Quantitative isoperimetric inequalities for log-convex probability measures on the line

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdr\'e). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) to any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type

Investigating the migration of immiscible contaminant fluid flow in homogeneous and heterogeneous aquifers with high-precision numerical simulations

Numerical modeling of the migration of three-phase immiscible fluid flow in variably saturated zones is challenging due to the different behavior of the system between unsaturated and saturated zones. This behavior results in the use of different numerical methods for the numerical simulation of the fluid flow depending on whether it is in the unsaturated or saturated zones. This paper shows that using a high-resolution shock-capturing conservative method to resolve the nonlinear governing coupled partial differential equations of a three-phase immiscible fluid flow allows the numerical simulation of the system through both zones providing a unitary vision (and resolution) of the migration of an immiscible contaminant problem within a porous medium. In particular, using different initial scenarios (including impermeable “lenses” in heterogeneous aquifers), three-dimensional numerical simulation results are presented on the temporal evolution of the contaminant migration following the saturation profiles of the three-phases fluids flow in variably saturated zones. It is considered either light nonaqueous phase liquid with a density less than the water, or dense nonaqueous phase liquid, which has densities greater than the water initially released in unsaturated dry soil. Our study shows that the fate of the migration of immiscible contaminants in variably saturated zones can be accurately described, using a unique mathematical conservative model, with different evolution depending on the value of the system’s physical parameters, including the contaminant density, and accurately tracking the evolution of the sharp (shock) contaminant front

High-resolution shock-capturing numerical simulations of three-phase immiscible fluids from the unsaturated to the saturated zone

Numerical modeling of immiscible contaminant fluid flow in unsaturated and saturated porous aquifers is of great importance in many scientific fields to properly manage groundwater resources. We present a high-resolution numerical model that simulates three-phase immiscible fluid flow in both unsaturated and saturated zone in a porous aquifer. We use coupled conserved mass equations for each phase and study the dynamics of a multiphase fluid flow as a function of saturation, capillary pressure, permeability, and porosity of the different phases, initial and boundary conditions. To deal with the sharp front originated from the partial differential equations’ nonlinearity and accurately propagate the sharp front of the fluid component, we use a high-resolution shock-capturing method to treat discontinuities due to capillary pressure and permeabilities that depend on the saturation of the three different phases. The main approach to the problem’s numerical solution is based on (full) explicit evolution of the discretized (in-space) variables. Since explicit methods require the time step to be sufficiently small, this condition is very restrictive, particularly for long-time integrations. With the increased computational speed and capacity of today’s multicore computer, it is possible to simulate in detail contaminants’ fate flow using high-performance computing

Towards an efficient segmentation of small rodents brain: a short critical review

One of the most common tasks in small rodents MRI pipelines is the voxel-wise segmentation of the volume in multiple classes. While many segmentation schemes have been developed for the human brain, fewer are available for rodent MRI, often by adaptation from human neuroimaging. Common methods include atlas-based and clustering schemes. The former labels the target volume by registering one or more pre-labeled atlases using a deformable registration method, in which case the result depends on the quality of the reference volumes, the registration algorithm and the label fusion approach, if more than one atlas is employed. The latter is based on an expectation maximization procedure to maximize the variance between voxel categories, and is often combined with Markov Random Fields and the atlas based approach to include spatial information, priors, and improve the classification accuracy. Our primary goal is to critically review the state of the art of rat and mouse segmentation of neuro MRI volumes and compare the available literature on popular, readily and freely available MRI toolsets, including SPM, FSL and ANTs, when applied to this task in the context of common pre-processing steps. Furthermore, we will briefly address the emerging Deep Learning methods for the segmentation of medical imaging, and the perspectives for applications to small rodents

Omental well-differentiated liposarcoma: US, CT and MR findings

Liposarcomas are the most common of sarcoma tumours, they are usually located in the lower limbs, retroperitoneum, or abdominal cavity; up to date, only a few cases of omental liposarcoma with different histotype have been described. We present a case of omental well-differentiated liposarcoma and discuss imaging findings on ultrasound, computed tomography, and magnetic resonance to differentiate omental liposarcomas from other abdominal tumour entities