and 95th percentile was 1.56×10–2–4.38mday–1 with a median of 2.26×10–1 mday–1. These modelled hydraulic conductivity data were used to develop a suite of stochastic 3D simulations conditioned to existing 3D representations of lithology. Ten per cent of the input data were excluded from the modelling process for use in a split-sample validation test, which demonstrated the effectiveness of this approach compared with non-spatial or lithologically unconstrained models. Our spatial model reduces the mean squared error between the estimated and observed values at the excluded data locations over those predicted using a simple homogeneous model by 73 %. The resulting 3D hydraulic conductivity model is of a much higher resolution than would have been possible from using only direct measurements, and will improve understanding of groundwater flow in Glasgow and reduce the spatial uncertainty of hydraulic parameters in groundwater process models. The methodology employed could be replicated in other regions where significant volumes of suitable geotechnical and site investigation data are available to predict ground conditions in areas with complex superficial deposits.
Introduction: Gastroenteritis accounts for 1.7 million emergency department visits by children annually in the United States. We conducted a double-blind trial to determine whether twice daily probiotic administration for 5 days, improves outcomes. Methods: 886 children aged 348 months with gastroenteritis were enrolled in six Canadian pediatric emergency departments. Participants were randomly assigned to twice daily Lactobacillus rhamnosus R0011 and Lactobacillus helveticus R0052, 4.0 x 109 CFU, in a 95:5 ratio or placebo. Primary outcome was development of moderate-severe disease within 14 days of randomization defined by a Modified Vesikari Scale score 9. Secondary outcomes included duration of diarrhea and vomiting, subsequent physician visits and adverse events. Results: Moderate-severe disease occurred in 108 (26.1%) participants administered probiotics and 102 (24.7%) participants allocated to placebo (OR 1.06; 95%CI: 0.77, 1.46; P=0.72). After adjustment for site, age, and frequency of vomiting and diarrhea, treatment assignment did not predict moderate-severe disease (OR, 1.11, 95%CI, 0.80 to 1.56; P=0.53). In the probiotic versus placebo groups, there were no differences in the median duration of diarrhea [52.5 (18.3, 95.8) vs. 55.5 (20.2, 102.3) hours; P=0.31], vomiting [17.7 (0, 58.6) vs. 18.7 (0, 51.6) hours; P=0.18], physician visits (30.2% vs. 26.6%; OR 1.19; 95% CI0.87. 1.62; P=0.27), or adverse events (32.9% vs. 36.8%; OR 0.83; 95%CI 0.62. 1.11; P=0.21). Conclusion: In children presenting to an emergency department with gastroenteritis, twice daily administration of 4.0 x 109 CFU of a Lactobacillus rhamnosus/helveticus probiotic does not prevent development of moderate-severe disease or improvements in other outcomes measured.