Sequences of Exact Analytical Solutions for Plane-Waves in Graded Media

We present a new method for building sequences of solvable profiles of the electromagnetic (EM) admittance in lossless isotropic materials with 1D graded permittivity and permeability (in particular profiles of the optical refractive-index). These solvable profiles lead to analytical closed-form expressions of the EM fields, for both TE and TM modes. The Property-and-Field Darboux Transformations method, initially developed for heat diffusion modelling, is here transposed to the Maxwell equations in the optical-depth space. Several examples are provided, all stemming from a constant seed-potential, which makes them based on elementary functions only. Solvable profiles of increasingly complex shape can be obtained by iterating the process or by assembling highly flexible canonical profiles. Their implementation for modelling optical devices like matching layers, rugate filters, Bragg gratings, chirped mirrors or 1D photonic crystals, offers an exact and cost-effective alternative to the classical approachesComment: 74 pages, 20 figures, Corrected typos in Annex

Multipurpose S-shaped solvable profiles of the refractive index: application to modeling of antireflection layers and quasi-crystals

A class of four-parameter solvable profiles of the electromagnetic admittance has recently been discovered by applying the newly developed Property & Field Darboux Transformation method (PROFIDT). These profiles are highly flexible. In addition, the related electromagnetic-field solutions are exact, in closed-form and involve only elementary functions. In this paper, we focus on those who are S-shaped and we provide all the tools needed for easy implementation. These analytical bricks can be used for high-level modeling of lightwave propagation in photonic devices presenting a piecewise-sigmoidal refractive-index profile such as, for example, antireflection layers, rugate filters, chirped filters and photonic crystals. For small amplitude of the index modulation, these elementary profiles are very close to a cosine profile. They can therefore be considered as valuable surrogates for computing the scattering properties of components like Bragg filters and reflectors as well. In this paper we present an application for antireflection layers and another for 1D quasicrystals (QC). The proposed S-shaped profiles can be easily manipulated for exploring the optical properties of smooth QC, a class of photonic devices that adds to the classical binary-level QC.Comment: 14 pages, 18 fi

The computation of the axisymmetric squeeze film problem in elastohydrodynamic lubrication

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Joint asymptotics for semi-nonparametric regression models with partially linear structure

We consider a joint asymptotic framework for studying semi-nonparametric regression models where (finite-dimensional) Euclidean parameters and (infinite-dimensional) functional parameters are both of interest. The class of models in consideration share a partially linear structure and are estimated in two general contexts: (i) quasi-likelihood and (ii) true likelihood. We first show that the Euclidean estimator and (pointwise) functional estimator, which are re-scaled at different rates, jointly converge to a zero-mean Gaussian vector. This weak convergence result reveals a surprising joint asymptotics phenomenon: these two estimators are asymptotically independent. A major goal of this paper is to gain first-hand insights into the above phenomenon. Moreover, a likelihood ratio testing is proposed for a set of joint local hypotheses, where a new version of the Wilks phenomenon [Ann. Math. Stat. 9 (1938) 60-62; Ann. Statist. 1 (2001) 153-193] is unveiled. A novel technical tool, called a joint Bahadur representation, is developed for studying these joint asymptotics results.Comment: Published at http://dx.doi.org/10.1214/15-AOS1313 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear Data: Theory and Algorithms

Techniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications

Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques

A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated