Sharp minimax tests for large covariance matrices and adaptation

We consider the detection problem of correlations in a $p$-dimensional Gaussian vector, when we observe $n$ independent, identically distributed random vectors, for $n$ and $p$ large. We assume that the covariance matrix varies in some ellipsoid with parameter $\alpha >1/2$ and total energy bounded by $L>0$. We propose a test procedure based on a U-statistic of order 2 which is weighted in an optimal way. The weights are the solution of an optimization problem, they are constant on each diagonal and non-null only for the $T$ first diagonals, where $T=o(p)$. We show that this test statistic is asymptotically Gaussian distributed under the null hypothesis and also under the alternative hypothesis for matrices close to the detection boundary. We prove upper bounds for the total error probability of our test procedure, for $\alpha>1/2$ and under the assumption $T=o(p)$ which implies that $n=o(p^{ 2 \alpha})$. We illustrate via a numerical study the behavior of our test procedure. Moreover, we prove lower bounds for the maximal type II error and the total error probabilities. Thus we obtain the asymptotic and the sharp asymptotically minimax separation rate $\tilde{\varphi} = (C(\alpha, L) n^2 p )^{- \alpha/(4 \alpha + 1)}$, for $\alpha>3/2$ and for $\alpha >1$ together with the additional assumption $p= o(n^{4 \alpha -1})$, respectively. We deduce rate asymptotic minimax results for testing the inverse of the covariance matrix. We construct an adaptive test procedure with respect to the parameter $\alpha$ and show that it attains the rate $\tilde{\psi}= ( n^2 p / \ln\ln(n \displaystyle\sqrt{p}) )^{- \alpha/(4 \alpha + 1)}$

Arsenic in Food: Chronology of Development of Rules and Regulations

openThis thesis explores the pervasive issue of arsenic contamination in food, focusing on its impact on public health within the European Union (EU). It begins with a comprehensive introduction to arsenic, covering its properties, common uses, various exposure routes, and its presence in the environment. Notably, the thesis examines arsenic's occurrence in drinking water, air, soil, and sediments, along with its presence in food items. The toxicity of arsenic to humans is also scrutinized, encompassing both acute and chronic health effects. The research's objectives and aims are detailed, emphasizing the significance of a thorough understanding of arsenic contamination in the food supply chain, with a specific focus on the regulatory framework within the EU. The thesis further delves into the domain of food contaminants management in the EU. It investigates the definition of 'Food' under Regulation EC 178/2002, elucidating its scope and exclusions, thereby providing insight into the boundaries of EU food safety regulations. The role of risk analysis in EU food safety and contaminant regulations is highlighted, underscoring the importance of robust safety measures. Additionally, the study examines the pivotal role played by the European Commission in regulating food safety and contaminants within the European Union. Furthermore, it explores the contributions of the European Food Safety Authority (EFSA) in assessing and mitigating food contaminants. The research also considers the influence of the Food and Agriculture Organization (FAO) and the Joint Expert Committee on Food Additives (JECFA) in shaping food safety regulations and conducting contaminant assessments. In summary, this thesis provides a comprehensive analysis of arsenic contamination in food and its intersection with EU food safety regulations. By shedding light on these critical aspects, the research aims to contribute to the improvement of food safety standards and the safeguarding of public health in the European Union.This thesis explores the pervasive issue of arsenic contamination in food, focusing on its impact on public health within the European Union (EU). It begins with a comprehensive introduction to arsenic, covering its properties, common uses, various exposure routes, and its presence in the environment. Notably, the thesis examines arsenic's occurrence in drinking water, air, soil, and sediments, along with its presence in food items. The toxicity of arsenic to humans is also scrutinized, encompassing both acute and chronic health effects. The research's objectives and aims are detailed, emphasizing the significance of a thorough understanding of arsenic contamination in the food supply chain, with a specific focus on the regulatory framework within the EU. The thesis further delves into the domain of food contaminants management in the EU. It investigates the definition of 'Food' under Regulation EC 178/2002, elucidating its scope and exclusions, thereby providing insight into the boundaries of EU food safety regulations. The role of risk analysis in EU food safety and contaminant regulations is highlighted, underscoring the importance of robust safety measures. Additionally, the study examines the pivotal role played by the European Commission in regulating food safety and contaminants within the European Union. Furthermore, it explores the contributions of the European Food Safety Authority (EFSA) in assessing and mitigating food contaminants. The research also considers the influence of the Food and Agriculture Organization (FAO) and the Joint Expert Committee on Food Additives (JECFA) in shaping food safety regulations and conducting contaminant assessments. In summary, this thesis provides a comprehensive analysis of arsenic contamination in food and its intersection with EU food safety regulations. By shedding light on these critical aspects, the research aims to contribute to the improvement of food safety standards and the safeguarding of public health in the European Union

Multiple Axisymmetric Solutions for Axially Traveling Waves in Solid Rocket Motors

