7,518 research outputs found
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 -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 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 , many
important questions have been related to the study of linear statistics of
eigenvalues , where is a known
function. We study here truncated linear statistics where the sum is restricted
to the largest eigenvalues: .
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 . Using the Coulomb gas
technique, we study the limit with fixed. We show that the
constraint that 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
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, . 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 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
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