# On the modelling of semi-insulating GaAs including surface tension and bulk stresses

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank

*2010 Mathematics Subject Classification*

- 74N20 74A15 74N05 74B99

*2008 Physics and Astronomy Classification Scheme*

- 82.60.-s 61.72.Bb 61.72.Qq 64.10.+h 64.30.-t 64.70.Dv

*Keywords*

- thermodynamic, phase transition, phase diagrams, precipitates, surface stress, deviatoric stress, chemical potentials, elasticity, GaAs

*DOI*

*Abstract*

Necessary heat treatment of single crystal semi-insulating Gallium Arsenide (GaAs), which is deployed in micro- and opto- electronic devices, generate undesirable liquid precipitates in the solid phase. The appearance of precipitates is influenced by surface tension at the liquid/solid interface and deviatoric stresses in the solid.

The central quantity for the description of the various aspects of phase transitions is the chemical potential, which can be additively decomposed into a chemical and a mechanical part. In particular the calculation of the mechanical part of the chemical potential is of crucial importance. We determine the chemical potential in the framework of the St. Venant--Kirchhoff law which gives an appropriate stress/strain relation for many solids in the small strain regime. We establish criteria, which allow the correct replacement of the St. Venant--Kirchhoff law by the simpler Hooke law.

The main objectives of this study are: (*i*) We develop a thermo-mechanical model that describes diffusion and interface motion, which both are strongly influenced by surface tension effects and deviatoric stresses. (*ii*) We give an overview and outlook on problems that can be posed and solved within the framework of the model. (*iii*) We calculate non-standard phase diagrams, i.e. those that take into account surface tension and non-deviatoric stresses, for GaAs above 786°C, and we compare the results with classical phase diagrams without these phenomena.

*Appeared in*

- Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008) pp. 2693-2720.

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# On the Becker/Döring theory of nucleation of liquid droplets in solids

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank

*2010 Mathematics Subject Classification*

- 74A25 74A15 80A30

*2008 Physics and Astronomy Classification Scheme*

- 8.60.Nh 82.20.Db 82.20.-w 61.72.Bb 61.72.Qq 64.70.Dv

*Keywords*

- nucleation, kinetic of phase transition, metastability, surface stress, GaAs, elasticity

*DOI*

*Abstract*

Nucleation of liquid precipitates in semi-insulating GaAs is accompanied by deviatoric stresses resulting from the liquid/solid misfit. A competition of surface tension and stress deviators at the interface determines the nucleation barrier. The evolution of liquid precipitates in semi-insulating GaAs is due to diffusional processes in the vicinity of the droplet. The diffusion flux results from a competition of chemical and mechanical driving forces. The size distribution of the precipitates is determined by a Becker/Döring system. The study of its properties in the presence of deviatoric stresses is the subject of this study. The main tasks of this study are: (*i*) We propose a new Becker/Döring model that takes thermomechanical coupling into account. (*ii*) We compare the current model with already existing models from the literature. Irrespective of the incorporation of mechanical stresses, the various models differ by different environments where the evolution of precipitates takes place. (*iii*) We determine the structure of equilibrium solutions according to the Becker/Döring model, and we compare these solutions with those that result from equilibrium thermodynamics.

*Appeared in*

- J. Statist. Phys., 123 (2006) pp. 55--87.

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# Diffusion in the vicinity of an evolving spherical arsenic droplet

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank
- Qamar, Shamsul

*2010 Mathematics Subject Classification*

- 74N25 82B24 82C26

*2008 Physics and Astronomy Classification Scheme*

- 82.20.Wt 61.72.Bb 61.72.Qq 64.70.Dv

*Keywords*

- phase transition, diffusion, thermomecanical coupling, GaAs, precipitates

*DOI*

*Abstract*

We study the diffusion problem of liquid droplets in single crystal semi-insulating Gallium Arsenide (GaAs). This problem is posed by an industrial application, where the droplets, also called precipitates, appear during a necessary heat treatment of GaAs wafer. The subsequent dissolution of the droplets is mandatory, in order to use the wafer after the heat treatment as a substrate material for micro- and opto- electronic devices.

In this study we consider a single droplet in a solid matrix, which is in contact with an arsenic gas, so that the arsenic can cross the solid/gas interface. The model equations have been derived by the authors. They consist of a nonlinear diffusion equation with diffusion controlled and kinetic boundary conditions, respectively, at the liquid/solid interface. Furthermore we study at the solid/gas interface alternatively zero flux and Dirichlet conditions.

Surface tension at the liquid/solid interface and deviatoric stresses in the solid are taken into account. The latter appear due to different densities of liquid and solid GaAs. There is a large influence of these effects on diffusion, interface motion and phase diagrams, which are used to identify regions, where coexistence of liquid and solid phases is possible.

In order to study the evolution of the droplet, and in particular possibilities to enforce its dissolution, we solve several initial and boundary value problems for the diffusion system.

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# Current coupling of drift-diffusion models and dissipative Schrödinger--Poisson systems: Dissipative hybrid models

*Authors*

- Baro, Michael
- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47B44 47E05 35J05

*2008 Physics and Astronomy Classification Scheme*

- 73.23.-b, 73.23.Ad, 73.63.-b, 73.63.Nm

*Keywords*

- semi-conductors, quantum-classical coupling, hybrid models, drift-diffusion models, dissipative Schrödinger systems, Poisson equation, current coupling

*DOI*

*Abstract*

A 1D coupled drift-diffusion dissipative Schrödinger model (hybrid model), which is capable to describe the transport of electrons and holes in semi-conductor devices in a non-equilibrium situation, is mathematically analyzed. The device domain is split into a part where the transport is well-described by the drift-diffusion equations (classical zone) and a part where a quantum description via a dissipative Schrödinger system (quantum zone) is used. Both system are coupled such that the continuity of the current densities is guaranteed. The electrostatic potential is self-consistently determined by Poisson's equation on the whole device. We show that the hybrid model is well-posed, prove existence of solutions and show their uniform boundedness provided the distribution function satisfy a so-called balance condition. The current densities are different from zero in the non-equilibrium case and uniformly bounded.

*Appeared in*

- SIAM J. Math. Anal., Vol. 37, No. 3, pp. 941-981, 2005

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# Conditional excursion representation for a class of interacting superprocesses

*Authors*

- Li, Zenhu
- Wang, Hao
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60J80 60G57 60J35

*Keywords*

- superprocess, interaction, immigration, non-linear SPDE, conditional log-Laplace functional, excursion law

*DOI*

*Abstract*

A class of interacting superprocesses, called superprocesses with dependent spatial motion (SDSMs), has been introduced and characterized in Wang citeWang98 and Dawson et al. citeDLW01. In this paper, we give a construction or an excursion representation of the non-degenerate SDSM with immigration by making use of a Poisson system associated with the conditional excursion laws of the SDSM. As pointed out in Wang citeWang98, the multiplicative property or summable property is lost for SDSMs and immigration SDSMs. However, summable property is the foundation of excursion representation. This raises a sequence of technical difficulties. The main tool we used is the conditional log-Laplace functional technique that gives the conditional summability, the conditional excursion law, and the Poisson point process for the construction of the immigration SDSMs.

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# Stability of infinite dimensional control problems with pointwise state constraints

*Authors*

- Hinze, Michael
- Meyer, Christian

*2010 Mathematics Subject Classification*

- 49K20 49N10 49M20

*Keywords*

- Optimal control of semi-linear elliptic equations, pointwise state constraints, finite element approximation

*DOI*

*Abstract*

A general class of nonlinear infinite dimensional optimization problems is considered that covers semi-linear elliptic control problems with distributed control as well as boundary control. Moreover, pointwise inequality constraints on the control and the state are incorporated. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. These class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data.

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# Time step truncation in direct simulation Monte Carlo for semiconductors

*Authors*

- Muscato, Orazio
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82D37 65C05

*Keywords*

- Boltzmann-Poisson equations, electronic devices, Monte Carlo simulations

*DOI*

*Abstract*

A homogeneous (bulk) silicon semiconductor is studied by using the Direct Simulation Monte Carlo (DSMC). Two DSMC algorithms are considered, the self scattering technique (SST) and the constant time technique (CTT). First, the results obtained by CTT are shown to converge (with vanishing time step) to the results obtained by SST. The truncation error of CTT turns out to be of first order with respect to the time step. Second, the efficiency of both algorithms is compared. It is found that SST is more efficient if a high precision (relative error less then three percent) of the results is needed.

*Appeared in*

- Math. Comput. Modelling, 42 (2005) pp. 683-700.

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# A mathematical model for case hardening of steel

*Authors*

- Fasano, Antonio
- Hömberg, Dietmar
- Panizzi, Lucia

*2010 Mathematics Subject Classification*

- 35K60 35R05 82B26

*Keywords*

- Heat treatment, phase transitions, coupled PDE, case hardening

*DOI*

*Abstract*

A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.

*Appeared in*

- Math. Models Methods Appl. Sci., 19 (2009) pp. 2101--2126.

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# Complete damage in elastic and viscoelastic media and its energetics

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš
- Zeman, Jan

*2008 Physics and Astronomy Classification Scheme*

- 46.15.-x 62.20.-x

*Keywords*

- Inelastic damage, small strain, energetic formulation

*DOI*

*Abstract*

A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 199 (2010) pp. 1242--1253.

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# Discrete random walk on large spherical grids generated by spherical means for PDEs

*Authors*

- Sabelfeld, Karl
- Shalimova, Irina
- Levykin, Alexander I.

*2010 Mathematics Subject Classification*

- 65C05 76F99

*DOI*

*Abstract*

A new general stochastic-deterministic approach for a numerical solution of boundary value problems of potential and elasticity theories is suggested. It is based on the use of the Poisson-like integral formulae for overlapping spheres. An equivalent system of integral equations is derived and then approximated by a system of linear algebraic equations. We develop two classes of special Monte Carlo iterative methods for solving these systems of equations which are a kind of stochastic versions of the Chebyshev iteration method and successive overrelaxation method (SOR). In the case of classical potential theory this approach accelerates the convergence of the well known Random Walk on Spheres method (RWS). What is however much more important, this approach suggests a first construction of a fast convergent finite-variance Monte Carlo method for the system of Lamé equations.

*Appeared in*

- Monte Carlo Methods Appl., vol 10 (2004), no. 3-4, pp. 559-574

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# Chaotic bound state of localized structures in the complex Ginzburg--Landau equation

*Authors*

- Turaev, Dmitry
- Vladimirov, Andrei
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 78A60 35Q60 35B32

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf 05.45.-a 42.65.Tg

*Keywords*

- dissipative soliton, pulse interaction, homoclinic bifurcation

*DOI*

*Abstract*

A new type of stable dynamic bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.

*Appeared in*

- Phys. Rev. E, 75 (2007) pp. 045601(R).

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# Conditional log-Laplace functionals of immigration superprocesses with dependent spatial motion

*Authors*

- Li, Zenghu
- Wang, Hao
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60J80 60G57 60J35

*Keywords*

- branching particle system, superprocess, dependent spatial motion, immigration process, non-linear SPDE, conditional log-Laplace functional

*DOI*

*Abstract*

A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional log-Laplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained.

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# On the unique solvability of a nonlocal phase separation problem for multicomponent systems

*Authors*

- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 35K45 45K05 47J35 35D10

*Keywords*

- Nonlocal phase separation models; Cahn--Hilliard equation; Initial boundary value problems; Nonlinear evolution equations;Regularity theory

*DOI*

*Abstract*

A nonlocal model of phase separation in multicomponent systems is presented. It is derived from conservation principles and minimization of free energy containing a nonlocal part due to particle interaction. In contrast to the classical Cahn--Hilliard theory with higher order terms this leads to an evolution system of second order parabolic equations for the particle densities, coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in standard Hilbert spaces for evolution systems. Moreover, using some regularity theory for parabolic boundary value problems in Hoelder spaces we get the unique solvability of our problem. We conclude our considerations with the presentation of simulation results for a ternary system.

*Appeared in*

- Banach Center Publ., 66 (2004) pp. 153-164.

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# Asymptotic convergence results for a system of partial differential equations with hysteresis

*Authors*

- Eleuteri, Michela
- Krejčí, Pavel

*2010 Mathematics Subject Classification*

- 35K55 47J40 35B40

*Keywords*

- partial differential equations, hysteresis, asymptotic convergence, Preisach operator

*DOI*

*Abstract*

A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.

*Appeared in*

- Commun. Pure Appl. Anal., 6 (2007) pp. 1131-1143.

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# On an evolutionary model for complete damage based on energies and stresses

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K65 35K85 49S05 74C05 74R05

*Keywords*

- Weak energetic solution, rate independent energetic system, complete damage, Gamma convergence

*DOI*

*Abstract*

A recent model allows for complete damage, such that the deformation is not well-defined. The evolution can be described in terms of energy densities and stresses. We introduce the notion of *weak energetic solution* and show how the existence theory can be generalized to convex, but non-quadratic elastic energies.

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# Variation of constants formula for hyperbolic systems

*Authors*

- Lichtner, Mark

*2010 Mathematics Subject Classification*

- 35L90 37L05 35B30 58D25 37L50 34K20 34K19

*Keywords*

- Semilinear hyperbolic systems, variations of constants formula, sun star calculus, smooth dependence on data, linarized stability, semigroup theory

*DOI*

*Abstract*

A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space $X$ of continuous functions together with its sun dual space $X^odot ast$. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in $X$. This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup.

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# Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

*Authors*

- Meyer, Christian
- Yousept, Irwin

*2010 Mathematics Subject Classification*

- 35J60 49K20 49M05 65K10

*Keywords*

- Nonlinear optimal control, nonlocal radiation interface conditions, state constraints, first-order necessary conditions, second-order sufficient conditions, Moreau-Yosida approximation

*DOI*

*Abstract*

A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.

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# A higher gradient theory of mixtures for multi-component materials with numerical examples for binary alloys

*Authors*

- Böhme, Thomas
- Dreyer, Wolfgang
- Duderstadt, Frank
- Müller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 74A15 74A50 74N15 74N25 80A17 80A20 82D35

*Keywords*

- Thermodynamics, structured surfaces and interfaces, coexistent phases, analysis of microstructure, transformations involving diffusion

*DOI*

*Abstract*

A theory of mixture for multi-component materials is presented based on a novel, straightforward method for the exploitation of the Second Law of thermodynamics. In particular the constitutive equations for entropy, heat and diffusion flux as well as the stress tensor are formulated as a consequence of the non-negative entropy production. Furthermore we derive the established Gibbs equation as well as the Gibbs Duhem relation which also follow from the formalism. Moreover, it is illustrated, how local mechanical strains due to eigenstrains or external loadings, modify the free energy and, consequently, change the chemical potentials of the components. All consecutive steps are illustrated, first, for simple mixtures and, second, for a system containing two different phases. So-called higher gradients of the concentrations are considered, which take the nonuniform composition into account. It will also become apparent that more/other variables of modified/different physical pr oblems beyond the illustrated ones can be easily treated within the presented framework. This work ends with the specification to binary alloys and with the presentation of various numerical simulations.

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# Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D05 35J60

*Keywords*

- non local boundary condition, right-hand side in $L^p (p geq 1)$

*DOI*

*Abstract*

Accurate modeling of heat transfer in high-temperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L-1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a right-hand side in L-p with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with non-local boundary conditions and right-hand side in L-p with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the L-p-norm of the heat-sources, and to pass to the limit.

*Appeared in*

- Math. Methods Appl. Sci., 32 (2008) pp. 135 - 166.

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# Universality of REM-like ageing in mean field spin glasses

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Černý, Jiři

*2008 Physics and Astronomy Classification Scheme*

- 75.10.Nr, 75.10.Jm, 75.10.Hk, 05.30.-d

*Keywords*

- aging, universality, spin glasses, SK model, random walk

*DOI*

*Abstract*

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model

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# Monotonicity properties of the quantum mechanical particle density

*Authors*

- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47H05 47B10

*Keywords*

- Trace functionals, trace class operators, monotonicity

*DOI*

*Abstract*

An elementary proof of the anti-monotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.

*Appeared in*

- Monatsh. Math., 158 (2009) pp. 179--185.

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# An inverse problem for fluid-solid interaction

*Authors*

- Elschner, Johannes
- Hsiao, George C.
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 35R30 76Q05 35J05 35J20 70G75

*Keywords*

- Acoustic and elastic waves, inverse scattering, gradients, Gauss-Newton method

*DOI*

*Abstract*

Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the domain occupied by the obstacle with the Helmholtz equation in the exterior. The far field pattern is obtained applying an integral operator to the scattered field function restricted to a simple smooth surface surrounding the obstacle. The subject of our paper is the inverse problem, where the shape of the elastic body represented by a parametrization of its boundary is to be reconstructed from a finite number of measured far field patterns. We define a family of objective functionals depending on a non-negative regularization parameter such that, for regularization parameter zero, the shape of the sought elastic obstacle is a minimizer of the functional. For any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem in its weak formulation may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.

*Appeared in*

- Inverse Probl. Imaging, 2 (2008) pp. 83--120.

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# Effect of higher-order dispersion on modulation instability, soliton propagation and pulse splitting

*Authors*

- Demircan, Ayhan
- Pietrzyk, Monika
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.81.Dp, 42.65.Sf

*Keywords*

- nonlinear fibers, pulse splitting, third-order dispersion, Modulation instability

*DOI*

*Abstract*

By solving numerically the extended nonlinear Schrödinger equation we investigate the influence of higher-order dispersion effects on the propagation of optical pulses in highly nonlinear fibers. In the anomalous dispersion regime third-order dispersion can, in general, induce soliton fission and yields asymmetric spectra, whereas modulation instability can be slightly suppressed. In the normal dispersion regime we demonstrate pulse splitting by third-order dispersion, as well as its later suppression by fourth-order dispersion.

*Appeared in*

- Opt. Quantum Electron., 40 (2008) pp. 455-460.

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# The effect of time-dependent coupling on non-equilibrium steady states

*Authors*

- Cornean, Horia D.
- Neidhardt, Hagen
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 46N55 47N55 47A40 35L90 47E05

*Keywords*

- non-equilibrium steady states, Landauer-Büttiker formula, Landau-Lifschitz formula, quantum Liouville equation, wave and scattering operator

*DOI*

*Abstract*

Consider (for simplicity) two one-dimensional semi-infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau-Lifschitz and Landauer-Büttiker formulas.