In this article, we consider the vorticoacoustic flowfield arising in a rightcylindrical porous chamber with uniform sidewall injection. Such configuration is often used to simulate the internal gaseous environment of a solid rocket motor (SRM). Assuming closed-closed acoustic conditions at both fore and aft ends of the domain, the introduction of small disturbances in the mean flow give rise to an axially traveling vortico-acoustically dominated wave structure that our study attempts to elucidate. Although this problem has been formulated before, it is reconsidered here in the context of WKB perturbation expansions in the reciprocal of the crossflow Reynolds number. This enables us to uncover multiple distinguished limits along with new asymptotic solutions that are presented for the first time. Among them are WKB approximations of type II and III that are systematically evaluated and discussed. The WKB solutions are shown to exhibit a peculiar singularity that warrants the use of matched asymptotic expansions to produce uniformly valid representations. Our solutions are obtained for any characteristic mean flow function satisfying Berman’s similarity condition for porous tubes. They are also derived to an arbitrary level of precision using a recursive formulation that can reproduce each of the asymptotic solutions to any prescribed order. Finally, our solutions are verified numerically over a wide range of physical parameters and through limiting process approximations

Linear stability analysis of subaqueous bedforms using direct numerical simulations

We present results on the formation of ripples from linear stability analysis. The analysis is coupled with direct numerical simulations of turbulent open-channel flow over a fixed sinusoidal bed. The presence of the sediment bed is accounted for using the immersed boundary method. The simulations are used to extract the bed shear stress and consequently the sediment transport rate. The approach is different from traditional linear stability analysis in the sense that the phase lag between the bed topology and the sediment flux is obtained from the three-dimensional turbulent simulations. The stability analysis is performed on the Exner equation, whose input, the sediment flux, is provided from the simulations. We ran 11 simulations at a fixed shear Reynolds number of 180, but for different sediment bed wavelengths. The analysis allows us to sweep a large range of physical and modelling parameters to predict their effects on linear growth. The Froude number appears to be the critical controlling parameter in the early linear development of ripples, in contrast with the dominant role of particle Reynolds number during the equilibrium stage. We also present results from a wave packet analysis using a one-dimensional Gaussian ridge

Sediment Patterns from Fluid-Bed Interactions: A Direct Numerical Simulations Study on Fluvial Turbulent Flows

We present results on the initial formation of ripples from an initially flattened erodible bed. We use direct numerical simulations (DNS) of turbulent open channel flow over a fixed sinusoidal bed coupled with hydrodynamic stability analysis. We use the direct forcing immersed boundary method to account for the presence of the sediment bed. The resolved flow provides the bed shear stress and consequently the sediment transport rate, which is needed in the stability analysis of the Exner equation. The approach is different from traditional linear stability analysis in the sense that the phase lag between the bed topology, and the sediment flux is obtained from the DNS. We ran 11 simulations at a fixed shear Reynolds number of 180, but for different sediment bed wavelengths. The analysis allows us to sweep a large range of physical and modelling parameters to predict their effects on linear growth. The Froude number appears to be the critical controlling parameter in the early linear development of ripples, in contrast with the dominant role of particle Reynolds number during the equilibrium stage

On the Role of Sidewalls in the Transition From Straight to Sinuous Bedforms

We present results from direct numerical simulation on the transition from straight-crested to sinuous-crested bedforms. The numerical setup is representative of turbulent open channel flow over an erodible sediment bed at a shear Reynolds number of Reτ = 180. The immersed boundary method accounts for the presence of the bed. The simulations are two-way coupled such that the turbulent flow can erode and modify the bed, and in turn, the bed modifies the overlying flow. Coupling from the flow to the bed occurs through the Exner equation, while back coupling from the bed to the flow is achieved by imposing the no-slip and no-penetration condition at the immersed boundary. The simulation setup is similar to that by Zgheib et al. (2018a, https://doi.org/10.1002/2017JF004398) except for the presence of sidewalls to better mimic laboratory flume conditions. Sidewalls are observed to significantly increase bedform sinuosity. Key Points Lateral sidewalls significantly increase crestline sinuosity Influence of lateral domain extent on sinuosity is small but noticeable Influence of lateral extent is amplified in the presence of sidewall

Spreading of non-planar non-axisymmetric gravity and turbidity currents

The dynamics of non-axisymmetric turbidity currents is considered here. The study comprises a series of experiments for which a finite volume of particle-laden solution is released into fresh water. A mixture of water and polystyrene particles of diameter 280<Dp<315μm and density ρc=1007Kg/m3 is initially confined in a hollow cylinder at the center of a large tank filled with fresh water. Cylinders with four different cross-sections are examined: a circle, a plus-shape, a rectangle and a rounded rectangle in which the sharp corners are smoothened. The time evolution of the front is recorded as well the spatial distribution of the thickness of the final deposit via the use of a laser triangulation technique. The dynamics of the front and final deposit are significantly influenced by the initial geometry, displaying substantial azimuthal variation especially for the rectangular case where the current extends farther and deposits more particles along the initial minor axis of the rectangular cross section. Interestingly, this departure from axisymmetry cannot be predicted by current theoretical methods such as the Box Model. Several parameters are varied to assess the dependence on the settling velocity, initial height aspect ratio, local curvature and mixture density