Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations (2013) http://dx.doi.org/10.1007/s40072-013-0014-

Multi-storm, multi-catchment investigation of rainfall spatial resolution requirements for urban hydrological applications

Rainfall estimates of the highest possible resolution are required for urban hydrological applications, given the small size and fast response which characterise urban catchments. While significant progress has been made over the last few decades in high resolution measurement of rainfall at urban scales and in the modelling of urban runoff processes, a number of questions as to the actual resolution requirements for input data and models remain to be answered. With the aim of answering some of these questions, this work investigates the impact of rainfall estimates of different spatial resolutions and structures on the hydraulic outputs of models of several urban catchments with different characteristics. For this purpose multiple storm events, including convective and stratiform ones, measured by a polarimetric X-band radar located in Cabauw (NL) were selected for analysis. The original radar estimates, at 100 m and 1 min resolutions, were aggregated to coarser spatial resolutions of up to 1000 m. These estimates were then applied to the high-resolution semi distributed hydraulic models of four urban catchments of similar size (approx. 7 km2), but different morphological and land use characteristics; these are: the Herent catchment (Belgium), the Cranbrook catchment (UK), the Morée Sausset catchment (France) and the Kralingen District of Rotterdam (The Netherlands). When doing so, methodologies for standardising rainfall inputs and making results comparable were implemented. Moreover, the results were analysed considering different points at each catchment, while also taking into account the particular storm and catchment characteristics. The results obtained for the storms used in this study show that flat and less compact catchments (e.g. polder areas) may be more sensitive to the spatial resolution of rainfall estimates, as compared to catchments with higher slopes and compactness, which in general show little sensitivity to changes in spatial resolution. While this study provides interesting insights, further investigation is still required in order to obtain a more complete answer regarding rainfall resolution requirements for urban hydrological applications. Future work should include testing on higher resolution fully distributed hydro models, as well as the analysis of many more storm events

Optimal Protocol for Contrast-enhanced Free-running 5D Whole-heart Coronary MR Angiography at 3T.

Free-running 5D whole-heart coronary MR angiography (MRA) is gaining in popularity because it reduces scanning complexity by removing the need for specific slice orientations, respiratory gating, or cardiac triggering. At 3T, a gradient echo (GRE) sequence is preferred in combination with contrast injection. However, neither the injection scheme of the gadolinium (Gd) contrast medium, the choice of the RF excitation angle, nor the dedicated image reconstruction parameters have been established for 3T GRE free-running 5D whole-heart coronary MRA. In this study, a Gd injection scheme, RF excitation angles of lipid-insensitive binominal off-resonance RF excitation (LIBRE) pulse for valid fat suppression and continuous data acquisition, and compressed-sensing reconstruction regularization parameters were optimized for contrast-enhanced free-running 5D whole-heart coronary MRA using a GRE sequence at 3T. Using this optimized protocol, contrast-enhanced free-running 5D whole-heart coronary MRA using a GRE sequence is feasible with good image quality at 3T

Sensitivity of urban drainage models to the spatial-temporal resolution of rainfall inputs: A multi-storm, multi-catchment investigation

Urban hydrological applications require high resolution precipitation and catchment information in order to well represent the spatial variability, fast runoff processes and short response times of urban catchments (Berne et al., 2004). Although fast progress has been made over the last few decades in high resolution measurement of rainfall at urban scales, including increasing use of weather radars, recent studies suggest that the resolution of the currently available rainfall estimates (typically 1 x 1 km2 in space and 5 min in time) may still be too coarse to meet the stringent requirements of urban hydrology (Gires et al., 2012). What is more, current evidence is still insufficient to provide a concrete answer regarding the added value of higher resolution rainfall estimates and actual rainfall input resolution requirements for urban hydrological applications. With the aim of providing further evidence in this regard, a collaborative study was conducted which investigated the impact of rainfall input resolutions on the outputs of the operational urban drainage models of four urban catchments in the UK and Belgium (Figure 1)

The need for high resolution data to improve urban flood 1 risk assessment 2

Abstract 12 Cities are particularly vulnerable to rainfall-generated floods that are typically characterised 13 by their rapid onset and localised nature. This implies that precipitation and catchment 14 information need to be available at high resolution to reliably predict hydrological response 15 and potential flooding. On the contrary, urban areas constitute a major knowledge gap as most 16 flood risk studies have concentrated on natural basins and records of rain gauges and water 17 level gauges in cities are scarce. While increase in intense precipitation as a result of climate 18 change is expected in many areas around the world, it is at present not possible to assess how 19 this will affect urban pluvial flood risk. Collection of reliable, high resolution data in cities 20 needs to start urgently to build up datasets in support of urban flood risk assessment and to 21 enable detection of changes in flood risk whether these are induced by climate change, 22 urbanisation or other future developments. This study shows how implementation of 23 polarimetric X-band radar can contribute to filling the knowledge gap of flood risk 24 quantification in cities. 2

Fractal analysis of urban catchments and their representation in semi-distributed models: imperviousness and sewer system

Fractal analysis relies on scale invariance and the concept of fractal dimension enables one to characterize and quantify the space filled by a geometrical set exhibiting complex and tortuous patterns. Fractal tools have been widely used in hydrology but seldom in the specific context of urban hydrology. In this paper, fractal tools are used to analyse surface and sewer data from 10 urban or peri-urban catchments located in five European countries. The aim was to characterize urban catchment properties accounting for the complexity and inhomogeneity typical of urban water systems. Sewer system density and imperviousness (roads or buildings), represented in rasterized maps of 2m  ×  2m pixels, were analysed to quantify their fractal dimension, characteristic of scaling invariance. The results showed that both sewer density and imperviousness exhibit scale-invariant features and can be characterized with the help of fractal dimensions ranging from 1.6 to 2, depending on the catchment. In a given area consistent results were found for the two geometrical features, yielding a robust and innovative way of quantifying the level of urbanization. The representation of imperviousness in operational semi-distributed hydrological models for these catchments was also investigated by computing fractal dimensions of the geometrical sets made up of the sub-catchments with coefficients of imperviousness greater than a range of thresholds. It enables one to quantify how well spatial structures of imperviousness were represented in the urban hydrological models

Evolutionary expansion of the Ras switch regulatory module in eukaryotes

Ras proteins control many aspects of eukaryotic cell homeostasis by switching between active (GTP-bound) and inactive (GDP-bound) conformations, a reaction catalyzed by GTPase exchange factors (GEF) and GTPase activating proteins (GAP) regulators, respectively. Here, we show that the complexity, measured as number of genes, of the canonical Ras switch genetic system (including Ras, RasGEF, RasGAP and RapGAP families) from 24 eukaryotic organisms is correlated with their genome size and is inversely correlated to their evolutionary distances from humans. Moreover, different gene subfamilies within the Ras switch have contributed unevenly to the module’s expansion and speciation processes during eukaryote evolution. The Ras system remarkably reduced its genetic expansion after the split of the Euteleostomi clade and presently looks practically crystallized in mammals. Supporting evidence points to gene duplication as the predominant mechanism generating functional diversity in the Ras system, stressing the leading role of gene duplication in the Ras family expansion. Domain fusion and alternative splicing are significant sources of functional diversity in the GAP and GEF families but their contribution is limited in the Ras family. An evolutionary model of the Ras system expansion is proposed suggesting an inherent ‘decision making’ topology with the GEF input signal integrated by a homologous molecular mechanism and bifurcation in GAP signaling propagation