*Appeared in*

- Ann. Henri Poincare, 10 (2009) pp. 61--93.

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# Diffusion Tensor Imaging: Structural adaptive smoothing

*Authors*

- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Voss, Henning U.

*2010 Mathematics Subject Classification*

- 62P10 92C55 62G05

*Keywords*

- diffusion tensor imaging, structural adaptive smoothing

*DOI*

*Abstract*

Diffusion Tensor Imaging (DTI) data is characterized by a high noise level. Thus, estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of DTI in clinical or neuroscience applications. Besides pulse sequence optimization, noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of DTI. Here, we suggest an anisotropic structural adaptive smoothing procedure, which is based on the Propagation-Separation method and preserves the structures seen in DTI and their different sizes and shapes. It is applied to artificial phantom data and a brain scan. We show that this method significantly improves the quality of the estimate of the diffusion tensor and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking.

*Appeared in*

- NeuroImage, 39 (2008) pp. 1763--1773.

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# Reverse approximation of energetic solutions to rate-independent processes

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rindler, Filip

*2010 Mathematics Subject Classification*

- 49J40 49S05 65J15 74C05 74H15

*Keywords*

- Rate-independent processes, energetic solutions, approximate incremental problems, Gamma convergence

*DOI*

*Abstract*

Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.

*Appeared in*

- NoDEA Nonlinear Differential Equations Appl., 16 (2009) pp. 17--40.

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# A descent method for the free energy of multicomponent systems

*Authors*

- Gajewski, Herbert
- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 90C26 82B26 94A08

*Keywords*

- Nonconvex functionals, Cahn-Hilliard equation, Lyapunov function, asymptotic behaviour, phase separation, image segmentation

*DOI*

*Abstract*

Equilibrium distributions of a multicomponent system minimize the free energy functional under the constraint of mass conservation of the components. However, since the free energy is not convex in general, one tries usually to characterize and to construct equilibrium distributions as steady states of an adequate evolution equation (for example, the nonlocal Cahn-Hilliard equation for binary alloys). In this work a direct descent method for nonconvex functionals is established and applied to phase separation problems in multicomponent systems and image segmentation.

*Appeared in*

- Discrete Contin. Dyn. Syst., 15 (2006) pp. 505--528.

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# Simulation of microwave circuits and laser structures including PML by means of FIT

*Authors*

- Hebermehl, Georg
- Schefter, Jürgen
- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q60 65N22 65F15 65F10

*Keywords*

- Microwave device, Semiconductor laser, Simulation, Maxwell's equations, Boundary value problem, PML boundarycondition, Eigenvalue problem, Linearalgebraic equations, Rectangular grids, Tetrahedral nets

*DOI*

*Abstract*

Field-oriented methods which describe the physical properties of microwave circuits and optical structures are an indispensable tool to avoid costly and time-consuming redesign cycles. Commonly the electromagnetic characteristics of the structures are described by the scattering matrix which is extracted from the orthogonal decomposition of the electric field. The electric field is the solution of an eigenvalue and a boundary value problem for Maxwell's equations in the frequency domain. We discretize the equations with orthogonal grids using the Finite Integration Technique (FIT). Maxwellian grid equations are formulated for staggered nonequidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells. The interesting modes of smallest attenuation are found solving a sequence of eigenvalue problems of modified matrices. To reduce the execution time for high-dimensional problems a coarse and a fine grid is used. The calculations are carried out, using two levels of parallelization. The discretized boundary value problem, a large-scale system of linear algebraic equations with different right-hand sides, is solved by a block Krylov subspace method with various preconditioning techniques. Special attention is paid to the Perfectly Matched Layer boundary condition (PML) which causes non physical modes and a significantly increased number of iterations in the iterative methods.

*Appeared in*

- Ädvances in Radio Science", vol. 2, pp. 107--112, (2004)

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# On the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields

*Authors*

- Cherkas, Leonid
- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C05 34C23

*Keywords*

- Multiple limit cycle, degenerate Hopf bifurcation, continuation method

*DOI*

*Abstract*

For plane vector fields depending on three parameters we describe an algorithm to construct a curve in the parameter space such that to each point of this curve there belongs a vector field possessing a limit cycle of multiplicity three. One point of this curve is related to the bifurcation of a limit cycle of multiplicity three from an equilibrium point. The underlying procedure is a continuation method.

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# Asymptotic statistical equivalence for ergodic diffusions

*Authors*

- Dalalyan, Arnak
- Reiß, Markus

*2010 Mathematics Subject Classification*

- 62B15 62C05 62G20 62M99

*Keywords*

- Asymptotic equivalence, statistical experiment, Le Cam distance, ergodic diffusion, mixed Gaussian white noise

*DOI*

*Abstract*

For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam's deficiency between statistical experiments is considered for long-time asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation.

*Appeared in*

- Probab. Theory Related Fields

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# Trace formulae for dissipative and coupled scattering systems

*Authors*

- Behrndt, Jussi
- Malamud, Mark
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A40 47A55 47B44

*Keywords*

- Scattering system, scattering matrix, boundary triplet, Titchmarsh-Weyl function, spectral shift function, Krein-Birman formula

*DOI*

*Abstract*

For scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved.

*Appeared in*

- vol. 188 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2008, pp. 57--93

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# Varying coefficient GARCH versus local constant volatility modeling. Comparison of the predictive power

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 91B84 62P05

*Keywords*

- varying coefficient GARCH, adaptive weights

*DOI*

*Abstract*

GARCH models are widely used in financial econometrics. However, we show by mean of a simple simulation example that the GARCH approach may lead to a serious model misspecification if the assumption of stationarity is violated. In particular, the well known integrated GARCH effect can be explained by nonstationarity of the time series. We then introduce a more general class of GARCH models with time varying coefficients and present an adaptive procedure which can estimate the GARCH coefficients as a function of time. We also discuss a simpler semiparametric model in which the ( beta )-parameter is fixed. Finally we compare the performance of the parametric, time varying nonparametric and semiparametric GARCH(1,1) models and the locally constant model from Polzehl and Spokoiny (2002) by means of simulated and real data sets using different forecasting criteria. Our results indicate that the simple locally constant model outperforms the other models in almost all cases. The GARCH(1,1) model also demonstrates a relatively good forecasting performance as far as the short term forecasting horizon is considered. However, its application to long term forecasting seems questionable because of possible misspecification of the model parameters.

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# Monte Carlo evaluation of American options using consumption processes

*Authors*

- Belomestny, Denis
- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- American and Bermudan options, Lower and Upper bounds, Monte Carlo simulation, Variance reduction

*DOI*

*Abstract*

Here we develop a new approach for pricing both continuous-time and discrete-time American options which is based on the fact that an American option is equivalent to a European option with a consumption process involved. This approach admits construction of an upper bound (a lower bound) on the true price using a lower bound (an upper bound) by Monte Carlo simulation. A number of effective estimators of the upper and lower bounds with reduced variance are proposed. The results obtained are supported by numerical experiments which look promising.

*Appeared in*

- Int. J. Theor. Appl. Finance, 9 (2006) pp. 455--481.

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# A stochastic evolution equation arising from the fluctuation of a class of interacting particle systems

*Authors*

- Kurtz, Thomas G.
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60H15 60H35 60B12 60F17 60F25 60H10 93E11

*Keywords*

- Stochastic partial differential equations, interacting infinite particle system, central limit theorem, Euler scheme

*DOI*

*Abstract*

In an earlier paper, we studied the approximation of solutions $V(t)$ to a class of SPDEs by the empirical measure $V^n(t)$ of a system of $n$ interacting diffusions. In the present paper, we consider a central limit type problem, showing that $sqrt n(V^n-V)$ converges weakly, in the dual of a nuclear space, to the unique solution of a stochastic evolution equation. Analogous results in which the diffusions that determine $V^n$ are replaced by their Euler approximations are also discussed.

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# Robust risk management. Accounting for nonstationarity and heavy tails

*Authors*

- Chen, Ying
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62P20

*Keywords*

- exponential smoothing, spatial aggregation, heavy-tailed distribution

*DOI*

*Abstract*

In the ideal Black-Scholes world, financial time series are assumed 1) stationary (time homogeneous) or can be modelled globally by a stationary process and 2) having conditionally normal distribution given the past. These two assumptions have been widely-used in many methods such as the RiskMetrics, one risk management method considered as industry standard. However these assumptions are unrealistic. The primary aim of the paper is to account for nonstationarity and heavy tails in time series by presenting a local exponential smoothing approach, by which the smoothing parameter is adaptively selected at every time point and the heavy-tailedness of the process is considered. A complete theory addresses both issues. In our study, we demonstrate the implementation of the proposed method in volatility estimation and risk management given simulated and real data. Numerical results show the proposed method delivers accurate and sensitive estimates.

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# Efficient treatments of stationary free boundary problems

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut

*2010 Mathematics Subject Classification*

- 49Q10 49M15 65N38 49K20

*Keywords*

- free boundary problem, shape calculus, Newton method, boundary integral equations, multiscale methods, sufficient second order conditions

*DOI*

*Abstract*

In the present paper we consider the efficient treatment of free boundary problems by shape optimization. We reformulate the free boundary problem as shape optimization problem. A second order shape calculus enables us to realize a Newton scheme to solve this problem. In particular, all evaluations of the underlying state function are required only on the boundary of the domain. We compute these data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems. Furthermore, the stability of the solutions is investigated by treating the second order sufficient optimality conditions of the underlying shape problem.

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# Shape optimization for 3D electrical impedance tomography

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut

*2010 Mathematics Subject Classification*

- 49Q10 49M37 65N38 49K20

*Keywords*

- Electrical impedance tomography, Newton method, regularization, shape calculus, boundary integral equations, wavelets

*DOI*

*Abstract*

In the present paper we consider the identification of an obstacle or void of different conductivity included in a three-dimensional domain by measurements of voltage and currents at the boundary. We reformulate the given identification problem as a shape optimization problem. Since the Hessian is compact at the given hole we apply a regularized Newton scheme as developed by the authors (WIAS-Preprint No. 943). All information of the state equation required for the optimization algorithm can be derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results confirm that the proposed regularized Newton scheme yields a powerful algorithm to solve the considered class of problems.

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# Large deviation principle for the single point catalytic super-Brownian motion

*Authors*

- Fleischmann, Klaus
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60J80

*Keywords*

- Point catalyst, superprocess, large deviations, exponential moments, singular catalytic medium, log-Laplace equation, representation by excursion densities

*DOI*

*Abstract*

In the single point catalytic super-Brownian motion "particles" branch only if they meet the position of the single point catalyst. If the branching rate tends to zero, the model degenerates to the heat flow. We are concerned with large deviation probabilities related to this law of large numbers. To this aim the well-known explicit representation of the model by excursion densities is heavily used. The rate function is described by the Fenchel-Legendre transform of log-exponential moments described by a log-Laplace equation.

*Appeared in*

- Markov Process. Related Fields, 11 (2005), pp. 519-533.

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# The universal Airy$_1$ and Airy$_2$ processes in the totally asymmetric simple exclusion process

*Authors*

- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, universality, Airy process, random matrices

*DOI*

*Abstract*

In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy$_1$ and Airy$_2$ processes. The Airy$_2$ process is an universal limit process occurring also in other models: in a stochastic growth model on $1+1$-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy$_1$ and Airy$_2$ processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.

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# Sensitivities for Bermudan options by regression methods

*Authors*

- Belomestny, Denis
- Milstein, Grigori N.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- Monte Carlo simulation, regression method, conditional probabilistic representations, optimal stopping times, American and Bermudan options, deltas

*DOI*

*Abstract*

In this article we propose several pathwise and finite difference based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations which allows, in combination with a regression approach, an efficient simultaneous computation of sensitivities at all initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods.

*Appeared in*

- Decis. Econ. Finance, 33 (2010) pp. 117--138.

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# Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity

*Authors*

- Knees, Dorothee
- Neff, Patrizio

*2010 Mathematics Subject Classification*

- 35B65 74C05 49N60 74A35 74G40

*Keywords*

- Polar materials, perfect plasticity, higher global regularity, quasilinear elliptic systems, error estimates, time-increments

*DOI*

*Abstract*

In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.

*Appeared in*

- SIAM J. Math. Anal., 40 (2008) pp. 21--43.

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# On the Landau--Levich problem for non-Newtonian liquids

*Authors*

- Afanasiev, Konstantin
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34B15 35G25 35K55 35Q35

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e

*Keywords*

- Lubrication models, non-Newtonian flow, fluid dynamics, phase plane analysis

*DOI*

*Abstract*

In this paper the drag-out problem for shear-thinning liquids at variable inclination angle is considered. For this free boundary problem dimension-reduced lubrication equations are derived for the most commonly used viscosity models, namely, the power-law, Ellis and Carreau model. For the resulting lubrication models a system of ordinary differential equation governing the steady state solutions is obtained. Phase plane analysis is used to characterize the type of possible steady state solutions and their dependence on the rheological parameters.

*Appeared in*

- Phys. Rev. E, 76 (2007) pp. 036307/1--036307/12.

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# On a class of partial differential equations with hysteresis arising in magnetohydrodynamics

*Authors*

- Eleuteri, Michela

*2010 Mathematics Subject Classification*

- 35K55 47J40 76W05

*Keywords*

- partial differential equations, hysteresis, Preisach operator, magnetohydrodynamics

*DOI*

*Abstract*

In this paper we deal with a class of parabolic partial differential equations containing a continuous hysteresis operator. We get an existence result by means of a technique based on an implicit time discretization scheme and we also analyse the dependence of the solution on the data. This model equation appears in the context of magnetohydrodynamics.

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# Expansion of random boundary excitations for elliptic PDEs

*Authors*

- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Pt

*Keywords*

- White noise, generalized random processes, Karhunen-Loève expansion, Poisson integral formula, random boundary excitations, Laplace equations, biharmonic equations, Lamé equations

*DOI*

*Abstract*

In this paper we deal with elliptic boundary value problems with random boundary conditions. Solutions to these problems are inhomogeneous random fields which can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Loève (K-L) series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous random fields. We study the correlation structure of solutions to some classical elliptic equations in respond to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to Laplace equation, biharmonic equation, and to the Lamé system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an excellent benchmark for developing numerical methods, e.g., Monte Carlo simulations, stochastic volume and boundary element methods.

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# Monte Carlo Greeks for financial products via approximative Greenian kernels

*Authors*

- Kampen, Jörg
- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H10 62G07 65C05

*Keywords*

- American options, Sensitivities, Monte-Carlo methods, WKB expansions

*DOI*

*Abstract*

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.

*Appeared in*

- SIAM J. Sci. Comput. Vol. 31, 1, pp. 1-22, 2008 under new title: Monte Carlo Greeks for financial products via approximative transition densities

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# GEOMS: A software package for the numerical integration of general model equations of multibody systems

*Authors*

- Steinbrecher, Andreas

*2010 Mathematics Subject Classification*

- 70E55 65L80

*Keywords*

- differential-algebraic equations, equations of motion, multibody system, numerical integration, simulation

*DOI*

*Abstract*

In this paper we present the new numerical algorithm GEOMS for the numerical integration of the most general form of the equations of motion of multibody systems, including nonholonomic constraints and possible redundancies in the constraints, as they may appear in industrial applications. Besides the numerical integration it offers some additional features like stabilization of the model equations, use of different decomposition strategies, or checking and correction of the initial values with respect to their consistency. Furthermore, GEOMS preserves hidden constraints and (possibly) existing solution invariants if they are provided as equations. We will also demonstrate the performance and the applicability of GEOMS for two mechanical examples of different degrees of complexity.

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# A stochastic volatility Libor model and its robust calibration

*Authors*

- Belomestny, Denis
- Mathew, Stanley
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G51 62G20 60H05 60H10 90A09 91B28

*Keywords*

- Libor modelling, stochastic volatility, CIR processes, calibration

*DOI*

*Abstract*

In this paper we propose a Libor model with a high-dimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement.

*Appeared in*

- Monte Carlo Methods Appl., 15 (2009) pp. 285-310 as "Multiple stochastic volatility extension of the Libor market model and its implementation".

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# A general asymptotic model for Lipschitzian curved rods

*Authors*

- Tiba, Dan
- Vodák, Rostislav

*2010 Mathematics Subject Classification*

- 74K10 74B99

*Keywords*

- curved rods, low geometrical regularity, 1D-asymptotic model

*DOI*

*Abstract*

In this paper we show that the asymptotic methods provide an advantageous approach to obtain models of thin elastic bodies under minimal regularity assumptions on the geometry. Our investigation is devoted to clamped curved rods with a nonsmooth line of centroids and the obtained model is a generalization of results already available in the literature.

*Appeared in*

- Adv. Math. Sci. Appl., vol. 15 (2005), no. 1, pp. 137-198.

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# A general asymptotic dynamic model for Lipschitzian curved rods

*Authors*

- Vodák, Rostislav

*2010 Mathematics Subject Classification*

- 74K10 35L15 74B99

*Keywords*

- curved rods, low geometrical regularity, evolution equation, asymptotic analysis

*DOI*

*Abstract*

In this paper we study the asymptotic behaviour of solutions to the linear evolution problem for clamped curved rods with the small thickness $epsilon $ under minimal regularity assumptions on the geometry. In addition, non-constant density of the curved rods is considered.

*Appeared in*

- J. Appl. Math. 2005, no. 4, 425--451

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# Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

*Authors*

- Bovier, Anton
- den Hollander, F.
- Nardi, F. R.

*2010 Mathematics Subject Classification*

- 60K35 82B43 82C43 82C80

*Keywords*

- Lattice gas, Kawasaki dynamics, metastability, critical droplet, discrete isoperimetric inequalities, potential theory, Dirichlet form, capacity

*DOI*

*Abstract*

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one in the limit of low temperature and low density, express the proportionality constant in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.

*Appeared in*

- Probab. Theory Related Fields, 135 (2006) pp. 265--310.

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# An optimal control approach to curved rods

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49J20 74K10

*Keywords*

- Curved rods, control variational methods, generalized Naghdi model

*DOI*

*Abstract*

In this paper, a new approach to the generalized Naghdi model for the deformation of three-dimensional curved rods is studied. The method is based on optimal control theory.

