Lipschitz regularity for elliptic equations with random coefficients

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched $L^2$ estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.Comment: 85 pages, minor revisio

Truncated linear statistics associated with the top eigenvalues of random matrices

Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(\lambda_1,\ldots,\lambda_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=\sum_{i=1}^Nf(\lambda_i)$, where $f(\lambda)$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1<N$ largest eigenvalues: $\tilde{L}=\sum_{i=1}^{N_1}f(\lambda_i)$. Motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with $f(\lambda)=\sqrt{\lambda}$. Using the Coulomb gas technique, we study the $N\to\infty$ limit with $N_1/N$ fixed. We show that the constraint that $\tilde{L}=\sum_{i=1}^{N_1}f(\lambda_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. Assuming that $f(\lambda)$ is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal.Comment: LaTeX, 30 pages, 20 pdf figures. Updated version: a typo has been corrected in Eq. (3.30) and more details are provided in the Appendi

Characteristic Functions Describing the Power Absorption Response of Periodic Structures to Partially Coherent Fields

Many new types of sensing or imaging surfaces are based on periodic thin films. It is explained how the response of those surfaces to partially coherent fields can be fully characterized by a set of functions in the wavenumber spectrum domain. The theory is developed here for the case of 2D absorbers with TE illumination and arbitrary material properties in the plane of the problem, except for the resistivity which is assumed isotropic. Sum and difference coordinates in both spatial and spectral domains are conveniently used to represent the characteristic functions, which are specialized here to the case of periodic structures. Those functions can be either computed or obtained experimentally. Simulations rely on solvers based on periodic-boundary conditions, while experiments correspond to Energy Absorption Interferometry (EAI), already described in the literature. We derive rules for the convergence of the representation versus the number of characteristic functions used, as well as for the sampling to be considered in EAI experiments. Numerical examples are given for the case of absorbing strips printed on a semi-infinite substrate.Comment: Submitted to JOSA

Modeling customer loyalty using customer lifetime value.

The definition and modeling of customer loyalty have been central issues in customer relationship management since many years. Recent papers propose solutions to detect customers that are becoming less loyal, also called churners. The churner status is then defined as a function of the volume of commercial transactions. In the context of a Belgian retail financial service company, our first contribution will be to redefine the notion of customer's loyalty by considering it from a customer-centric point-of-view instead of a product-centric point-of-view. We will hereby use the customer lifetime value (CLV) defined as the discounted value of future marginal earnings, based on the customer's activity. Hence, a churner will be defined as someone whose CLV, thus the related marginal profit, is decreasing. As a second contribution, the loss incurred by the CLV decrease will be used to appraise the cost to misclassify a customer by introducing a new loss function. In the empirical study, we will compare the accuracy of various classification techniques commonly used in the domain of churn prediction, including two cost-sensitive classirfiers. Our final conclusion is that since profit is what really matters in a commercial environment, standard statistical accuracy measures or prediction need to be revised and a more profit oriented focus may be desirable.Churn prediction; Classification; Customer lifetime value; Prediction models;

Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences

In this work we clarify some properties of the one-loop IR divergences in non-Abelian gauge field theories on non-commutative 4-dimensional Moyal space. Additionally, we derive the tree-level Slavnov-Taylor identities relating the two, three and four point functions, and verify their consistency with the divergent one-loop level results. We also discuss the special case of two dimensions.Comment: 21 pages, 3 figures; v2: minor corrections and references adde

Increasing the imaging capabilities of multimode fibers by exploiting the properties of highly scattering media

We present a novel design that exploits the focusing properties of scattering media to increase the resolution and the working distance of multimode fiber based imaging devices. Placing a highly scattering medium in front of the distal tip of the multimode fiber enables the formation of smaller sized foci at increased working distances away from the fiber tip. We perform a parametric study of the effect of the working distance and the separation between the fiber and the scattering medium on the focus size. We experimentally demonstrate submicron focused spots as far away as 800{\mu}m with 532nm light.Comment: 4 pages, 3 figure

Jensen-Feynman approach to the statistics of interacting electrons

Faussurier et al. [Phys. Rev. E 65, 016403 (2001)] proposed to use a variational principle relying on Jensen-Feynman (or Gibbs-Bogoliubov) inequality in order to optimize the accounting for two-particle interactions in the calculation of canonical partition functions. It consists in a decomposition into a reference electron system and a first-order correction. The procedure appears to be very efficient in order to evaluate the free energy and the orbital populations. In this work, we present numerical applications of the method and propose to extend it using a reference energy which includes the interaction between two electrons inside a given orbital. This is possible thanks to our efficient recursion relation for the calculation of partition functions. We also show that a linear reference energy, however, is usually sufficient to achieve a good precision and that the most promising way to improve the approach of Faussurier et al. is to apply Jensen's inequality to a more convenient convex function.Comment: submitted to Physical Review

Characterization of Power Absorption Response of Periodic 3D Structures to Partially Coherent Fields

In many applications of absorbing structures it is important to understand their spatial response to incident fields, for example in thermal solar panels, bolometric imaging and controlling radiative heat transfer. In practice, the illuminating field often originates from thermal sources and is only spatially partially coherent when reaching the absorbing device. In this paper, we present a method to fully characterize the way a structure can absorb such partially coherent fields. The method is presented for any 3D material and accounts for the partial coherence and partial polarization of the incident light. This characterization can be achieved numerically using simulation results or experimentally using the Energy Absorption Interferometry (EAI) that has been described previously in the literature. The absorbing structure is characterized through a set of absorbing functions, onto which any partially coherent field can be projected. This set is compact for any structure of finite extent and the absorbing function discrete for periodic structures

Fluctuations of observables for free fermions in a harmonic trap at finite temperature

We study a system of 1D noninteracting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of certain observables of the form of sums of a function of the fermions' positions, $\mathcal{L}=\sum_n h(x_n)$. Such observables are also called linear statistics of the positions. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations. We discuss the universality of the results with respect to the nature of the confinement.Comment: 36 pages, 6 pdf figure