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When it comes to the safety of cosmetic products, compliance with regulatory standards is crucialto guarantee consumer protection against the risks of skin irritation. Toxicologists must thereforebe fully conversant with all risks. This applies not only to their day-to-day work, but also to allthe algorithms they integrate into their routines. Recognizing this, ensuring the reproducibility ofalgorithms becomes one of the most crucial aspects to address.However, how can we prove the robustness of an algorithm such as the random forest, that reliesheavily on randomness? In this report, we will discuss the strategy of integrating random forest intoocular tolerance assessment for toxicologists.We will compare four packages: randomForest and Ranger (R packages), adapted in Python via theSKRanger package, and the widely used Scikit-Learn with the RandomForestClassifier() function.Our goal is to investigate the parameters and sources of randomness affecting the outcomes ofRandom Forest algorithms.By setting comparable parameters and using the same Pseudo-Random Number Generator (PRNG),we expect to reproduce results consistently across the various available implementations of therandom forest algorithm. Nevertheless, this exploration will unveil hidden layers of randomness andguide our understanding of the critical parameters necessary to ensure reproducibility across all fourimplementations of the random forest algorithm.

Ion adsorption and dynamics in porous carbons is crucial for many technologies such as energy storage and desalination. Nuclear Magnetic Resonance (NMR) spectroscopy is a key method to investigate such systems thanks to the possibility to distinguish adsorbed (in-pore) and bulk (ex-pore) species in the spectra. However, the large variety of magnetic environments experienced by the ions adsorbed in the particles and the existence of dynamic exchange between the inside of the particles and the bulk renders the intepretation of the NMR experiments very complex. In this work, we optimise and apply a mesoscopic model to simulate NMR spectra of ions in systems where carbon particles of different sizes can be considered. We demonstrate that even for monodisperse systems, complex NMR spectra, with broad and narrow peaks, can be observed. We then show that the inclusion of polydispersity is essential to recover some experimentally observed features, such as the co-existence of peaks assigned to in-pore, exchange and bulk. Indeed, the variety of exchange rates between in-pore and ex-pore environments, present in experiments but not taken into account in analytical models, is necessary to reproduce the complexity of experimental NMR spectra.

Molecular dynamics simulations have been used extensively to determine equilibrium properties of the electrode-electrolyte interface in supercapacitors held at various potentials. While such studies are essential to understand and optimize the performance of such energy storage systems, investigations of the dynamics of adsorption during the charge of the supercapacitors is also necessary. Dynamical properties are especially important to get an insight into the power density of supercapacitors, one of their main assets. In this work, we propose a new method to analyze the trajectories of adsorbing ions. We focus on pure 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide in contact with planar carbon electrodes. We characterize the evolution of the ion orientation and ion-electrode distance during adsorption and show that ions reorientate as they adsorb. We then determine the forces experienced by the adsorbing ions and demonstrate that Coulomb forces dominate at long range while van der Waals forces dominate at short range. We also show that there is an almost equal contribution from the two forces at an intermediate distance, explaining the peak of ion density close to the electrode surface.

We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure $\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on a parameter $\theta\in(0,1]$ and the set of functions with finite energy grows with $\theta$. For $\theta<1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in $\Omega_1\times\Omega_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White's rectifiability criterion. When $\theta=1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1\,$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.

This paper deals with a variant of the optimal transportation problem. Given f $\in$ L 1 (R d , [0, 1]) and a cost function c $\in$ C(R d x R d) of the form c(x, y) = k(y -- x), we minimise $\int$ c d$\gamma$ among transport plans $\gamma$ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 -- f. Denoting by $\Upsilon$(f) the infimum of this problem, we then consider the maximisation problem sup{$\Upsilon$(f) : $\int$ f = m} where m > 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m.

Characterizing ion adsorption and diffusion in porous carbons is essential to understand the performance of such materials in a range of key technologies such as energy storage and capacitive deionisation. Nuclear Magnetic Resonance (NMR) spectroscopy is a powerful technique to get insights in these systems thanks to its ability to distinguish between bulk and adsorbed species and to its sensitivity to dynamic phenomena. Nevertheless, a clear interpretation of the experimental results is sometimes rendered difficult by the various factors affecting NMR spectra. A mesoscopic model to predict NMR spectra of ions diffusing in carbon particles is adapted to include dynamic exchange between the intra-particle space and the bulk electrolyte surrounding the particle. A systematic study of the particle size effect on the NMR spectra for different distributions of magnetic environments in the porous carbons is conducted. The model demonstrates the importance of considering a range of magnetic environments, instead of a single chemical shift value corresponding to adsorbed species, and of including a range of exchange rates (between in and out of the particle), instead of a single timescale, to predict realistic NMR spectra. Depending on the pore size distribution of the carbon particle and the ratio between bulk and adsorbed species, both the NMR linewidth and peak positions can be largely influenced by the particle size.

In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.

A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties:(1) The tangent planes to $\mu_A$ split as $T_x\mu_A=L^1(x)\times L^2(x)$ for some $k_1$-plane $L^1(x)\subset\mathbb{R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset\mathbb{R}^{n_2}$.(2) $A=A_{\vert\Sigma^1\times\Sigma^2}$ for some sets $\Sigma^1\subset\mathbb{R}^{n_1}$, $\Sigma^2\subset\mathbb{R}^{n_2}$ such that $\Sigma^1$ is $k_1$-rectifiable and $\Sigma^2$ is $k_2$-rectifiable (we say that $A$ is $(k_1,k_2)$-rectifiable).The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ which generalize Fleming's $G$-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable.

We introduce the notion of set-decomposition of a normal G-flat chain. We show that any normal rectifiable $G$-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their ``measure theoretic'' connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer.As opposed to previous results, we do not assume that G is boundedly compact. Therefore we cannot rely on the compactness of sequences of chains with uniformly bounded N-norms. We deduce instead the result from a new abstract decomposition principle. As in earlier proofs a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h-mass to replace integrality.

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of global weak solutions to the problem. For this, a semi-discrete in time minimizing movements scheme \`a la De Giorgi is introduced. First, the existence of solutions to the scheme is established and then, using a priori estimates, it is proved that as the time step goes to zero these solutions converge up to extraction towards a weak solution to the free boundary model.

In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary which is uniform as the parameter $\varepsilon$ goes to $0$. Since the two terms in the energy are of the same order when $\varepsilon$ is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for $\varepsilon$ small enough, volume-constrained minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable $G$ with sufficiently fast decay at infinity.

Interfaces between aqueous electrolytes and nanoporous carbons are involved in a number of technological applications such as energy storage and capacitive deionization. The disordered nature of the carbon materials makes it challenging to characterize ion adsorption and relationships between materials properties and performance. Nuclear magnetic spectroscopy can be very helpful in that respect thanks to its nuclei specificity and ability to distinguish between ions in the bulk and in pores. Nevertheless, several factors can affect the measured chemical shifts making it difficult to interpret experimental results. We use complementary methods, namely density functional theory calculations, molecular dynamics simulations and a mesoscopic model, to investigate various factors affecting the chemical shifts of aqueous electrolyte species. We show that the relative importance of these factors depend on the ion nature. In particular, the hydration shell has a much more pronounced effect for large polarizable ions such as Rb$^+$ and Cs$^+$ compared to Li$^+$.

We report on the study of a detection scheme based on a Bloch separator in two-photon Bragg interferometers. We increase the spatial separation between the two output ports of the interferometer by selectively imparting 30 Bloch oscillations to one of them before their detection via time of flight. This method allows increasing the duration of the interferometer by reducing the time for discriminating the ports of the interferometer at detection. We study in detail the impact of this separator on the performance of a dual gravity sensor, and in particular on its measurement sensitivities to gravity acceleration and gravity gradients.

We are interested in a reduced model for corrosion of iron, in which ferric cations and electrons evolve in a fixed oxide layer subject to a self-consistent electrostatic potential. Reactions at the boundaries are modeled thanks to Butler-Volmer formulas, whereas the boundary conditions on the electrostatic potential model capacitors located at the interfaces between the materials. Our model takes inspiration in existing papers, to which we bring slight modifications in order to make it consistent with thermodynamics and its second principle. Building on a free energy estimate, we establish the global in time existence of a solution to the problem without any restriction on the physical parameters, in opposition to previous works. The proof further relies on uniform estimates on the chemical potentials that are obtained thanks to Moser iterations. Numerical illustrations are finally provided to highlight the similarities and the differences between our new model and the one previously studied in the literature.

Atom interferometry offers new perspectives for geophysics and inertial sensing. We present the industrial prototype of a new type of quantum-based instrument: a compact, transportable, differential quantum gravimeter capable of measuring simultaneously the absolute values of both gravitational acceleration, $g$, and its vertical gradient, $\Gamma_{zz}$. While the sensitivity to g is competitive with the best industrial gravimeters, the sensitivity on $\Gamma_{zz}$ reaches the limit set by quantum projection noise-leading to an unprecedented long-term stability of 0.1 E ($1E=1\times 10^{-9}s^{-2}$). This unique, dual-purpose instrument, paves the way for new applications in geophysics, civil engineering, and gravity-aided navigation, where accurate mapping of the gravitational field plays an important role.

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.