*Appeared in*

- SIAM J. Control. Optim, 47 (2009), pp. 3220-3236, in extended form as "The control variational approach for differential systems"

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# A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems

*Authors*

- Eymard, Robert
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65M12

*Keywords*

- Finite Volume Methods, Convergence, Nonlinear parabolic PDEs

*DOI*

*Abstract*

In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of parabolic partial differential equation with nonlinear diffusion and convection terms a 1D, 2D or 3D domain. The nonlinear diffusion term be bounded away from zero except a finite number of values. The method is based on the solution, at each interface between two control volumes, of a nonlinear elliptic two point boundary value problem derived from the original equation with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes.

*Appeared in*

- Numer. Math., 102 (2006) pp. 463--495.

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# Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems

*Authors*

- Wolfrum, Matthias
- Ehrt, Julia

*2010 Mathematics Subject Classification*

- 35B25 34C30 35K57

*Keywords*

- Pulse interaction, singular perturbation theory

*DOI*

*Abstract*

In this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions.

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# Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D05 35K05 35K15 35K55

*Keywords*

- Nonlinear parabolic equation, nonlocal boundary condition, right-hand side in L-p with p>=1

*DOI*

*Abstract*

It is known that the time-dependent heat equation with nonlocal radiation boundary conditions possesses a unique weak solution if the heat sources are in L-2. In this paper, we generalize the known existence and uniqueness results to the case that the right-hand side belongs to an arbitrary L-p space (p >= 1). This is the continuation of the results that we recently proved for the stationary problem. The purpose of both papers is to obtain energy estimates that involve only the L-p norm of the heat sources for some exponent p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating).

*Appeared in*

- Appl. Math., 55 (2010) pp. 111--149.

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# Local limit theorems for ladder moments

*Authors*

- Vatutin, Vladimir
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60G50 60G40

*Keywords*

- Random walk, ladder moment, Spitzer condition

*DOI*

*Abstract*

Let $S_0=0,S_n_ngeq1$ be a random walk generated by a sequence of i.i.d. random variables $X_1,X_2,...$ and let $tau^-:=minleft ngeq1: S_nleq0right $ and $tau^+:=minleft ngeq 1: S_n>0right $. Assuming that the distribution of $X_1$ belongs to the domain of attraction of an $alpha$-stable law$,alphaneq1,$ we study the asymptotic behavior of $mathbbP(tau^pm=n)$ as $nrightarrowinfty.$

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# The behaviour of aging functions in one-dimensional Bouchaud's trap model

*Authors*

- Černý, Jiri

*2010 Mathematics Subject Classification*

- 82D30 60K37 82C41

*Keywords*

- Trap models, aging, Levy processes, singular diffusions

*DOI*

*Abstract*

Let tau_x be a collection of i.i.d. positive random variables with distribution in the domain of attraction of alpha-stable law with alpha <1. The symmetric Bouchaud's trap model on Z is a Markov chain X(t) whose transition rates are given by w_xy=(2tau_x)^-1 if x, y are neighbours in Z. We study the behaviour of two correlation functions: P[X(t_w+t)=X(t_w)] and P[X(t')=X(t_w)forall t'in[t_w,t_w+t]]. It is well known that for any of these correlation functions a time-scale t=f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model.

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# Convergence of Fourier-wavelet models for Gaussian random processes

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 60G15

*2008 Physics and Astronomy Classification Scheme*

- 05.10.Ln

*Keywords*

- Fourier-Wavelet model, stationary Gaussian random process, Meyer's wavelets, Nikolskiui-Besov space, convergence in probability, convergence in mean square

*DOI*

*Abstract*

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolskiui-Besov classes cannot be improved.

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# Inhomogeneous dependence modelling with time varying copulae

*Authors*

- Giacomini, Enzo
- Härdle, Wolfgang
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62P20

*Keywords*

- adaptive estimation, nonparametric estimation, Value-at-Risk

*DOI*

*Abstract*

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for non-Gaussian dependence. The correct modelling of non-Gaussian dependences is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of portfolios and show their better performance over the RiskMetrics approach, a widely used methodology for VaR estimation.

*Appeared in*

- J. Bus. Econom. Statist., 27 (2009) pp. 224--234.

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# On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

*Authors*

- Henrion, René
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Electricity markets, bidding, noncooperative games, equilibrium constraint, EPEC, optimality condition, co-derivative, random demand

*DOI*

*Abstract*

Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata [17]. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived (Proposition 3.2). Later the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.

*Appeared in*

- Appl. Math., 522 (2007) pp. 473--494.

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# Polyhedral inclusion-exclusion

*Authors*

- Bukszar, Jozsef
- Henrion, René
- Hujter, Mihaly
- Szantai, Tamas

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Inclusion-Exclusion, polyhedron

*DOI*

*Abstract*

Motivated by numerical computations to solve probabilistic constrained stochastic programming problems, we derive a new identity claiming that many terms are cancelled out in the inclusion--exclusion formula expressing the complement of a Euclidean polyhedron.

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# Stationary solutions of driven fourth- and sixth-order Cahn--Hilliard type equations

*Authors*

- Korzec, Maciek D.
- Evans, Peter L.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34E05 74K35 65P99

*Keywords*

- convective Cahn-Hilliard equation, quantum dots, exponential asymptotics, matching, dynamical systems

*DOI*

*Abstract*

New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

*Appeared in*

- SIAM J. Appl. Math., 69 (2008) pp. 348-374.

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# On estimation and detection of smooth high-dimensional function

*Authors*

- Ingster, Yuri I.
- Suslina, Irina

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- high-dimensional estimation, high-dimensional signal detection, minimax hypothesis testing, separation rates, Sobolev norms, lattice problem

*DOI*

*Abstract*

Observing an unknown $n$-variables function $f(t), tin [0,1]^n$ in the white Gaussian noise of a level $e>0$. We suppose that there exist $1$-periodical (in each variable) $sigma$-smooth extensions of functions $f(t)$ on $R^n$ and $f$ belongs to a Sobolev ball, i.e., $ f _sigma,2leq 1$, where $ cdot _sigma,2$ is a Sobolev norm (we consider two variants of one). We study two problem: estimation of $f$ and testing of the null hypothesis $H_0: f=0$ against alternatives $ f _2geq r_e$. We study the asymptotics (as $eto 0, ntoinfty$) of the minimax risk for square losses, for estimation problem, and of minimax error probabilities and of minimax separation rates in the detection problem. We show that of $ntoinfty$, then there exist ``sharp separation rates'' in the detection problem. The asymptotics of minimax risks of estimation and of separation rates of testing are of different type for $nll loge^-1$ and for $ngg loge^-1$. The problems under consideration are closely related with ``lattice problem'' in the numerical theory.

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# Computing likelihoods for coalescents with multiple collisions in the infinitely-many-sites model

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60G09 60G52 60J75 60J85

*Keywords*

- $Lambda$-coalescent, likelihood-based inference, infinitely-many-sites, population genetics, Monte-Carlo method

*DOI*

*Abstract*

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised $Lambda$-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called $Lambda$-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (1987) and Griffiths and Tavaré (1994, 1995) to obtain a likelihood based inference method for general $Lambda$-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of $Lambda$-coalescents, the parametrisable sub-family of Beta$(2-alpha, alpha)$-coalescents, where $alpha in (1,2]$, are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

*Appeared in*

- J. Math. Biol., 57 (2008) pp. 435--465.

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# Inference for $Lambda$-coalescents

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60G09 60G52 60J75 60J85

*Keywords*

- Lambda-coalescent, inference, infinitely-many-sites model, mathematical population genetics, Fleming-Viot process, multiple collisions, frequency spectrum, Monte-Carlo simulation

*DOI*

*Abstract*

One of the main problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of a classical Fleming-Viot process, then the standard means to describe the corresponding genealogy is Kingman's coalescent. For this process, powerful inference methods are well-established. An important feature of this class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents. Moreover, Eldon & Wakeley (2006) have provided evidence that such more general coalescents, which allow m ultiple collisions, might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier & Griffiths (1987) and Griffiths & Tavaré (1994) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta$(2-alpha,alpha)$-coalescents, where $alpha in (1,2]$, are of particular biological relevance. We apply our method in this case to simulated and real data (taken from Árnason (2004)). We conclude that for populations with extreme reproductive behaviour, the Kingman-coalescent as standard model might have to be replaced by more general coalescents, in particular by Beta$(2-alpha,alpha)$-coalescents.

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# Bildsegmentation zur Untersuchung von Streulichtbildern bei der laseroptischen Diagnose von rheumatoider Arthritis

*Authors*

- Gajewski, Herbert
- Griepentrog, Jens André
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Beuthan, J.
- Zabarylo, U.
- Minet, O.

*2010 Mathematics Subject Classification*

- 90C26 82B26 94A08

*2008 Physics and Astronomy Classification Scheme*

- 42.62.Be 87.63.Lk 42.30.Va 07.05.Pj 02.30.Sa 02.60.Lj

*Keywords*

- Image segmentation, Rheumatoid arthritis, Optical diagnostics, Laser transillumination, Non-convex energy functionals

*DOI*

*Abstract*

Optical imaging in biomedicine is governed by the light absorption and scattering interaction on microscopic and macroscopic constituents in the medium. Therefore, light scattering characteristics of human tissue correlates with the stage of some diseases. In the near infrared range the scattering event with the coefficient approximately two orders of magnitude greater than absorption plays a dominant role. The potential of an experimental laser diode based setup for the transillumination of rheumatoid finger joints and the pattern of the stray light detection are demonstrated. For evaluating the scattering light images a new non-local image segmentation method is presented. Regarding a noisy picture as a multicomponent mixture of gray scaled particles, this method minimizes a non-convex free energy functional under the constraint of mass conservation of the components. Contrary to constructing equilibrium distributions as steady states of an adequate evolution equation, a direct descent method for the free energy is used to separate the components of the image.

*Appeared in*

- Mathematics -- Key Technology for the Future, W. JÄGER, H.-J. KREBS, eds., Springer, Heidelberg, 2008, english version ``Image Segmentation for the Investigation of Scattered-Light Images when Laser-Optically Diagnosing Rheumatoid Arthritis'', pp. 149--161.

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# Optimal control problems with delays in state and control and mixed control-state constraints

*Authors*

- Göllmann, Laurenz
- Kern, Daniela
- Maurer, Helmut

*2010 Mathematics Subject Classification*

- 49K15 49K25

*Keywords*

- Retarded optimal control problems, delays in state and control, mixed control-state inequality constraints, Pontryagin's minimum principle, discretization methods, optimal control of a CSTR reactor, optimal fishing

*DOI*

*Abstract*

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods for the delayed control problem are discussed which amount to solving a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and two numerical examples from chemical engineering and economics illustrate the results.

*Appeared in*

- Optimal Control Appl. Methods, 30 (2009) pp. 341--365.

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# Discrepancy distances and scenario reduction in two-stage stochastic integer programming

*Authors*

- Henrion, René
- Küchler, Christian
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Stochastic programming, two-stage, mixed-integer, chance constraints,, scenario reduction, discrepancy, Kolmogorov metric

*DOI*

*Abstract*

Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.

*Appeared in*

- J. Indust. Management Optim., 4 (2008) pp. 363--384.

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# Spatially adaptive regression estimation: Propagation-separation approach

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- adaptive weights; local structure, local polynomial regression, propagation, separation

*DOI*

*Abstract*

Polzehl and Spokoiny (2000) introduced the adaptive weights smoothing (AWS) procedure in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is fully adaptive and dimension free. Simulations with artificial images show that AWS is superior to classical smoothing techniques especially when the underlying image function is discontinuous and can be well approximated by a piecewise constant function. However, the latter assumption can be rather restrictive for a number of potential applications. Here we present a new method based on the ideas of propagation and separation which extends the AWS procedure to the case of an arbitrary local linear parametric structure. We also establish some important results about properties of the new `propagation-separation' procedure including rate optimality in the pointwise and global sense. The performance of the procedure is illustrated by examples for local polynomial regression and by applications to artificial and real images.

*Appeared in*

- Probab. Theory Related Fields, 135 (2006) pp. 335--362.

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# Flow solution properties in full-zone thermocapillary liquid bridges

*Authors*

- Davis, Dominic
- Smith, Frank

*2010 Mathematics Subject Classification*

- 76D05 76E30 35B40 53A05 65M60

*Keywords*

- liquid bridges, floating-zone, thermocapillarity, finite-element, Navier--Stokes equations, transient, three-dimensional, mode interaction, nonlinearity, midzone

*DOI*

*Abstract*

Properties of low Prandtl number flows in slender cylindrical liquid bridges driven by interfacial thermocapillary forces are addressed here in a theoretical and computational light. Both `outward' (positive Marangoni number Ma) and `inward' (negative Ma) flow along the liquid-gas interface are considered. In previous investigations (Davis & Smith 2003), a solution curve for steady, axisymmetric flow was determined from asymptotic theory in the context of slender bridges. It indicated both the non-existence of solutions beyond a positive, cut-off value of the scaled Marangoni number and a double branch in the case of solvability (although with only one `attractor'). In the present study full numerical simulation (using a finite-element iterative solver, described herein) reveals the unsteady, three-dimensional nature of the flow solution beyond the cut-off value. Attention is paid to the case where the radius-to-height aspect ratio is 0.5, from which the (nonlinearly-coupled) azimuthal modes m=1 and m=2 are seen to dominate. The branch behaviour is then examined for Ma<0, and asymptotic analysis reveals that a critical value of the scaled Marangoni number exists, on approach to which the pressure gradient across the midzone becomes large and negative. Full computational solutions on the attractor branch for Ma<0 are subsequently presented, and these show encouraging agreement with asymptotic predictions (as well as slender-flow midzone computations) near the critical Marangoni number. The critical value moreover is shown to correspond to the onset of `lemonheads' (non-convex radial velocity profiles near the midzone), in precisely the same manner as the cut-off value for positive Ma.

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# Large scale localization of a spatial version of Neveu's branching process

*Authors*

- Fleischmann, Klaus
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60G57

*Keywords*

- Neveu's continuous-state branching, infinite mean branching superprocess, large scale concentration in one point, log-Laplace product formula, small epsilon asymptotics

*DOI*

*Abstract*

Recently a spatial versions of Neveu's (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process.

*Appeared in*

- Electron. J. Probab., 11 (2006) pp. 723-767.

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# SIMEX and TLS: An equivalence result

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Zwanzig, Silvelyn

*2010 Mathematics Subject Classification*

- 62F12 62J05

*Keywords*

- Errors-in-variables, SIMEX, Moment estimator, Total Least Squares

*DOI*

*Abstract*

SIMEX was introduced by Cook and Stefanski (1994) as a simulation type estimator in errors-in-variables models. The idea of the SIMEX procedure is to compensate for the effect of the measurement errors while still using naive regression estimators. Polzehl and Zwanzig (2004) defined a symmetrized version of this estimator. In this paper we establish some results relating these two simulation-extrapolation-type estimators to well known consistent estimators like the total least squares estimator (TLS) and the moment estimator (MME) in the context of errors-in-variables models. We further introduce an adaptive SIMEX (ASIMEX), which is calculated like SIMEX, but based on an estimated variance. The main result of this paper is that SYMEX, ASIMEX are equivalent to TLS. Additionally we see that SIMEX is equivalent to the moment estimator.

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# A finite element method for surface diffusion: The parametric case

*Authors*

- Bänsch, Eberhard
- Morin, Pedro
- Nochetto, Ricardo H.

*2010 Mathematics Subject Classification*

- 35K55 65M12 65M15 65M60 65Z05

*Keywords*

- Surface diffusion, fourth-order parabolic problem, finite elements, Schur complement, smoothing effect, pinch-off

*DOI*

*Abstract*

Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for parametric surfaces with or without boundaries. The method is semi-implicit, requires no explicit parametrization, and yields a linear system of elliptic PDE to solve at each time step. We next develop a finite element method, propose a Schur complement approach to solve the resulting linear systems, and show several significant simulations, some with pinch-off in finite time. We introduce a mesh regularization algorithm, which helps prevent mesh distortion, and discuss the use of time and space adaptivity to increase accuracy while reducing complexity.

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# Control and removing of modulational instabilities in low dispersion photonic crystal fiber cavities

*Authors*

- Tlidi, Mustapha
- Mussot, Arnaud
- Louvergneaux, Eric
- Kozyreff, Gregory
- Vladimirov, Andrei G.
- Taki, Abdelmajid

*2010 Mathematics Subject Classification*

- 78A60 37K45

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.65.Wi, 42.65.Sf

*Keywords*

- Photonic crystal fibers, modulational instability, forth order dispersion

*DOI*

*Abstract*

Taking up to fourth order dispersion effects into account, we show that fiber resonators become stable for large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for arbitrary small but nonzero fourth order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and opens the way to experimental implementation.

*Appeared in*

- Optics Letters, 32 (2007) pp. 662-664.

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# Transport behaviour of a Bose--Einstein condensate in a bichromatic optical lattice

*Authors*

- Bhattacharjee, Aranya
- Pietrzyk, Monika

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 03.75.Lm 03.75.Kk 32.80.Lg

*Keywords*

- 0:0:optical lattice, Bose Einstein condensate, Gross-Pitaevskii equation

*DOI*

*Abstract*

The Bloch and dipole oscillations of a Bose Einstein condensate (BEC) in an optical superlattice is investigated. We show that the effective mass increases in an optical superlattice, which leads to localization of the BEC, in accordance with recent experimental observations [17]. In addition, we find that the secondary optical lattice is a useful additional tool to manipulate the dynamics of the atoms.

*Appeared in*

- Cent. Eur. J. Math., 6 (2008) pp. 26-32.

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# A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 60J65 47G10 47G30 35S30 82C31 35C15

*Keywords*

- Fokker-Planck equation, general Brownian motion, dimension-reduction, pseudodifferential operator

*DOI*

*Abstract*

The Brownian motion of a classical particle can be described by a Fokker-Planck-like equation. Its solution is a probability density in phase space.By integrating this density w.r.t. the velocity, we get the spatial distribution or concentration. We reduce the 2n-dimensional problem to an n-dimensional diffusion-like equation in a rigorous way, i.e., without further assumptions in the case of general Brownian motion, when the particle is forced by linear friction and homogeneous random (non-Gaussian) noise. Using a representation with pseudodifferential operators, we derive a reduced diffusion-like equation, which turns out to be non-autonomous and can become elliptic for long times and hyperbolic for short times, although the original problem was time homogeneous. Moreover, we consider some examples: the classical Brownian motion (Gaussian noise), the Cauchy noise case (which leads to an autonomous diffusion-like equation), and the free particle case.

