Nonlinear eigenvalue problem for optimal resonances in optical cavities

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <= e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities e_1 and e_2. To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.Comment: 16 page

Generalized Newton's Method based on Graphical Derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations

Ultramodular functions.

We study the properties of ultramodular functions, a class of functions that generalizes scalar convexity and that naturally arises in some economic and statistical applications.

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Optimal designs for the methane flux in troposphere

The understanding of methane emission and methane absorption plays a central role both in the atmosphere and on the surface of the Earth. Several important ecological processes, e.g., ebullition of methane and its natural microergodicity request better designs for observations in order to decrease variability in parameter estimation. Thus, a crucial fact, before the measurements are taken, is to give an optimal design of the sites where observations should be collected in order to stabilize the variability of estimators. In this paper we introduce a realistic parametric model of covariance and provide theoretical and numerical results on optimal designs. For parameter estimation D-optimality, while for prediction integrated mean square error and entropy criteria are used. We illustrate applicability of obtained benchmark designs for increasing/measuring the efficiency of the engineering designs for estimation of methane rate in various temperature ranges and under different correlation parameters. We show that in most situations these benchmark designs have higher efficiency.Comment: 25 pages, 4 figure

Projective structures, grafting, and measured laminations

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective (\CP^1) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.Comment: 31 pages, 4 figure