*Appeared in*

- J. Mathematical Physics Analysis Geometry (MAG), 12 (2005) pp. 187-202.

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# Rate independent Kurzweil processes

*Authors*

- Krejčí, Pavel
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 49J40 49J53 74C15

*Keywords*

- Rate independent process, Kurzweil integral, variational inequality

*DOI*

*Abstract*

The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

*Appeared in*

- Appl. Math., 54 (2009) pp. 117--145.

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# Tortuosity and objective relative acceleration in the theory of porous materials

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 80A17 74A20 74F10 74J10

*Keywords*

- Thermodynamics, Biot's model, poroelastic materials, acoustic waves

*DOI*

*Abstract*

The aim of this work is twofold. We show the construction of an objective relative acceleration for a two-component mixture and prove that its incorporation in the momentum source requires additional terms in partial stresses and in the energy. This may be interpreted as an influence of tortuosity in the theory of saturated poroelastic materials and a connection of tortuosity with fluctuations of the kinetic energy on a mesoscopic level of observation. The linearization of such a model yields Biot's equations used in poroacoustics. We demonstrate as well that results for the propagation of acoustic waves in saturated poroelastic media are qualitatively similar for Biot's model and for the simple mixture model in which both the tortuosity and the Biot's coupling between partial stresses are neglected. It is also indicated that the coupling constant of Biot's model obtained by means of the Gassmann relation may be too large as it leads to very small differences in the speed of propagation of the P1-wave for small and large frequencies which contradicts the data for soils.

*Appeared in*

- Proc. Roy. Soc. London Ser. A 461 (2005), pp. 1533--1561.

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# Existence of bounded discrete steady state solutions of the van Roosbroeck system on boundary conforming Delaunay grids

*Authors*

- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65M12 35K65

*Keywords*

- Reaction-diffusion systems, discrete bounded solutions, Delaunay grids, discrete weak maximum principle

*DOI*

*Abstract*

The classic van Roosbroeck system describes the carrier transport in semiconductors in a drift diffusion approximation. Its analytic steady state solutions fulfill bounds for some mobility and recombination/generation models. The main goal of this paper is to establish the identical bounds for discrete in space, steady state solutions on 3d boundary conforming Delaunay grids and the classical Scharfetter-Gummel-scheme. Together with a uniqueness proof for small applied voltages and the known dissipativity (continuous as well as space and time discrete) these discretization techniques carry over the essential analytic properties to the discrete case. The proofs are of interest for deriving averaging schemes for space or state dependent material parameters, which are preserving these qualitative properties, too. To illustrate the properties of the scheme 1, 4, 16 elementary cells of a modified CoolMOS like structure are depleted by increasing the applied voltage until steady state avalanche breakdown occurs.

*Appeared in*

- SIAM J. Sci. Comput., 31 (2009) pp. 1347--1362.

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# A complete-damage problem at small strains

*Authors*

- Bouchitté, Guy
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K65 35K85 49S05 74C05 74R05

*Keywords*

- Inelastic damage, small strain, variational inequality, energetic formulation

*DOI*

*Abstract*

The complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

*Appeared in*

- Z. Angew. Math. Phys., 60 (2009) pp. 205--236.

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# Numerical analysis of monochromatic surface waves in a poroelastic medium

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 74J15 76S05 74S99

*Keywords*

- Surface waves, flows in porous media, numerical analysis of dispersion relation.

*DOI*

*Abstract*

The dispersion relation for surface waves on the boundary between a fully saturated poroelastic medium and a vacuum is investigated numerically in the whole range of frequencies. A linear model of a two-component poroelastic medium similar to but simpler than the classical Biot's model is used. In the whole range of frequencies there exist two modes of surface waves corresponding to the classical Rayleigh and Stoneley waves. The numerical results for phase velocities, group velocities and attenuations of these waves are shown for different values of the bulk permeability coefficient.

*Appeared in*

- Trends in Applications of Mathematics to Mechanics. Proceedings of the XIVth International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM'2004), Seeheim, Germany, 22--28 August 2004, K. Hutter, Y. Wang, eds., Berichte aus der Mathematik, Shaker, Aachen, 2005, pp. 21--30.

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# A model for two phase flow with evaporation

*Authors*

- Krahl, Rolf
- Adamov, Miroslav
- Lozano Aviles, Miriam
- Bänsch, Eberhard

*2010 Mathematics Subject Classification*

- 80A20 76T10 76D05

*Keywords*

- evaporation, two phase flow, free capillary surface, phase change, model

*DOI*

*Abstract*

The dynamic behavior of a free gas--liquid phase boundary is often influenced by evaporation or condensation to an extend that may not be neglected. In this paper, we derive a general model for the dynamics of a two phase flow with evaporation, starting from the balance of mass, energy, and momentum. The model takes into account that the gas phase might consist of a mixture of vapor and inert gas. It is based on the incompressible Navier--Stokes equations in the bulk of the liquid and the gas phase, convection--diffusion equations for heat and vapor, and appropriate conditions for the transfer of mass, momentum, and energy through the phase boundary. As a simplification, the flow field in the liquid and gaseous phase can be decoupled, if the stress from the gas phase on the free surface is neglected. The special case of a gas phase containing only pure vapor is considered, which allows one to neglect the gas phase completely, leading to a single phase flow problem with a free boundary.

*Appeared in*

- G. P. Celata, P. Di Marco, A. Mariani, and R. K. Shah, eds., Two-Phase Flow Modelling and Experimentation 2004. Edizioni ETS, Pisa, 2004.

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# Numerical investigation of the non-isothermal contact angle

*Authors*

- Krahl, Rolf
- Bänsch, Eberhard

*2010 Mathematics Subject Classification*

- 76T10 76D45 80A20

*Keywords*

- two phase flow, free capillary surface, thermocapillarity, contact angle, Marangoni effect

*DOI*

*Abstract*

The influence of thermocapillary stress on the shape of the gas-liquid phase boundary is investigated numerically. We consider the case of a cold liquid meniscus at a heated solid wall in the absence of gravity. An äpparent contact angle" is defined geometrically and the deviation of this apparent contact angle from the prescribed static contact angle due to thermocapillary convection is studied.

*Appeared in*

- Microgravity - Science and Technology, 17(3), S. 39-44, 2005. Under new title: Impact of Marangoni effects on the apparent contact angle - a numerical investigation.

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# Elastic modelling of surface waves in single and multicomponent systems -- Lecture notes

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 74J10 74J15 74F10

*Keywords*

- Surface waves, waves in porous media, monochromatic waves

*DOI*

*Abstract*

The main aim of this article is to present a review of most important acoustic surface waves which are described by linear one- and two-component models. It has been written for the CISM-course: Surface waves in Geomechanics (Udine, September 6-10, 2004). Among the waves in one-component linear elastic media we present the classical Rayleigh waves on a plane boundary, Rayleigh waves on a cylindrical surface, Love waves, Stoneley waves (solid/solid and fluid/solid interface). In the second part of the article we discuss two two-component models of porous materials (Biot's model and a simple mixture model). We indicate basic differences of the models and demonstrate qualitative similarities. We introduce as well some fundamental notions yielding the description of surface waves in two-component systems (saturated porous materials) and review certain (porous materials with impermeable boundaries) asymptotic results for such waves. However, the full discussion of this subject including numerous results of computer calculations can be found in the article of B. Albers also included in this volume.

*Appeared in*

- Surface Waves in Geomechanics: Direct and Inverse Modelling for Soils and Rocks, C. Lai, K. Wilmanski, eds., CISM Courses and Lectures, Springer, Wien [u.a.], 2005, pp. 203--276.

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# On Eisenbud's and Wigner's R-matrix: A general approach

*Authors*

- Behrndt, Jussi
- Neidhardt, Hagen
- Racec, Roxana
- Racec, Paul N.
- Wulf, Ulrich

*2010 Mathematics Subject Classification*

- 47A40 34L25 81U20

*Keywords*

- Scattering, scattering matrix, R-matrix, symmetric and selfadjoint operators, extension theory, boundary triplets, Weyl function, ordinary differential operators

*DOI*

*Abstract*

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.

*Appeared in*

- J. Differential Equations, 244 (2008) pp. 2545--2577.

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# Multisymplectic analysis of the short pulse equation

*Authors*

- Pietrzyk, Monika
- Kanattšikow, Igor

*2010 Mathematics Subject Classification*

- 37K10 78A60 35Q60 35Q51

*2008 Physics and Astronomy Classification Scheme*

- 02.30.lk, 42.65.-k, 42.81.Gs

*Keywords*

- Multisymplectic formalism, multisymplectic integrator, Short Pulse Equation, ultrashort pulses, nonlinear optics

*DOI*

*Abstract*

The multisymplectic analysis of the Short Pulse Equation known in nonlinear optics is used in order to construct a geometric multisymplectic integrator of it. A brief comparison of its effectiveness relative to the pseudo-spectral integration scheme is presented.

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# When did the 2001 recession really start?

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Starica, Catalin

*2010 Mathematics Subject Classification*

- 62M10

*Keywords*

- business cycle, non-parametric smoothing, non-stationarity.

*DOI*

*Abstract*

The paper develops a non-parametric, non-stationary framework for business-cycle dating based on an innovative statistical methodology known as Adaptive Weights Smoothing (AWS). The methodology is used both for the study of the individual macroeconomic time series relevant to the dating of the business cycle as well as for the estimation of their joint dynamic. Since the business cycle is defined as the common dynamic of some set of macroeconomic indicators, its estimation depends fundamentally on the group of series monitored. We apply our dating approach to two sets of US economic indicators including the monthly series of industrial production, nonfarm payroll employment, real income, wholesale-retail trade and gross domestic product (GDP). We find evidence of a change in the methodology of the NBER's Business-Cycle Dating Committee: an extended set of five monthly macroeconomic indicators replaced in the dating of the last recession the set of indicators emphasized by the NBER's Business-Cycle Dating Committee in recent decades. This change seems to seriously affect the continuity in the outcome of the dating of business cycle. Had the dating been done on the traditional set of indicators, the last recession would have lasted one year and a half longer. We find that, independent of the set of coincident indicators monitored, the last economic contraction began in November 2000, four months before the date of the NBER's Business-Cycle Dating Committee.

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# Structural adaptive dimension reduction

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Sperlich, Stefan

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- Dimension-reduction, multi-index model, index space, structural adaptation, R

*DOI*

*Abstract*

The paper introduces and discusses different estimation methods for multi index models where the indices are parametric and the link function is nonparametric. More specific, the here introduced methods follow the idea of Hristache et al. (2001), modify and try to improve it. Moreover, they constitute alternatives to the so called MAVE-based methods (Xia et al, 2002). We concentrate on an intuitive presentation of what each procedure is doing to the data and its implementation. All methods considered here we have made freely available in R. We conclude with a comparative simulation study based on the provided package EDR.

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# Calmness of constraint systems with applications

*Authors*

- Henrion, René
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 90C30 49J53

*Keywords*

- calmness, constraint sets, nonsmooth calculus, value-at-risk

*DOI*

*Abstract*

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable to detect calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk.

*Appeared in*

- Math. Program., 104 (2005) pp. 437--464.

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# Uniqueness results for an inverse periodic transmission problem

*Authors*

- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 78A46 35R30

*Keywords*

- Diffraction grating, periodic Helmholtz equation, inverse transmission problem

*DOI*

*Abstract*

The paper is devoted to the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We show that measurements corresponding to a finite number of refractive indices above or below the grating profile, uniquely determine the periodic interface in the inverse TE transmission problem. If a priori information on the height of the diffraction grating is available, then we also obtain upper bounds of the required number of wavenumbers by using the Courant-Weyl min-max principle for a fourth-order elliptic problem. This extends uniqueness results by Hettlich and Kirsch [11] to the inverse transmission problem.

*Appeared in*

- Inverse Problems, 20 (2004) pp. 1841--1852.

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# Linear non-autonomous Cauchy problems and evolution semigroups

*Authors*

- Neidhardt, Hagen
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 35L90 34G10 47D06

*Keywords*

- linear evolution equations, evolution semigroups, perturbation theory, time-dependent Schrödinger operators, moving potentials

*DOI*

*Abstract*

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

*Appeared in*

- Adv. Differential Equations, 14 (2009) pp. 289--340.

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# Thermodynamics of simple two-component thermo-poroelastic media

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 80A17 74A20 74F10

*Keywords*

- Thermodynamics of multicomponent systems, thermo-poroelastic materials, simple mixtures

*DOI*

*Abstract*

The paper is devoted to the thermodynamic construction of a two-component model of poroelastic media undergoing, in contrast to earlier works on this subject, nonisothermal processes. Under the constitutive dependence on partial mass densities, deformation gradient of skeleton, relative velocity, temperature, temperature gradient and porosity (simple poroelastic material) as well as the assumption of small deviations from the thermodynamic equilibrium we construct explicit relations for fluxes, prove the splitting of the free energy into partial contributions without mechanical couplings, construct a chemical potential for the fluid component important for the formulation of boundary conditions on permeable boundaries. We discuss as well a modification of the porosity balance equation in which we account for time changes of equilibrium porosity. This modification yields the behavior of the model characteristic for granular materials.

*Appeared in*

- Trends and Applications of Mathematics to Mechanics. STAMM-2002, G. Romano, S. Rionero, eds., Springer, Wien [u.a.], 2005, pp. 293-306.

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# Local likelihood modelling via stagewise aggregation

*Authors*

- Belomestny, Denis
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62G07 62G08 62H30

*Keywords*

- aggregation, local likelihood, exponential family, density estimation, classification, spatial adaptivity

*DOI*

*Abstract*

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. Given a sequence of local likelihood estimates which we call "weak" estimates, the proposed method yields a new aggregated estimate whose pointwise risk does not exceed the smallest risk among all "weak" estimates multiplied by some logarithmic factor. We establish a number of important theoretical results concerning optimality of the aggregated estimate and show a good performance of the procedure in simulated examples.

*Appeared in*

- Ann. Statist., 25 (2007) pp. 2287--2311.

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# Exponential bounds for the minimum contrast with some applications

*Authors*

- Golubev, Yuri
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62F10 62F12 62F25

*Keywords*

- risk bound, quasi maximum likelihood, smooth contrast

*DOI*

*Abstract*

The paper studies parametric minimum contrast estimates under rather general conditions. The quality if estimation is measured by the rate function related to the contrast which allows for stating the results without specifying the particular parametric structure of the model. This approach permits also to go far beyond the classical i.i.d. case and to obtain nonasymptotic upper bounds for the risk. These bounds apply even for small or moderate samples. They also cover the case of misspecified parametric models. Another important feature of the approach is that it works well in the case when the parametric set can be unbounded and non-compact. In the case of a smooth contrast, the obtained exponential bounds do not rely on the covering numbers and can be easily computed. We also illustrate how these bound can be used for statistical inference: bounding the estimation risk, constructing the confidence sets for the underlying parameters, establishing the concentration properties of the minimum contrast estimate. The general results are specified to the case of a Gaussian contrast and of an i.i.d. sample. We also illustrate the approach by several popular examples including least squares and least absolute deviation contrasts and the problem of estimating the location of the change point. What we obtain in these examples slightly differs from usual asymptotic results known in the classical literature. This difference is due to the unboundness of the parameter set and a possible model misspecification.

*Appeared in*

- Electron. J. Stat., 3 (2009) pp. 712--746.

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# A regularized Newton method in electrical impedance tomography using shape Hessian information

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut

*2010 Mathematics Subject Classification*

- 49Q10 49M37 65N38 49K20

*Keywords*

- electrical impedance tomography, shape optimization, integral equation method, Newton type descent

*DOI*

*Abstract*

The present paper is concerned with the identification of an obstacle or void of different conductivity included in a two-dimensional domain by measurements of voltage and currents at the boundary. We employ a reformulation of the given identification problem as a shape optimization problem as proposed by Sokolowski and Roche. It turns out that the shape Hessian degenerates at the given hole which gives a further hint on the ill-posedness of the problem. For numerical methods, we propose a preprocessing for detecting the barycenter and a crude approximation of the void or hole. Then, we resolve the shape of the hole by a regularized Newton method.

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# Detection of transient generalized and mutual phase synchronization by clustering and application by single brain signals

*Authors*

- Hutt, Axel
- Schrauf, Michael

*2010 Mathematics Subject Classification*

- 62H11 62H30

*Keywords*

- cluster analysis, multivariate time series, synchronization

*DOI*

*Abstract*

The present work introduces an analysis framework for the detection of metastable signal segments in multivariate time series. It is shown that in case of linear data these segments represent transient generalized synchronization, while metastable segments in circular data reflect transient mutual phase synchronization. We propose a single segmentation approach for both types of data considering the space-time structure of the data. Applications to both event-related potentials and single evoked potentials obtained from an auditory oddball experiment reveal the lack of the component P300 in an experimental condition, indicates attention effects in component N100 and shows dramatic latency jitters in single trials. A comparison of the proposed method to a conventional index of mutual phase synchronization demonstrates the superiority of considering space-time data structures.

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# Simulation of microwave and semiconductor laser structures including PML: Computation of the eigenmode problem, the boundary value problem, and the scattering matrix

*Authors*

- Hebermehl, Georg
- Schefter, Jürgen
- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q60 65N22 65F15 65F10 78M25

*Keywords*

- Microwave device, Semiconductor laser, Simulation, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Rectangular grids, Tetrahedral nets

*DOI*

*Abstract*

The properties of microwave circuits and optical structures can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT). Maxwellian grid equations are formulated for staggered nonequidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells. The surface of the computation domain is assumed to be an electric or a magnetic wall, open-region problems require uniaxial Perfectly Matched Layer (PML) absorbing boundary conditions. Calculating the excitations at the ports, one obtains eigenvalue problems and then systems of linear algebraic equations.

*Appeared in*

- Proc. 5th International Workshop Scientific Computing in Electrical Engineering (SCEE), Capo D'Orlando, Italy, September 5--9, 2004, A. Anile, G. Ali, G. Mascali, eds., Scientific Computing in Electrical Engineering, Springer Verlag, 2006, pp. 203--214

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# Monte Carlo methods and numerical solutions

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- Rarefied gas dynamics, Direct Simulation Monte Carlo, Boltzmann equation, stochastic models

*DOI*

*Abstract*

The purpose of this paper is to illustrate that direct simulation Monte Carlo methods can often be considered as rigorous mathematical tools for solving nonlinear kinetic equations numerically. First a convergence result for Bird's DSMC method is recalled. Then some sketch of the history of stochastic models related to rarefied gas dynamics is given. The model introduced by Leontovich in 1935 provides the basis for a rigorous derivation of the Boltzmann equation from a stochastic particle system. The last part of the paper is concerned with some recent directions of study in the field of Monte Carlo methods for nonlinear kinetic equations. Models with general particle interactions and the corresponding limiting equations are discussed in some detail. In particular, these models cover rarefied granular gases (inelastic Boltzmann equation) and ideal quantum gases (Uehling-Uhlenbeck-Boltzmann equation). Problems related to the order of convergence, to the approximation of the steady state solution, and to variance reduction are briefly mentioned.

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# Poisson convergence in the restricted k-partioning problems

*Authors*

- Bovier, Anton
- Kurkova, Irina

*2010 Mathematics Subject Classification*

- 90C27 60G70

*Keywords*

- Number partioning, extreme values, Poisson process, Random Energy Model

*DOI*

*Abstract*

The randomized $k$-number partitioning problem is the task to distribute $N$ i.i.d. random variables into $k$ groups in such a way that the sums of the variables in each group are as similar as possible. The restricted $k$-partitioning problem refers to the case where the number of elements in each group is fixed to $N/k$. In the case $k=2$ it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case $k>2$ in the restricted problem and show that the vector of differences between the $k$ sums converges to a $k-1$-dimensional Poisson point process.

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# Impact of size, shape and composition on piezoelectric effects and the electronic properties of InGaAs/GaAs quantum dots

*Authors*

- Schliwa, Andrei
- Winkelnkemper, Momme
- Bimberg, Dieter

*Keywords*

- Electronic properties, Quantum dots, Piezolectricity

*DOI*

*Abstract*

The strain fields in and around self-organized In(Ga)As/GaAs quantum dots (QD) sensitively depend on QD geometry, average InGaAs composition and the In/Ga distribution profile. Piezoelectric fields of varying size are one result of these strain fields. We study systematically a large variety of realistic QD geometries and composition profiles, and calculate the linear and quadratic parts of the piezoelectric field. The balance of the two orders depends strongly on the QD shape and composition. For pyramidal InAs QDs with sharp interfaces a strong dominance of the second order fields is found. Upon annealing the first order terms become dominant, resulting in a reordering of the electron p- and d-states and a reorientation of the hole wavefunctions.

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# Linear stability analysis of a sharp-interface model for dewetting thin films

*Authors*

- King, John R.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76M45 34B15 65M06

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e

*Keywords*

- sharp-interface model, slippage, stability, dewetting, fingering, Lubrication models, rim

*DOI*

*Abstract*

The topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.

*Appeared in*

- J. Engrg. Math., 63 (2009) pp. 177--195.

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# Small strain oscillations of an elastoplastic Kirchhoff plate

*Authors*

- Guenther, Ronald B.
- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74C05 35Q72 47J40

*Keywords*

- elastoplastic plate, hysteresis operators, vector Prandtl-Ishlinskii model, von Mises model

*DOI*

*Abstract*

The two dimensional equation for transversal vibrations of an elastoplastic plate is derived from a general three dimensional system with a single yield tensorial von Mises plasticity model in the five dimensional deviatoric space. It leads after dimensional reduction to a multiyield three dimensional Prandtl-Ishlinskii hysteresis model whose weight function is explicitly given. The resulting partial differential equation with hysteresis is solved by means of viscous approximations and a monotonicity argument.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 88 (2008) pp. 199--217.

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# Linear sound waves in poroelastic materials: Simple mixture vs. Biot's model

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 74J10 76S05 74L05

*Keywords*

- Bulk waves, porous media, geophysics.

*DOI*

*Abstract*

The work contains the comparison of speeds and attenuations of P1-, S-, and P2-waves in poroelastic materials obtained within Biot's model and simple mixture model.

*Appeared in*

- Trends in Applications of Mathematics to Mechanics. Proceedings of the XIVth International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM'2004), Seeheim, Germany, 22--28 August 2004, K. Hutter, Y. Wang, eds., Berichte aus der Mathematik, Shaker, Aachen, 2005, pp. 21--30.

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# Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay

*Authors*

- Hutt, Axel

*2010 Mathematics Subject Classification*

- 45J05 92C20

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Sk 05.45.Xt 05.10.-a

*Keywords*

- nonlocal neural activity, traveling wave front, constant feedback delay

*DOI*

*Abstract*

The work introduces a model for reciprocal connections in neural fields by a nonlocal feedback mechanism, while the neural field exhibits nonlocal interactions and intra-areal transmission delays. We study the speed of traveling fronts with respect to the transmission delay, the spatial feedback range and the feedback delay for general axonal and feedback connectivity kernels. In addition, we find a novel shape of traveling fronts due to the applied feedback and criteria for its occurence are derived.

*Appeared in*

- Phys. Rev. E *70*, 052902 (2004)

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# Integral manifolds for slow-fast differential systems loosing their attractivity in time

*Authors*

- Schneider, Klaus R.
- Shchetinina, Ekaterina
- Sobolev, Vladimir

*2010 Mathematics Subject Classification*

- 34C45 34D15 34E15

*Keywords*

- integral manifolds; slow-fast systems; change of attractivity

*DOI*

*Abstract*

The work is devoted to the investigation of the integral manifolds of the nonautonomous slow-fast systems, which change their attractivity in time. The method used here is based on gluing attractive and repulsive integral manifolds by using an additional function.

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# Modelling of surface waves in poroelastic saturated materials by means of a two component continuum -- Lecture notes

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 74J15 76S05 74L05 74S99

*Keywords*

- Surface waves, porous media, geophysics, numerical analysis of dispersion relation

*DOI*

*Abstract*

These lecture notes are devoted to an overview of the modelling and the numerical analysis of surface waves in two-component saturated poroelastic media. This is an extension to the part of the lecture notes by K. Wilmanski (WIAS-Preprint No. 945) which is primarily concerned with the classical surface waves in single component media. We use the ''simple mixture model'' which is a simplification of the classical Biot's model for poroelastic media. Two interfaces are considered here: firstly the interface between a porous half space and a vacuum and secondly the interface between a porous halfspace and a fluid halfspace. For both problems we show how a solution can be constructed and a numerical solution of the dispersion relation can be found. We discuss the results for phase and group velocities and attenuations, and compare some of them to the high and low frequency approximations. For the interface porous medium/vacuum there exist in the whole range of frequencies two surface waves - a leaky Rayleigh wave and a true Stoneley wave. For the interface porous medium/fluid one more surface wave appears - a leaky Stoneley wave. For this boundary velocities and attenuations of the waves are shown in dependence on the surface permeability. The true Stoneley wave exists only in a limited range of this parameter. At the end we have a look on some results of other authors and a glance on a logical continuation of this work, namely the description of the structure and the acoustic behavior of partially saturated porous media.

*Appeared in*

- Surface waves in Geomechanics: Direct and Inverse Modelling for Soils and Rocks, C. Lai, K. Wilmanski, eds., CISM Courses and Lectures, Springer, Wien [u.a.], 2005, pp. 277-323.

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# Self-pulsation and excitability of blue-violet InGaN lasers

*Authors*

- Tronciu, Vasile Z.
- Yamada, Minoru
- Abram, Richard
- Kawakami, Toshiyuki
- Ito, Shigetoshi
- Ohno, Tomoki
- Taneya, Mototaka

*2010 Mathematics Subject Classification*

- 78A60 34C60

*Keywords*

- self-pulsation, excitability, blue lasers

*DOI*

*Abstract*

This article gives a review of our latest results on the self-pulsation and excitability properties of blue-violet lasers. A number of investigations of the phenomena in InGaN lasers with different designs are described. The bifurcations, which are the origin of the phenomena, are identified and the effects of the lasers parameters on device dynamics are discussed. It is shown how different laser structures can be used to control device behaviour and the dependence of self-pulsation and excitability behaviour on laser geometry is discussed. Finally, agreement between the results of numerical calculations and experimental measurements on self-pulsation is demonstrated.

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# Thermally driven phase transformation in shape-memory alloys

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Petrov, Adrien

*2010 Mathematics Subject Classification*

- 49J40 74C05 74F05 74M05 74N30

*Keywords*

- Shape-memory materials, doubly nonlinear differential inclusion, rate-independent processes, energetic formulation, temperature-induced phase transformation

*DOI*

*Abstract*

This paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.

*Appeared in*

- Adv. Math. Sci. Appl., 17 (2007) pp. 667--685.

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# A metric approach to a class of doubly nonlinear evolution equations and applications

*Authors*

- Rossi, Riccarda
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 35K55 49Q20 58E99

*Keywords*

- Doubly nonlinear equations, analysis in metric spaces, existence and approximation results

*DOI*

*Abstract*

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.

*Appeared in*

- Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), VII (2008) pp. 97--169.

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# Thermodynamics-based modeling edge-emitting quantum well lasers

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Gajewski, Herbert
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 78A60 35G25 35Q60 80A20 76M10

*Keywords*

- semiconductor lasers, continuity equations, Poisson equation, wave guide equations, photon rate equations, heat flow equation, entropy balance equation, discretization, iteration scheme

*DOI*

*Abstract*

This paper describes the modeling and the simulation of edge-emitting quantum well lasers, based on the drift-diffusion equations and equations for the optical field. By applying fundamental thermodynamic principles as the maximum entropy principle and the principle of local thermal equilibrium we derive a self-consistent energy transport model which can be proven to meet the thermodynamic requirements. It's numerical solution is discussed explicitly, by starting from the discretization procedure and by ending up with the iteration scheme. As an example, we demonstrate the simulation of a long-wavelength ridge-waveguide multi-quantum well laser.

*Appeared in*

- U. Bandelow, H. Gajewski, R. Hünlich, Chapter 3: Fabry--Perot Lasers: Thermodynamics-based Modeling, in: Optoelectronic Devices --- Advanced Simulation and Analysis, J. Piprek, ed., Springer, New York, 2005, pp. 63-85

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# On the stability of elastic-plastic systems with hardening

*Authors*

- Martins, João A.C.
- Monteiro Marques, Manuel D.P.
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 34A60 47H06 73H99

*Keywords*

- Differential inclusions, plasticity, hardening, existence, stability

*DOI*

*Abstract*

This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.

*Appeared in*

- J. Math. Anal. Appl., 343 (2008) pp. 1007--1021.

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# Computational comparison between the Taylor--Hood and the conforming Crouzeix--Raviart element

*Authors*

- Krahl, Rolf
- Bänsch, Eberhard

*2010 Mathematics Subject Classification*

- 65N30 76M10

*Keywords*

- Taylor-Hood element, Crouzeix-Raviart element, incompressible fluid flow, preconditioners for Quasi-Stokes

*DOI*

*Abstract*

This paper is concerned with the computational performance of the P₂ P₁ Taylor-Hood element and the conforming P₂^{+} P_{-1} Crouzeix-Raviart element in the finite element discretization of the incompressible Navier-Stokes equations. To this end various kinds of discretization errors are computed as well as the behavior of two different preconditioners to solve the arising systems are studied.

*Appeared in*

- A. Handlovičová, Z. Krivá, K. Mikula und D. Ševčovič, Hrsg., ALGORITMY 2005 - 17th Conference on Scientific Computing. Slovak University of Technology, Bratislava, 2005, S. 369-379.

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# Nonlinear evolution inclusions arising from phase change models

*Authors*

- Colli, Pierluigi
- Krejčí, Pavel
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 47J35 35G25 82B26 74H40

*Keywords*

- nonlinear and nonlocal evolution equations, Cahn-Hilliard type dynamics, phase transitions models, existence, uniqueness, long-time behaviour

*DOI*

*Abstract*

This paper is devoted to the analysis of an abstract evolution inclusion with a non-invertible operator, motivated by problems arising in nonlocal phase separation modeling. Existence, uniqueness, and long-time behaviour of the solution to the related Cauchy problem are discussed in detail.

*Appeared in*

- Czechoslovak Math. J., 57 (2007) pp. 1067-1098.

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# On existence of a bounded solution in a problem with a control parameter

*Authors*

- Shchetinina, Ekaterina

*2010 Mathematics Subject Classification*

- 34A34 34C11

*Keywords*

- bounded solutions, delayed loss of stability, gluing

*DOI*

*Abstract*

This paper is devoted to the problem of existence of bounded solutions for non-autonomous differential equations in the case when the linear part has a pair of simple complex conjugate eigenvalues crossing the imaginary axis for increasing $t$. By introducing a control parameter into the system we derive conditions for the existence of a global uniformly bounded solution.

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# Estimation of time dependent volatility via local change point analysis

*Authors*

- Mercurio, Danilo
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62M10 62P20

*Keywords*

- volatility model, adaptive estimation, local homogeneity, change point

*DOI*

*Abstract*

This paper offers a new procedure for estimation and forecasting of the volatility of financial time series. The approach is based on the assumption of local homogeneity: for every time point there exists an interval of time homogeneity in which the volatility parameter can be well approximated by a constant. The procedure recovers this interval from the data using the local change point analysis. Afterwards the estimate of the volatility can be simply obtained by local averaging. We investigate the performance of the procedure both from the theoretical point of view and through Monte Carlo simulations. Then the new procedure is applied to some data sets and a comparison with the LAVE procedure from Mercurio and Spokoiny (2004) and with a standard GARCH model is also provided. Finally we apply the new method for the The numerical results demonstrate a very reasonable performance of the new method.

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# Experimental and mathematical study of the influence of growth factors and the kinetics of adult human articular chondrocytes

*Authors*

- Barbero, Andrea
- Palumberi, Viviana
- Wagner, Barbara
- Sader, Robert
- Grote, Markus J.
- Martin, Ivan

*2010 Mathematics Subject Classification*

- 92D25 92C37

*Keywords*

- chondrocytes, cell expansion, growth kinetics, delay model

*DOI*

*Abstract*

This study aimed at determining how kinetic parameters of adult human articular chondrocytes (AHAC) growth are modulated by the growth factor combination underline TGF$beta$1, underline FGF-2, and underline PDGF BB (TFP), recently shown to stimulate AHAC proliferation. AHAC, isolated from cartilage biopsies of three individuals, were cultured in medium without (CTR) or with TFP. For growth curves, AHAC were seeded at 1000 cells/cm$^2$ and cultured for 12 days, with cell numbers measured fluorimetrically in the same wells every 12 hours. For microcolony tests, AHAC were seeded at 2.5 cells/cm$^2$ and cultured for 6 days, with cell numbers determined for each microcolony by phase contrast microscopy every 8 hours. A mathematical model combining delay and logistic equations was developed to capture the growth kinetic parameters and to enable the description of the complete growth process of the cell culture. As compared to CTR medium, the presence of TFP increased the number of cells/well starting from the fifth day of culture, and a 4-fold larger cell number was reached at confluency. For single microcolonies, TFP reduced the time for the first cell division by 26.6%, the time for subsequent cell divisions (generation time) by 16.8%, and the percentage of quiescent cells by 42.5%. The mathematical model fitted well the experimental data of the growth kinetic. Finally, using both microcolony tests and the mathematical model, we determined that prolonged cell expansion induces an enrichment of AHAC with shorter first division time, but not of those with shorter generation time.

*Appeared in*

- J. Cellular Physiology, 204 (2005) pp. 830--838.

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# Stochastic Eulerian model for the flow simulation in porous media. Unconfined aquifers

*Authors*

- Kolyukhin, Dmitry R.
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Hydraulic conductivity, Lognormal random field, small fluctuations, Darcy law, randomized spectral representation

*DOI*

*Abstract*

This work deals with a stochastic unconfined aquifer flow simulation in statistically isotropic saturated porous media. This approach is a generalization of the 3D model we developed in citeks. In this paper we deal with a 2D model obtained via depth-averaging of the 3D model. The average hydraulic conductivity is assumed to be a random field with a lognormal distribution. Assuming the fluctuations in the hydraulic conductivity to be small we construct a stochastic Eulerian model for the flow as a Gaussian random field with a spectral tensor of a special structure derived from Darcy's law. A randomized spectral representation is then used to simulate this random field. A series of test calculations confirmed the high accuracy and computational efficiency of the method.

*Appeared in*

- Monte Carlo Methods Appl. vol 10 (2004), no. 3-4, pp. 345-357

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# Analysis of nonlocal neural fields for both general and gamma-distributed connectivities

*Authors*

- Hutt, Axel
- Atay, Fatihcan M.

*2010 Mathematics Subject Classification*

- 45J05 92C2

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Sk 05.45.Xt 05.10.-a

*Keywords*

- Nonlocal neural activity, space-dependent delay, stability analysis

*DOI*

*Abstract*

This work studies the stability of spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation considers synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation. In case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition-lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.

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# Quantifying hydrodynamic slip: A comprehensive analysis of dewetting profiles

*Authors*

- Fetzer, Renate
- Münch, Andreas
- Wagner, Barbara
- Rauscher, Markus
- Jacobs, Karin

*2010 Mathematics Subject Classification*

- 76D08 76E17 74A55

*Keywords*

- Polymer melts, slip boundary effects, interfacial and free surface flows, lubrication models, Stokes model

*DOI*

*Abstract*

To characterize non-trivial boundary conditions of a liquid flowing past a solid, the slip length is commonly used as a measure. From the profile of a retracting liquid front as measured, e.g., with atomic force microscopy, the slip length as well as the capillary number can be extracted by the help of the Stokes model for a thin liquid film dewetting from a solid substrate. Specifically, we use a lubrication model derived from the Stokes model for strong slippage and linearize the film profile around the flat, unperturbed film, and, for small slip lengths a Taylor approximation of the linearisation for the full Stokes model. Furthermore, from the capillary number and the knowledge of the liquid front velocity and the surface tension, we can obtain the viscosity of the fluid film. We compare theoretical and experimental results, test the consistency and the validity of the models/approximations, and give an easy-to-follow manual of how they can be used to analyze experiments.

*Appeared in*

- Langmuir, 23 (2007) pp. 10559-10566.

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# Numerical bifurcation analysis of traveling wave model of multisection semiconductor lasers

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 37L65 35B32 78M25 65P30

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 05.45.-a

*Keywords*

- Numerical bifurcation analysis, traveling wave model, mode approximations, quality of pulsations

*DOI*

*Abstract*

Traveling wave equations are used to model the dynamics of multisection semiconductor lasers. To perform a bifurcation analysis of this system of 1-D partial differential equations its low dimensional approximations are constructed and considered. Along this paper this analysis is used for the extensive study of the pulsations in a three section distributed feedback laser. Namely, stability of pulsations, different bifurcation scenaria, tunability of the pulsation frequency and its locking by the frequency of electrical modulation are considered. All these pulsation qualities are highly important when applying lasers in optical communication systems.

*Appeared in*

- Physica D, 213 (1), pp. 98-112, 2006.

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# On the left tail asymptotics for the limit law of supercritical Galton--Watson processes in the Böttcher case

*Authors*

- Fleischmann, Klaus
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F10

*Keywords*

- Lower deviation probabilities, Schröder case, Böttcher case, logarithmic asymptotics, fine asymptotics, precise asymptotics, tiny oscillations

*DOI*

*Abstract*

Under a well-known scaling, supercritical Galton-Watson processes $Z$ converge to a non-degenerate non-negative random limit variable $W.$ We are dealing with the left tail (i.e. lose to the origin) asymptotics of its law. In the Bötcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing tiny oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the Böttcher case, we improve a recent lower deviation probability result by describing the precise asymptotics under a logarithmic scaling (Theorem 3). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 4).

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# Stability and sensitivity of optimization problems with first order stochastic dominance constraints

*Authors*

- Dentcheva, Darinka
- Henrion, René
- Ruszczynski, Andrzej

*2010 Mathematics Subject Classification*

- 90C15 90C34 90C48

*Keywords*

- stochastic programming, stochastic ordering, semi-infinite optimization, chance constraints, Lipschitz stability, metric regularity, directional differentiability

*DOI*

*Abstract*

We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.

*Appeared in*

- SIAM J. Optim., 18 (2007) pp. 322--337.

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# Non-nested multi-grid solvers for mixed divergence-free Scott--Vogelius discretizations

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Matthies, Gunar
- Tobiska, Lutz

*2010 Mathematics Subject Classification*

- 76D05 65F10

*2008 Physics and Astronomy Classification Scheme*

- 47.11.-j

*Keywords*

- Non-Nested Multi-Grid, Stabilized Finite Elements, Navier-Stokes Equations, LBB-Stability

*DOI*

*Abstract*

We apply the general framework developed by John et al. in V. John, P. Knobloch, G. Matthies, L. Tobiska: Non-nested multi-level solvers for finite element discretisations of mixed problems, Computing 2002, to analyze the convergence of multi-level methods for mixed finite element discretizations of the generalized Stokes problem using the Scott-Vogelius element. Having in mind that semi-implicit operator splitting schemes for the Navier-Stokes equations lead to this class of problems, we take symmetric stabilization operators into account. The use of the class of Scott-Vogelius elements seems to be promising since discretely divergence-free functions are pointwise divergence-free. However, to satisfy the Ladyzhenskaya-Babuška-Brezzi stability condition, we have to deal in the multi-grid analysis with non-nested families of meshes which are derived from nested macro element triangulations.

*Appeared in*

- Computing, 83 (2008) pp. 87--107.

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# Optimal superhedging under nonconvex constraints -- A BSDE approach

*Authors*

- Bender, Christian
- Kohlmann, Michael

*2010 Mathematics Subject Classification*

- 91B28 91B24 93E20 60H10

*Keywords*

- BSDE, Constraints, Penalization, Superhedging

*DOI*

*Abstract*

We apply theoretical results of S. Peng on supersolutions for BSDEs to the problem of finding optimal superhedging strategies in a Black-Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be nonconvex, time-dependent, and random. Constraints on the portfolio may e.g. be formulated in terms of the amount of money invested, the portfolio proportion, or the number of shares held.

*Appeared in*

- Int. J. Theor. Appl. Finance, 11 pp. 363--380.

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# New slip regimes and the shape of dewetting thin liquid films

*Authors*

- Konrad, Renate
- Jacobs, Karin
- Münch, Andreas
- Wagner, Barbara
- Witelski, Thomas

*2010 Mathematics Subject Classification*

- 76A55 76E17 76E17

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e, 47.20.Ma, 47.54.+r, 68.37.Ps

*Keywords*

- slippage, lubrication approximation, interfacial instability

*DOI*

*Abstract*

We compare the dewetting behavior of liquid polymer films on silicon/silicon oxide wafers that have been coated with either Octadecyltrichlorosilane (OTS) or Dodecyltrichlorosilane (DTS). Our experiments show that the dewetting rates for DTS are significantly larger than for OTS. We also compare the profile of the rim that forms as the film dewets and find that it develops a spatially decaying oscillatory structure on the side facing the undisturbed film if an OTS coated wafer is used, but is monotonically decaying for DTS. We argue that for this situation only the friction coefficient can be different, suggesting that slippage plays a role in this transition. For the first time, we show here that this transition is in fact captured by a lubrication model that can be derived from the Navier-Stokes equations with a Navier-slip boundary condition at the liquid/solid interface, and accounts for large slip lengths.

*Appeared in*

- Phys.Rev. Lett., 95 (2005), pp. 127801/1--127801/4.

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# Weak-convergence methods for Hamiltonian multiscale problems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35B27 74Q10 37Kxx 37K60 70Hxx

*Keywords*

- Homogenization, infinite-dimensional Hamiltonian and Lagrangian, effective Hamiltonian, wave equation, oscillator chain, Gamma convergence, recovery operators

*DOI*

*Abstract*

We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

*Appeared in*

- Discrete Contin. Dyn. Syst., 20 (2008) pp. 53--79.

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# Optimal control of a semilinear PDE with nonlocal radiation interface conditions

*Authors*

- Meyer, Christian
- Philip, Peter
- Tröltzsch, Fredi

*2010 Mathematics Subject Classification*

- 49K20 35J65 49J20 80M50

*Keywords*

- Optimal control, semilinear elliptic equations, nonlocal interface conditions, boundedness of solutions

*DOI*

*Abstract*

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize the temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based on a minimum principle for the semilinear equation as well as L-infinity estimates for the weak solution, we establish the existence of an optimal solution as well as necessary optimality conditions. The theoretical results are illustrated by results of numerical computations.

*Appeared in*

- SIAM Journal On Control and Optimization 45 (2006), pp.699-721.

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# State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions

*Authors*

- Meyer, Christian
- Yousept, Irwin

*2010 Mathematics Subject Classification*

- 35J60 49K20 49M05 65K10

*Keywords*

- Nonlinear optimal control, nonlocal radiation interface conditions, state constraints, first-order necessary conditions, second-order sufficient conditions

*DOI*

*Abstract*

We consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state-constraints represent the particular features of this problem. To deal with the state-constraints, continuity of the state is shown which allows to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an $L^2$-neighborhood.

*Appeared in*

- SIAM J. Control Optim., 48 (2009) pp. 734--755

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# Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

*Authors*

- Giannoulis, Johannes
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Sparber, Christof

*2010 Mathematics Subject Classification*

- 81Q20 34E13 34E20 35Q55

*Keywords*

- Nonlinear Schrödinger equation, Bloch eigenvalue problem, two scale asymptotics, modulation equations, four-wave interaction

*DOI*

*Abstract*

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

*Appeared in*

- J. Differential Equations, 245 (2008) pp. 939--963.

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# Positivity and time behavior of a general linear evolution system, non-local in space and time

*Authors*

- Stephan, Holger
- Khrabustovskyi, Andrii

*2010 Mathematics Subject Classification*

- 35B27 35K60

*Keywords*

- diffusion-reaction systems, positive solutions, maximum principle, homogenization, Riemannian manifold

*DOI*

*Abstract*

We consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.

*Appeared in*

- Math. Methods Appl. Sci., 31 (2008) pp. 1809--1834.

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# Quasiperiodic regimes in multisection semiconductor lasers

*Authors*

- Gonchenko, Sergey V.
- Schneider, Klaus R.
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 34C29 78A60 34C20 34C60

*Keywords*

- multisection semiconductor laser, mode approximation, quasiperiodic regime, normal form, stability

*DOI*

*Abstract*

We consider a mode approximation model for the longitudinal dynamics of a multisection semiconductor laser which represents a slow-fast system of ordinary differential equations for the electromagnetic field and the carrier densities. Under the condition that the number of active sections $q$ coincides with the number of critical eigenvalues we introduce a normal form which admits to establish the existence of invariant tori. The case $q=2$ is investigated in more detail where we also derive conditions for the stability of the quasiperiodic regime.

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# Large time asymptotics of growth models on space-like paths I: PushASEP

*Authors*

- Borodin, Alexei
- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, space-like universality, KPZ class, Airy processes

*DOI*

*Abstract*

We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

*Appeared in*

- Electron. J. Probab., 13 (2008) pp. 1380--1418.

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# A 1D coupled Schrödinger drift-diffusion model including collisions

*Authors*

- Baro, Michael
- Ben Abdallah, Naoufel
- Degond, Pierre
- El Ayyadi, Asma

*2010 Mathematics Subject Classification*

- 65Z05 82D37 78A35 82C70 34L40 34L30 34L25

*Keywords*

- quantum-classical coupling, Schroedinger equation, scattering states, Pauli master equation, drift-diffusion, interface conditions

*DOI*

*Abstract*

We consider a one-dimensional coupled stationary Schroedinger drift-diffusion model for quantum semiconductor device simulations. The device domain is decomposed into a part with large quantum effects (quantum zone) and a part where quantum effects are negligible (classical zone). We give boundary conditions at the classic-quantum interface which are current preserving. Collisions within the quantum zone are introduced via a Pauli master equation. To illustrate the validity we apply the model to three resonant tunneling diodes.

*Appeared in*

- Journal of Computational Physics, 2005, Vol. 203, no. 1, pp. 129-153

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# Mott law as lower bound for a random walk in a random environment

*Authors*

- Faggionato, Alessandra

ORCID: 0000-0002-6168-3517 - Schulz-Baldes, H.
- Spehner, D.

*2010 Mathematics Subject Classification*

- 60D05 60K3 88C44

*Keywords*

- Disordered systems, Mott law, random walk in random environment, point processes, percolation

*DOI*

*Abstract*

We consider a random walk on the support of a stationary simple point process on $RR^d$, $dgeq 2$ which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localisation. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.

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# Universality of the REM for dynamics of mean-field spin glasses

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Černý, Jiři

*2010 Mathematics Subject Classification*

- 82C44 60K35 60G70

*Keywords*

- aging, universality, spin glasses, SK model, random walk

*DOI*

*Abstract*

We consider a version of a Glauber dynamics for a $p$-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the $N$-dimensional hypercube. We show that, for any $p geq 3$ and any inverse temperature $beta>0$, there exist constants $g_0>0$, such that for all exponential time scales, $exp(gamma N)$, with $gleq g_0$, the properly rescaled emphclock process (time-change process), converges to an $a$-stable subordinator where $a=g/b^2<1$. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this hbox$alpha$-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of $p$-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case $p=2$) seems to belong to a different universality class.

*Appeared in*

- J. Stat. Mech. Theory Exp., 4 (2008) pp. L04003/1--L04003/8.

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# Temporal decorrelation for branching random walks with state dependent branching rate

*Authors*

- Birkner, Matthias

*2010 Mathematics Subject Classification*

- 60K35

*Keywords*

- State dependent branching, temporal decorrelation

*DOI*

*Abstract*

We consider branching random walks in $d ge 3$ with a Lipschitz branching rate function and show that the system, starting either in a Poisson field or in equilibrium, decorrelates over long time horizons, and employ this to obtain an ergodic theorem. We use coupling and a stochastic representation of the Palm distribution.

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# Energy estimates for continuous and discretized electro-reaction-diffusion systems

*Authors*

- Glitzky, Annegret
- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 35B40 35K57 78A35 35R05 65M12

*Keywords*

- Reaction-diffusion systems, drift-diffusion processes, motion of charged particles, energy estimates, thermodynamic equilibria, asymptotic behaviour, time and space discretization

*DOI*

*Abstract*

We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

*Appeared in*

- Nonlinear Anal., 70 (2009) pp. 788--805.

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# Asymptotic stability of continual sets of periodic solutions to systems with hysteresis

*Authors*

- Brokate, Martin

ORCID: 0000-0003-4660-9180 - Pokrovskii, Alexei
- Rachinskii, Dmitri

*2010 Mathematics Subject Classification*

- 34C55 34D20 34D10

*Keywords*

- Hysteresis perturbations of ODE, Preisach hysteresis nonlinearity, periodic solutions, stability

*DOI*

*Abstract*

We consider hysteresis perturbations of the system of ODEs which has an asymptotically stable periodic solution $z_*$. It is proved that if the oscillation of the appropriate projection of $z_*$ is smaller than some threshold number defined by the hysteresis nonlinearity, then the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of $z_*$. The main result is the theorem on stability of this continuum.

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# Nonlinear estimation for linear inverse problems with error in the operator

*Authors*

- Hoffmann, Marc
- Reiß, Markus

*2010 Mathematics Subject Classification*

- 65J20 62G07

*Keywords*

- statistical inverse problem, Galerkin projection method, wavelet thresholding, minimax rate, degree of ill-posedness, matrix compression

*DOI*

*Abstract*

We consider nonlinear estimation methods for statistical inverse problems in the case where the operator is not exactly known. For a canonical formulation a Gaussian operator white noise framework is developed. Two different nonlinear estimators are constructed, which correspond to the different order of the linear inversion and nonlinear smoothing step. We show that both estimators are rate-optimal over a wide range of Besov smoothness classes. The construction is based on the Galerkin projection method and wavelet thresholding schemes for the data and the operator.

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# Scaling limit and aging for directed trap models

*Authors*

- Zindy, Olivier

*2010 Mathematics Subject Classification*

- 60K37 60G50 60G52 60F17 82D30

*Keywords*

- Directed trap model, random walk, scaling limit, subordinator, aging

*DOI*

*Abstract*

We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Černý for a large class of graphs.

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# On the approximation of the limit cycles function

*Authors*

- Cherkas, Leonid
- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C05 34C07 65L99

*Keywords*

- Family of limit cycles, multiple limit cycle, Liénard system

*DOI*

*Abstract*

We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincaré function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Liénard system. The obtained result supports a conjecture by Lins, de Melo and Pugh.

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# Properties of stationary states of delay equations with large delay and applications to laser dynamics

*Authors*

- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34K20 34K60 34K06

*Keywords*

- delay equations, large delay, map, Lang-Kobayashi system, laser

*DOI*

*Abstract*

We consider properties of periodic solutions of the differential-delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential-delay system with fixed large delay $tau$. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit $tautoinfty$.

*Appeared in*

- Math. Meth. Appl. Sci. (2005) 28:363-377.

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# Regular polynomial interpolation and approximation of global solutions of linear partial differential equations

*Authors*

- Kampen, Jörg

*2010 Mathematics Subject Classification*

- 65D05 35G05

*Keywords*

- extended Newtonian interpolation, linear systems of partial differential equations, error estimates

*DOI*

*Abstract*

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

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# Change of the type of contrast structures in parabolic Neumann problems

*Authors*

- Nefedov, Nikolai N.
- Radziunas, Mindaugas
- Schneider, Klaus R.
- Vasil'eva, Adelaida B.

*2010 Mathematics Subject Classification*

- 35B25 35K50 35K57

*Keywords*

- Singularly perturbed parabolic problems, contrast structures

*DOI*

*Abstract*

We consider some class of singularly perturbed nonlinear parabolic problems in the case when a solution with an interior layer changes into a solution having only boundary layers. Analytical results on this phenomenon are compared with numerical studies of some examples.

*Appeared in*

- Comp. Math. and Math. Physics *45(1)*, pp. 37-51, 2005

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# Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Eymard, Robert
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65N25 65N30 81Q10

*Keywords*

- Schrödinger operator, eigenvalues, finite volume schemes.

*DOI*

*Abstract*

We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally.

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# Solutions to muscle fiber equations and their long time behaviour

*Authors*

- Krejč'ı, Pavel
- Sante-Marie, Jacques
- Sorine, Michel
- Urquiza, Jose M.

*2010 Mathematics Subject Classification*

- 35K60 74L15 35B35 35B40 92C10

*Keywords*

- nonlinear initial boundary value problem, existence and uniqueness of solutions, asymptotic behaviour, muscle and cardiac mechanics

*DOI*

*Abstract*

We consider the nonlinear initial-boundary value problem governing the dynamical displacements of a one dimensional solid body with specific stress-strain law. This constitutive law results from the modelization of the mechanisms that rules the electrically activated mechanical behaviour of cardiac muscle fibers at the microscopic level. We prove global existence and uniqueness of solutions and we study their asymptotic behaviour in time. In particular we show that under vanishing external forcing solutions asymptotically converge to an equilibrium.

*Appeared in*

- Nonlinear Anal. Real World Appl., 7 (2006) pp. 535-558.

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# Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP

*Authors*

- Borodin, Alexei
- Ferrari, Patrik
- Sasamoto, Tomohiro

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, space-like universality, KPZ class, Airy processes

*DOI*

*Abstract*

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.

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# Minimax and Bayes estimation in deconvolution problem

*Authors*

- Ermakov, Mikhail S.

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- deconvolution, minimax estimation, Bayes estimation, Wiener filtration

*DOI*

*Abstract*

We consider the problem of estimation of solution of convolution equation on observations blurred a random noise. The noise is a product of Gaussian stationary process and a weight function $epsilon h in L_2(R1)$ with constant $epsilon > 0$. The presence of weight function $h$ makes the power of noise finite on $R1$. This allows to suppose that the power of solution is also finite. For this model we find asymptotically minimax and Bayes estimators. The solution is supposed infinitely differentiable. The model with solutions having finite number of derivatives was studied in [5].

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# Recovering edges of an image from noisy tomographic data

*Authors*

- Goldenshluger, Alexander
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G20 62C20 94A08

*Keywords*

- Radon transform, optimal rates of convergence, support function, edge detection, minimax estimation

*DOI*

*Abstract*

We consider the problem of recovering edges of an image from noisy tomographic data. The original image is assumed to have a discontinuity jump (edge) along the boundary of a compact convex set. The Radon transform of the image is observed with noise, and the problem is to estimate the edge. We develop an estimation procedure which is based on recovering support function of the edge. It is shown that the proposed estimator is nearly optimal in order in a minimax sense. Numerical examples illustrate reasonable practical behavior of the estimation procedure.

*Appeared in*

- IEEE Trans. Inform. Theory, 4 (2006) pp. 1322--1334.

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# Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D10 35J55 35Q60

*Keywords*

- Maxwell's equations, natural interface conditions, Lorentz force, higher integrability

*DOI*

*Abstract*

We consider the stationary Maxwell system in a domain filled with different materials. The magnetic permeability being only piecewise smooth, we have to take into account the natural interface conditions for the electromagnetic fields. We present two sets of hypotheses under which we can prove the existence of weak solutions to the Maxwell system such that the Lorentz force jxB is integrable to a power larger than 6/5. This property is important for the investigation of problems in magnetohydrodynamics, with many industrial applications such as crystal growth.

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# Transition between Airy$_1$ and Airy$_2$ processes and TASEP fluctuations

*Authors*

- Borodin, Alexei
- Ferrari, Patrik
- Sasamoto, Tomohiro

*2010 Mathematics Subject Classification*

- 82C22 60K35 15A52

*Keywords*

- Simple exclusion process, universality, KPZ class, Airy process, random matrices

*DOI*

*Abstract*

We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. F or large time $t$, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy$_1$ and Airy$_2$ processes, whose one-point distributions are the GOE and GUE Tracy-Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one-point distribution is a new interpolati on between GOE and GUE edge distributions.

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# Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

*Authors*

- Enriquez, N.
- Sabot, C.
- Zindy, O.

*2010 Mathematics Subject Classification*

- 60K37 60G50 60J45 82D30

*Keywords*

- Random walk in random environment; aging; quenched, localization

*DOI*

*Abstract*

We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of 'valleys' of height log $t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.

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# On some classes of limit cycles of planar dynamical systems

*Authors*

- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C07 34C05 37C27

*Keywords*

- plane vector fields, maximal number of limit cycles, weakend Hilbert 16-th problem

*DOI*

*Abstract*

We consider two-dimensional smooth vector fields $dx/dt = P(x,y), dy/dt = Q(x,y)$ and estimate the maximal number of limit cycles with special properties which are defined by means of generalized Dulac and Cherkas functions. In case that $P$ and $Q$ are polynomials we present results about the weakend 16-th problem of Hilbert.

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# Destabilization patterns in large regular networks

*Authors*

- Yanchuk, Serhiy
- Wolfrum, Matthias

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 02.30.Oz, 89.75.Fb, 82.40.Ck

*Keywords*

- Networks, coupled oscillators, bifurcations, Eckhaus instability

*DOI*

*Abstract*

We describe a generic mechanism for the destabilization in large regular networks of identical coupled oscillators. Based on a reduction method for the spectral problem, we first present a criterion for this type of destabilization. Then, we investigate the related bifurcation scenario, showing the existence of a large number of coexisting periodic solutions with different frequencies, spatial patterns, and stability properties. Even for unidirectional coupling this can be understood in analogy to the well-known Eckhaus scenario for diffusive systems.

*Appeared in*

- Phys. Rev. E, 77 (2008) pp. 026212/1-026212/7.

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# Alpha-stable branching and beta-coalescents

*Authors*

- Birkner, Matthias
- Blath, Jochen
- Capaldo, Marcella
- Etheridge, Alison
- Möhle, Martin
- Schweinsberg, Jason
- Wakolbinger, Anton

*2010 Mathematics Subject Classification*

- 60J80 60J70 60J25 60G09 60G52 92D25

*Keywords*

- Alpha-stable branching, coalescent, genealogy, lookdown construction

*DOI*

*Abstract*

We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $alpha$-stable branching mechanisms. The random ancestral partition is then a time-changed $Lambda$-coalescent, where $Lambda$ is the Beta-distribution with parameters $2-alpha$ and $alpha$, and the time change is given by $Z^1-alpha$, where $Z$ is the total population size. For $alpha = 2$ (Feller's branching diffusion) and $Lambda = delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $alpha =1$ and $Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent. We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999)

*Appeared in*

- Electron. J. Probab., 10 (9), 303-325 (electronic)

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# Stochastic simulation method for a 2D elasticity problem with random loads

*Authors*

- Sabelfeld, Karl
- Shalimova, Irina
- Levykin, Alexander I.

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*Keywords*

- Isotropic Random Fields, Spectral Tensor, Poisson integral formula, Random Walk on Fixed Spheres, Lamé equation, Successive Over Relaxation Method, Transverse and Longitudinal Correlations

*DOI*

*Abstract*

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper citesab-lev-shal-2006. The vector random field of loads which stands in the right-hand-side of the system of elasticity equations is simulated by the Randomization Spectral method presented in citesab-1991 and recently revised and generalized in citekurb-sab-2006. Comparative analysis of RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads.

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# Galerkin method for feedback controlled Rayleigh--Bénard convection

*Authors*

- Wagner, Barbara
- Münch, Andreas

*2010 Mathematics Subject Classification*

- 74D10 74F05 74F10 77N25

*Keywords*

- Galerkin approximation, Stability, Pattern Formation

*DOI*

*Abstract*

We employ a Galerkin approximation for the system of equations governing Rayleigh-Bénard convection. This approximation reduces the dimension of the problem by one, while it captures the nonlinear behavior even when only a few basis functions are used. We prove convergence of the method and finally demonstrate the effectiveness of this method for the problems of feedback controlled Rayleigh-Bénard convection in three dimensions and the complex dynamics of spiral-defect chaos.

*Appeared in*

- Nonlinearity, Volume 21 (2008) pp. 2625--2651.

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# On moderate deviation probabilities of empirical bootstrap measure

*Authors*

- Ermakov, Mikhail S.

*2010 Mathematics Subject Classification*

- 60F10

*Keywords*

- large deviations, moderate deviations, bootstrap, empirical measure

*DOI*

*Abstract*

We establish the moderate deviation principle for the common distribution of empirical measure and empirical bootstrap measure (empirical measure obtaining by the bootstrap procedure). For the most widespread statistical functionals depending on empirical measure (in particular differentiable and homogeneous functionals) we compare their asymptotic of moderate deviation probabilities with the asymptotic given by the bootstrap procedure.

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# Nonparametric volatility estimation on the real line from low-frequency data

*Authors*

- Reiß, Markus

*2010 Mathematics Subject Classification*

- 62M05 60H10

*Keywords*

- diffusion process, nonparametric inference, wavelet, spectral approximation, low-frequency observations

*DOI*

*Abstract*

We estimate the volatility function of a diffusion process on the real line on the basis of low frequency observations. The estimator is based on spectral properties of the estimated Markov transition operator of the embedded Markov chain. Asymptotic risk estimates for a growing number of observations are provided without assuming the observation distance to become small.

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# A simple method to study the transitional dynamics in endogenous growth models

*Authors*

- Bethmann, Dirk
- Reiß, Markus

*2010 Mathematics Subject Classification*

- 91B62 49L20 91B66

*Keywords*

- transitional dynamics, dynamic programming, Uzawa-Lucas model, human capital allocation, dimension reduction

*DOI*

*Abstract*

We introduce a simple method of analyzing the transitional dynamics of the Uzawa-Lucas endogenous growth model with human capital externalities. We use the value function approach to solve both the social planner's optimization problem in the centralized economy and the representative agent's optimization problem in the decentralized economy. The complexity of the Hamilton-Jacobi-Bellman equations is significantly reduced to an initial value problem for one ordinary differential equation. This approach allows us to find the optimal controls for the non-concave Hamiltonian in the centralized case and to identify the symmetric Nash equilibrium of the agents' optimal strategies in the decentralized case. For a wide range of the degree of the human capital externality we calculate the global transitional dynamics towards the balanced growth path. The U-shaped course of output growth rates is explained in detail. JEL Classifications: C61, O41, C72

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# Degree of ill-posedness of statistical inverse problems

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421

*2010 Mathematics Subject Classification*

- 65J20 62G20

*Keywords*

- Degree of ill--posedness, statistical inverse problem, variable Hilbert scale

*DOI*

*Abstract*

We introduce the notion of the degree of ill--posedness of linear operators in operator equations between Hilbert spaces. For specific assumptions on the noise this quantity can be computed explicitely. Next it is shown that the degree of ill--posedness as introduced explains the loss of accuracy when solving inverse problems in Hilbert spaces for a variety of instances.

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# A thermoelastic contact problem with a phase transition

*Authors*

- Hömberg, Dietmar
- Khludnev, Alexander

*2010 Mathematics Subject Classification*

- 35K85 74N25 74F05 74M15

*Keywords*

- Thermoelasticity, phase transition, contact problem

*DOI*

*Abstract*

We investigate a thermomechanical contact problem with phase transitions. The system of equations consists of a quasistatic momentum balance and a semilinear energy balance. The phase transition is described by an ordinary differential equation. Different mechanical properties of the respective phases are taken care of by a mixture ansatz. We prove the existence of a weak solution and a uniqueness result, the latter only being valid in one space dimension.

*Appeared in*

- IMA J. Appl. Math., 71 (2006) pp. 479--495.

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# On a thermomechanical model of phase transitions in steel

*Authors*

- Chełminski, Krzysztof
- Hömberg, Dietmar
- Kern, Daniela

*2010 Mathematics Subject Classification*

- 35Q72 74A15 74F05

*Keywords*

- Heat treatment, coupled partial differential equation, existence and uniqueness, phase transitions, linear thermoelasticity, thermomechanics, distortion

*DOI*

*Abstract*

We investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.

*Appeared in*

- Adv. Math. Sci. Appl., 18 (2008), pp. 119--140

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# Convexity of chance constraints with independent random variables

*Authors*

- Henrion, René
- Strugarek, Cyrille

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Chance constraints, probabilistic constraints, stochastic programming, convexity, random matrix

*DOI*

*Abstract*

We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.

*Appeared in*

- Computational Optimization and Applications 41 (2008) 263-276.

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# Dewetting rates of thin liquid films

*Authors*

- Münch, Andreas

*2010 Mathematics Subject Classification*

- 76A20 76D27

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e 47.15.Gf 68.03.Kn 68.03.Cd

*Keywords*

- Lubrication approximation, slippage

*DOI*

*Abstract*

We investigate the dewetting rates of thin liquid films using a lubrication model that describes the dewetting process of polymer melts on hydrophobized substrates. We study the effect of different boundary conditions at the liquid/solid interface, in particular, of the no-slip and the Navier slip boundary condition, and compare our numerical solutions for the no-slip and the slip dominated cases to available results that originate from scaling arguments, simplified flow assumptions and energy balances. We furthermore consider these issues for an extended lubrication model that includes nonlinear curvature.

*Appeared in*

- J. Phys.: Condensed Matter, 17 (2005), pp. S309--S318

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# Contact-line instability of dewetting thin films

*Authors*

- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76A20 76D27

*2008 Physics and Astronomy Classification Scheme*

- 97.10.Gz, 97.30.Qt, 97.80.Gm

*Keywords*

- Lubrication approximation, stability, slippage

*DOI*

*Abstract*

We investigate the linear stability of dewetting thin polymer films on hydrophobised substrates driven by Van-der-Waals forces, using a lubrication model. We focus on the role of slippage in the emerging instability at the three-phase contact-line and compare our results to the corresponding no-slip case. Our analysis shows that generically, small perturbations of the receding front are amplified, but in the slippage case by orders of magnitude larger than in the no-slip case. Moreover, while the perturbations become symmetrical in the no-slip case, they are asymmetrical in the slippage case. We furthermore extend our lubrication model to include effects of nonlinear curvature.

*Appeared in*

- Phys. D. 209 (2005), pp.178-190.

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# Universality of residence-time distributions in non-adiabatic stochastic resonance

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Ey 05.10.Gg 05.40.-a

*Keywords*

- Stochastic resonance, residence-time distribution, noise-induced exit, oscillating barrier, periodic driving, activated escape, metastability, cycling.

*DOI*

*Abstract*

We present a mathematically rigorous expression for the residence-time distribution of a periodically forced Brownian particle in a bistable potential. For a broad range of forcing frequencies and amplitudes, the distribution is close to a periodically modulated exponential one. Remarkably, the periodic modulation is governed by a universal function, depending on a single parameter related to the forcing period. The behaviour of the distribution and its moments is analysed, in particular in the low- and high-frequency limits.

*Appeared in*

- Europhys. Lett. 70 (1), pp. 1-7, (2005) under new title: Universality of first-passage- and residence-time distributions in non-adiabatic stochastic resonance

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# An iterative algorithm for multiple stopping: Convergence and stability

*Authors*

- Bender, Christian
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G40 62L15 91B28

*Keywords*

- optimal stopping, policy improvement, multiple callable financial derivatives

*DOI*

*Abstract*

We present a new iterative procedure for solving the discrete multiple stopping problem and discuss the stability of the algorithm. The algorithm produces monotonically increasing approximations of the Snell envelope, which coincide with the Snell envelope after finitely many steps. Contrary to backward dynamic programming, the algorithm allows to calculate approximative solutions with only a few nestings of conditionals expectations and is, therefore, tailor-made for a plain Monte-Carlo implementation.

*Appeared in*

- Advances in Applied Probability, Volume 38, Number 3 (2006) pp. 729--749.

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# Iterative construction of the optimal Bermudan stopping time

*Authors*

- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62L15 65C05 91B228

*Keywords*

- Bermudan options, optimal stopping, Monte Carlo simulation, LIBOR market model

*DOI*

*Abstract*

We present an iterative procedure for computing the optimal Bermudan stopping time. We prove convergence and, as a consequence, the method allows for approximation of the Snell envelope from below. By using duality, we then deduce a convergent procedure for approximating the Snell envelope from above as well. We provide numerical examples for Bermudan swaptions in the context of a LIBOR market model.

*Appeared in*

- Finance and Stochastics, Volume 10, Number 1 (Jan. 2006), pp. 27 - 49

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# Elliptic model problems including mixed boundary conditions and material heterogeneities

*Authors*

- Haller-Dintelmann, Robert
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35J25 35R05

*Keywords*

- Elliptic transmission problems, mixed boundary problems, $W^1,p$ regularity

*DOI*

*Abstract*

We present model problems in three dimensions, where the operator $-nabla cdot mu nabla$ maps the Sobolev space $W^1,p_Gamma(Omega)$ isomorphically onto $W^-1,p_Gamma(Omega)$ for a $p>3$. The emphasis is here on the case where different boundary conditions meet material heterogeneities.

*Appeared in*

- J. Math. Pures Appl., 89 (2008) pp. 25--48.

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# The opinion game: Stock price evolution from microscopic market modelling

*Authors*

- Bovier, Anton
- Černý, Jiri
- Hryniv, Ostap

*2010 Mathematics Subject Classification*

- 91B26 60K35 60J20

*Keywords*

- Stock prices, financial markets, statistical mechanics, stochastic dynamics

*DOI*

*Abstract*

We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.

*Appeared in*

- Int. J. Theor. Appl. Finance, 9, (2006) pp. 91--111.

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# Passive mode-locking with slow saturable absorber: A delay differential model

*Authors*

- Vladimirov, Andrei
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc,42.55.Px,42.60.Mi,42.65.Pc,42.60.Gd

*Keywords*

- semiconductor laser, mode-locking, delay differential equations, bifurcations

*DOI*

*Abstract*

We propose and study a new model describing passive mode-locking in a semiconductor laser - a set of differential equations with time delay. Analytical analysis of this model is performed under the slow saturable absorber approximation. Bifurcations responsible for the appearance and break-up of mode-locking regime are studied numerically.

*Appeared in*

- Phys. Rev. A, 72 (2005) pp. 033808/1-033808/13.

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# Much ado about Derrida's GREM

*Authors*

- Bovier, Anton
- Kurkova, Irina

*2010 Mathematics Subject Classification*

- 82B44

*Keywords*

- Gaussian processes, spin-glasses, Generalised random energy model, Poisson point processes, branching processes, coalescence

*DOI*

*Abstract*

We provide a detailed analysis of Derrida's Generalised Random Energy Model (GREM). In particular, we describe its limiting Gibbs measure in terms Ruelle's Poisson cascades. Next we introduce and analyse a more general class of Continuous Random Energy Models (CREMs) which differs from the well-known class of Sherrington-Kirkpatrick models only in the choice of distance on the space of spin configurations : the Hamming distance defines the later class while the ultrametric distance corresponds to the former one. We express explicitly the geometry of its limiting Gibbs measure in terms of genealogies of Neveu's Continuous State branching Process via an appropriate time change. We also identify the distances between replicas under the limiting CREM's Gibbs measure with those between integers of Bolthausen-Sznitman coalescent under the same time change.

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# Phase transition and hysteresis in a rechargeable lithium battery

*Authors*

- Dreyer, Wolfgang
- Gaberšček, Miran
- Jamnik, Janko

*2010 Mathematics Subject Classification*

- 74A15 74A50 74F25 74N20 74N25 74N30

*2008 Physics and Astronomy Classification Scheme*

- 81.30.Mh 82.56.Lz 82.45.Fk 82.60.Qr

*Keywords*

- Thermodynamics, Structured surfaces and interfaces, Solid-phase precipitation,, coexistent phases, Chemical and reactive effects, Dynamics of phase, boundaries, Transformations involving diffusion, Problems involving, hysteresis

*DOI*

*Abstract*

We represent a model which describes the evolution of a phase transition that occurs in some part of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate the hysteretic behavior of the voltage - charge characteristics. During discharging of the battery, the interstitial lattice sites of a small crystalline host system are filled up with lithium atoms and these are released again during charging. We show within the context of a sharp interface model that two mechanical phenomena go along with a phase transition that appears in the host system during supply and removal of lithium. At first the lithium atoms need more space than it is available by the interstitial lattice sites, which leads to a maximal relative change of the crystal volume of about $6%$. Furthermore there is an interface between two adjacent phases that has very large curvature of the order of magnitude 100 m, which evoke here a discontinuity of the normal component of the stress. In order to simulate the dynamics of the phase transitions and in particular the observed hysteresis we establish a new initial and boundary value problem for a nonlinear PDE system that can be reduced in some limiting case to an ODE system.

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# Measure-valued diffusions, general coalescents and population genetic inference

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 92D15 60J70 60G09 60G57

*Keywords*

- Generalised Fleming-Viot process, $Lambda$-coalescent, lookdown construction, mathematical population genetics, Monte-Carlo simulation

*DOI*

*Abstract*

We review recent progress in the understanding of the interplay between population models, measure-valued diffusions, general coalescent processes and inference methods for evolutionary parameters in population genetics. Along the way, we will discuss the powerful and intuitive (modified) lookdown construction of Donnelly and Kurtz, Pitman's and Sagitov's $Lambda$-coalescents as well as recursions and Monte Carlo schemes for likelihood-based inference of evolutionary parameters based on observed genetic types

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# Spectral characterisation of ageing: The REM-like trap model

*Authors*

- Bovier, Anton
- Faggionato, Alessandra

ORCID: 0000-0002-6168-3517

*2010 Mathematics Subject Classification*

- 60K35 82C4

*Keywords*

- disordered systems, random dynamics, trap models, ageing, spectral properties

*DOI*

*Abstract*

We review the ageing phenomenon in the context of simplest trap model, Bouchaud's REM-like trap model from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalised exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a 'grand canonical' representation of the model in terms of the Markov process on a Poisson point process . In this context we analyse the dynamics on various time scales.

*Appeared in*

- Ann. Appl. Probab. , 15, (2005) pp. 1997-2037

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# Marangoni-driven liquid films rising out of a meniscus onto a nearly horizontal substrate

*Authors*

- Münch, Andreas
- Evans, P. L.

*2010 Mathematics Subject Classification*

- 76D08 37N10 76D45 76B45 76A20 34E10 34B60 76D27

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e 47.20.Ky 47.15.Gf 68.03.Kn 68.03.Cd

*Keywords*

- Lubrication theory, Gravity and surface tension driven liquid flows, undercompressive waves, Landau-Levich drag-out problem, coating flows

*DOI*

*Abstract*

We revisit here the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the counter-acting parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in the case where the substrate is nearly horizontal, so there is a significant contribution from the normal component of gravity. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions found in [Münch, SIAM J. Appl. Math., 62(6):2045-2063, 2002]. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness.

*Appeared in*

- Phys. D, 209 (2005), pp. 164--177.

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# Lyapunov functions for positive linear evolution problems

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 47D07 47A63 35B40 35B50 37A35 46E10 82C31

*Keywords*

- Markov operator, Lyapunov function, Fokker-Planck equation, positive continuous semigroup, positive minimum principle, Radon measures

*DOI*

*Abstract*

We rigorously investigate the time monotonicity of Lyapunov functions for general positive linear evolution problems, including degenerate problems. This can be done by considering the problem in the convex set of probability measures and finding a general inequality for such Radon measures and Markov operators. For linear evolution problems (with discrete or continuous time), the existence of time monotone Lyapunov functions is not a consequence of any physical properties, but of the positivity and norm conservation of the equation. In some special cases the structure of such equations is given. Moreover, we describe completely the case of time constant Lyapunov functions - a property of deterministic dynamical systems.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 85 (2005) pp. 766-777

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# Rescaled stable generalised Fleming--Viot processes: Flickering random measures

*Authors*

- Birkner, Matthias
- Blath, Jochen

*2010 Mathematics Subject Classification*

- 60G57 60G17

*Keywords*

- Generalised Fleming-Viot process, measure-valued diffusion,, tightness, Skorohod topology, lookdown construction, wandering random measure, path properties,

*DOI*

*Abstract*

We show how Donnelly and Kurtz' (modified) lookdown construction for measure-valued processes can be used to analyse the longterm- and scaling properties of spatially stable generalised $Lambda$-Fleming Viot processes, exhibiting a rare ``natural'' example of a scaling family converging in f.d.d. sense, but not in any of Skorohod's topologies on path space. This completes results of Fleischmann and Wachtel (2004) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process. The lookdown construction provides an elegant machinery and clear intuition to describe the path properties of the family in terms of a ``flicker effect'', clarifying ``what can go wrong.''

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# Chaotic soliton walk in periodically modulated media

*Authors*

- Turaev, Dmitry
- Radziunas, Mindaugas
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37K45

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf, 05.45.-a, 42.65.Tg

*Keywords*

- nonlinear Shrödinger equation, soliton, chaotic motion

*DOI*

*Abstract*

We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger equation with saturable nonlinearity can result in nontrivial dynamics of radially symmetric solitons. In particular, in the case of hexagonal profile of the modulation the soliton moves chaotically.

*Appeared in*

- Phys. Rev. E, 77 (2008) pp. 06520/1--06520/4.

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# Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

*Authors*

- Gonchenko, Sergey V.
- Shilnikov, Leonid
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37J45 37J25 37C29

*Keywords*

- non-hyperbolic dynamics, global bifurcations, Hamiltonian chaos, elliptic islands

*DOI*

*Abstract*

We show that maps with infinitely many homoclinic tangencies of arbitrarily high orders are dense among real-analytic area-preserving diffeomorphisms in the Newhouse regions.

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# Longitudinal modes of multisection semiconductor lasers and their dynamics

*Authors*

- Radziunas, Mindaugas
- Wünsche, Hans-Jürgen

*2010 Mathematics Subject Classification*

- 65Z05 65N25 78A60 34L16 35P10

*Keywords*

- semiconductor laser, modes, spectral expansion, bifurcations, dynamics, self-pulsations

*DOI*

*Abstract*

We simulate and analyse a 1D-PDE model describing the dynamics of multisection semiconductor lasers. We demonstrate how a semi-analytical computation of the spectrum and the corresponding eigenfunction expansion of the computed solutions provides a useful information allowing to achieve a better understanding of the laser dynamics. Basic algorithms implemented into a corresponding software tool are described.

*Appeared in*

- Optoelectronic Devices - Advanced Simulation and Analysis/, pp. 121-150, ed. J. Piprek, Springer Verlag, New York, 2005. ISBN: 0-387-22659-1 .

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# Ostwald ripening of faceted two-dimensional islands

*Authors*

- Kaganer, Vladimir
- Braun, Wolfgang
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20

*2008 Physics and Astronomy Classification Scheme*

- 81.10.Aj,05.10.Ln,68.43.Jk,81.15.-z

*Keywords*

- Ostwald ripening, Lifshitz--Slyozov--Wagner theory, Becker--Döring equations, faceted crystalline droplets, Gibbs--Thomson formula, kinetic Monte Carlo simulations

*DOI*

*Abstract*

We study Ostwald ripening of two-dimensional adatom and advacancy islands on a crystal surface by means of kinetic Monte Carlo simulations. At large bond energies the islands are square-shaped, which qualitatively changes the coarsening kinetics. The Gibbs--Thomson chemical potential is violated: the coarsening proceeds through a sequence of `magic' sizes corresponding to square or rectangular islands. The coarsening becomes attachment-limited, but Wagner's asymptotic law is reached only after a very long transient time. The unusual coarsening kinetics obtained in the Monte Carlo simulations are well described by the Becker--Döring equations of nucleation kinetics. These equations can be applied to a wide range of coarsening problems.

*Appeared in*

- Phys. Rev. B., 76 (2007) pp. 075415 (11 pages).

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# Glauber dynamics on hyperbolic graphs: Boundary conditions and mixing time

*Authors*

- Bianchi, Alessandra

ORCID: 0000-0003-1566-6000

*2010 Mathematics Subject Classification*

- 82B20 82B43 82C80 60K35

*Keywords*

- Ising model, Glauber dynamics, hyperbolic graphs, Dirichlet form, spectral gap, mixing time

*DOI*

*Abstract*

We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an $n$-vertex ball of the hyperbolic graph $H(v,s)$, where $v$ is the number of neighbors of each vertex and $s$ is the number of sides of each face, conditioned on having $(+)$-boundary. If $v>4$, $s>3$ and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of $n$. This implies that the mixing time grows at most linearly in $n$, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature $b$. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time.

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# Mathematical results on existence for viscoelastodynamic problems with unilateral constraints

*Authors*

- Petrov, Adrien
- Schatzman, M.

*2010 Mathematics Subject Classification*

- 35L85 49J40 73D99 73V25

*Keywords*

- Viscoelasticity, Signorini conditions, penalty method, traces, variational inequality, convolution

*DOI*

*Abstract*

We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.

*Appeared in*

- SIAM J. Math. Anal., 40 (2009) pp. 1882--1904.

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# The equilibria of vapour-liquid systems revisited

*Authors*

- Dreyer, Wolfgang
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 82B26 49Q20

*2008 Physics and Astronomy Classification Scheme*

- 64.70.Fx

*Keywords*

- two-phase fluid, mechanical and phase equilibria, surface tension, mean curvature, contact angle

*DOI*

*Abstract*

We study equilibrium conditions of liquid-vapour phase transitions for a single substance at constant temperature. The phase transitions are modelled by a classical sharp interface model with boundary contact energy. We revisit this old problem mainly for the following reasons. Equilibria in a two-phase system can be established either under fixed external pressure or under fixed total volume. These two different settings lead to distinct equilibria, a fact that is usually ignored in the literature. In nature and in most technical processes, the approach of a two-phase system to equilibrium runs at constant pressure, whereas mathematicians prefer to study processes in constant domains, i.e. at constant volume. Furthermore, in the literature the sharp interface of the liquid and the vapour phase is usually described by a surface with high symmetry like a plane interface or a radially symmetric interface which has the shape of the boundary of a ball. In this paper we establish equilibrium conditions for pressure control as well as for volume control with arbitrary shapes of the interface. The results are derived by methods of differential geometry. Further, the common features and differences of pressure and volume control are worked out for some simple cases.

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# Existence and uniqueness results for reaction-diffusion processes of electrically charged species

*Authors*

- Gajewski, Herbert
- Skrypnik, Igor V.

*2010 Mathematics Subject Classification*

- 35B45 35K15 35K20 35K65

*Keywords*

- Nonlinear elliptic-parabolic systems, nonlocal drift, global bounded solutions, uniqueness, nonstandard assumptions, degenerate typ

*DOI*

*Abstract*

We study initial-- boundary value problems for elliptic--parabolic systems of nonlinear partial differential equations describing drift--diffusion processes of electrically charged species in N--dimensional bounded Lipschitzian domains. We include Fermi--Dirac statistics and admit nonsmooth material coefficients. We prove existence and uniqueness of bounded global solutions.

*Appeared in*

- Nonlinear elliptic and parabolic problems, 151--188, Prog. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, 2005

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# Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion

*Authors*

- Yulin, Alexey
- Skryabin, Dmitry
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37K45 37K40

*2008 Physics and Astronomy Classification Scheme*

- 2.65.-k,42.65.Tg,42.65.Sf,42.81.Qb,42.81.Dp

*Keywords*

- Discrete solitons, modulational instability, waveguide arrays

*DOI*

*Abstract*

We study temporal modulational instability of spatial discrete solitons in waveguide arrays with group velocity dispersion (GVD). For normal GVD we report existence of the strong 'neck'-type instability specific for the discrete solitons. For anomalous GVD the instability leads to formation of the mixed discrete-continuous spatio-temporal quasi-solitons. Feasibility of experimental observation of these effects in the arrays of silicon-on-insulator waveguides is discussed.

*Appeared in*

- Optics Express, 14 (2006) pp. 12347--12352.

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# Instabilities of stationary states in lasers with long-delay optical feedback

*Authors*

- Yanchuk, Serhiy
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34K60 34K20 34K26

*Keywords*

- Lang-Kobayashi system, laser with feedback, external cavity mode, singularly perturbed delay-differential equation, stability

*DOI*

*Abstract*

We study the Lang-Kobayashi model in the long-delay limit, focussing our attention on the stability properties of external cavity modes (ECMs) of this system. We show that ECMs can display different types of instabilities: strong instabilities and weak modulational-type instability. We explain the origin of these instabilities and show how they affect the complicated dynamics of the Lang-Kobayashi model.

*Appeared in*

- SIAM J. Appl. Dyn. Syst. Vol. 9, No. 2 (2010), 519-535.

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# On the inviscid limit of a model for crack propagation

*Authors*

- Knees, Dorothee
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Zanini, Chiara

*2010 Mathematics Subject Classification*

- 49J40 74R10 49L25 35K90 74B20 74G65

*Keywords*

- rate-indepentent problems, energetic formulation, energy release rate, Griffith fracture criterion, vanishing viscosity method

*DOI*

*Abstract*

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

*Appeared in*

- Math. Models Methods Appl. Sci., 18 (2008) pp. 1529--1569.

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# Long-term behavior for superprocesses over a stochastic flow

*Authors*

- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60G57 60H15 60J80

*Keywords*

- Superprocess, stochastic flow, log-Laplace equation, long-term behavior

*DOI*

*Abstract*

We study the limit of a superprocess controlled by a stochastic flow as $ttoinfty$. It is proved that when $dle 2$, this process suffers long-time local extinction, when $dge 3$, it has a limit which is persistent. The stochastic log-Laplace equation conjectured by Skoulakis and Adler [7] and studied by this author [12] plays a key role in the proofs like the one played by the log-Laplace equation in deriving long-term behavior for usual super-Brownian motion.

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# A reduction approximation method for curved rods

*Authors*

- Arnautu, Viorel
- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 65L20 74K10

*Keywords*

- Finite element approximation, locking problem, uniform convergence

*DOI*

*Abstract*

We study the numerical approximation of a general linear model for three-dimensional clamped curved rods. We introduce a modified system and we show that the convergence of the numerical discretization is independent of the small parameters entering the coefficients of the differential equations.

*Appeared in*

- Numer. Funct. Anal. Optim., Volume 26 (2005), pp. 139-155

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# Optimal investment strategy under saving/borrowing rates spread with partial information

*Authors*

- Xiong, Jie
- Yang, Zhaojun

*2010 Mathematics Subject Classification*

- 90A09

*Keywords*

- Investment, stochastic optimal control, nonlinear filtering, optimal strategy, utility function, explicit solution

*DOI*

*Abstract*

We study the optimal investment strategy for maximizing the expected utility of the terminal wealth with partial information. Under the assumption that the borrowing rate is higher than the saving rate and the utility function is $U(x)=log x$, we develop a new method to solve such problem and derive the explicit solutions that are easy to implement.

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# Some limit theorems for a particle system of single point catalytic branching random walks

*Authors*

- Vatutin, Vladimir
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60J80 60K25

*Keywords*

- Renewal equation, branching particle system, scaling limit

*DOI*

*Abstract*

We study the scaling limit for a catalytic branching particle system whose particles performs random walks on $ZZ$ and can branch at 0 only. Varying the initial (finite) number of particles we get for this system different limiting distributions. To be more specific, suppose that initially there are $n^be$ particles and consider the scaled process $Z^n_t(bullet)=Z_nt(sqrtn, bullet)$ where $Z_t$ is the measure-valued process representing the original particle system. We prove that $Z^n_t$ converges to 0 when $be

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# Q-switching instability in a mode-locked semiconductor laser

*Authors*

- Rachinskii, Dmitri
- Vladimirov, Andrei

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.55.Px 42.60.Mi 42.65.Pc 42.60.Gd

*Keywords*

- Semiconductor laser, mode-locking, Q-switching, delay differential equations, Neimark-Sacker bifurcation

*DOI*

*Abstract*

We suggest analytic estimates for the Q-switching instability boundary of the continuous-wave mode-locking regime domain for a ring cavity semiconductor laser. We use a differential delay laser model that allows to assume large gain and loss in the cavity, which is a typical situation for this laser class. The slow saturable absorber approximation is applied to derive a map that describes the transformation of the pulse parameters after a round trip in the cavity. The Q-switching instability boundary is then obtained as a Neimark-Sacker bifurcation curve of this map. We study the dependence of this boundary on laser parameters and compare it with the boundaries obtained by the New stability criterion and by direct numerical analysis of the original differential model. Further modification of our approach, based on the hyperbolic secant ansatz for the pulse shape, is used to estimate the width and repetition rate of the mode locking pulses.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 23 (2006) pp. 663-670.

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# Asymptotic analysis of elastic curved rods

*Authors*

- Vodák, Rostislav

*2010 Mathematics Subject Classification*

- 74K10 35J25 74B99

*Keywords*

- curved rods, low geometrical regularity, 1-D asymptotic model, asymptotic analysis

*DOI*

*Abstract*

We suppose a convergent sequence of curved rods made from an isotropic elastic material and clamped on the lower basis or on both bases, and the linearized elasticity system posed on the sequence of the curved rods. We study the asymptotic behaviour of the stress tensor and the solution to this system, when the radius of the domains tends to zero. The curved rods with a nonsmooth line of centroids are covered by the used asymptotic method as well.

*Appeared in*

- Math. Methods Appl. Sci., 30 (2007), pp. 43-75

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# Hopf bifurcations and simple structures of periodic solution sets in systems with the Preisach nonlinearity

*Authors*

- Brokate, Martin

ORCID: 0000-0003-4660-9180 - Rachinskii, Dmitri

*2010 Mathematics Subject Classification*

- 34C55 34D20 34D10

*Keywords*

- Hysteresis, forced periodic oscillations, cycles, one-parameter continuum of periodic regimes, Hopf bifurcation, Preisach nonlinearity

*DOI*

*Abstract*

We survey a number of recent results and suggest some new ones on periodic solutions of systems with hysteresis. The main focus of this work is the situation when simple one-parameter structures of periodic regimes appear. We consider forced oscillations, cycles of autonomous systems and Hopf bifurcations from the equilibrium and from infinity.

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# Non-Raman redshift by pulse splitting in the normal dispersion regime

*Authors*

- Demircan, Ayhan
- Kroh, Marcel
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*Keywords*

- Nonlinear Schrödinger equation, optical Fiber.

*DOI*

*Abstract*

While usually the generation of a Stokes component is attributed to Raman scattering, we present here experimentally and numerically a more fundamental mechanism which can be explained by the nonlinear Schrödinger equation alone. It can be employed to excite new frequency components on the red side, by using pulse splitting in the normal dispersion regime.

*Appeared in*

- Proceedings of the 7th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD '07, 24--27 September 2007, J. Piprek, D. Prather, eds., IEEE, Piscataway, NJ, 2007, pp. 99--100

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