# Hölder index for density states of (alpha,1,beta)-superprocesses at a given point

*Authors*

- Fleischmann, Klaus
- Mytnik, Leonid
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60G57

*Keywords*

- Hölder continuity at a given point, optimal exponent, multifractal, structure, Hausdorff dimension

*DOI*

*Abstract*

A Hölder regularity index at given points for density states of (alpha,1,beta)-superprocesses with alpha>1+beta is determined. It is shown that this index is strictly greater than the optimal index of local Holder continuity for those density states.

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# Stochastic simulation of flows and particle transport in porous tubes

*Authors*

- Sabelfeld, Karl
- Kurbanmuradov, Orazgeldi
- Levykin, Alexander

*2010 Mathematics Subject Classification*

- 65C05 76S05

*2008 Physics and Astronomy Classification Scheme*

- 02.70q 05.10.Ln

*Keywords*

- Darcy equation, random hydraulic conductivity, Lagrangian trajectory, Randomized spectral models, lognormal random fields

*DOI*

*Abstract*

A Monte Carlo method is developed for stochastic simulation of flows and particle transport in tubes filled with a porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modelled in a tube with prescribed boundary conditions. Numerical experiments are carried out by solving the random Darcy equation for each sample of the hydraulic conductivity by a SOR iteration method, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, diffusion coefficients, the mean and variance of Lagrangian trajectories, and discuss a ''stagnation" effect which was found in our simulations.

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# A stochastic fractal model of the universe related to the fractional Laplacian

*Authors*

- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 76S05

*2008 Physics and Astronomy Classification Scheme*

- 98.80.Jk, 95.75.Pq, 98.65.Dx

*Keywords*

- Stochastic fractals, fractal dimension of the Universe, fractional Laplace equation, boundary excitations, spectral and correlation functions

*DOI*

*Abstract*

A new stochastic fractal model based on a fractional Laplace equation is developed. Exact representation for the spectral and correlation functions under random boundary excitation are obtained. Randomized spectral expansion is constructed for simulation of the solution of the fractional Laplace equation. We present calculations for 2D and 3D spaces for a series of fractional parameters showing a strong memory effect: the decay of correlations is several order of magnitudes less compared to the conventional Laplace equation model.

*Appeared in*

- J. Cosmol. Astropart. Phys., 004 (2008) pp. 1475-7516.

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# Quasi-static contact problem with finitely many degrees of freedom and dry friction

*Authors*

- Schmid, Florian

*2010 Mathematics Subject Classification*

- 74M10 74M15 74H20 74B20

*Keywords*

- Coulomb friction, varying friction coefficient, unilateral contact, quasi-static, energetic formulation, nonlinear elasticity, existence, finite-dimensional, time-incremental approximation

*DOI*

*Abstract*

A quasi-static contact problem is considered for a non-linear elastic system with finitely many degrees of freedom. Coulomb's law is used to model friction and the friction coefficient may be anisotropic and may vary along the surface of the rigid obstacle. Existence is established following a time-incremental minimization problem. Friction is artificially decreased to resolve the discontinuity arising from making and losing contact.

*Appeared in*

- ZAMM, Volume 89, Issue 5, 2009, pp. 383-398

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# On the clustering property of the random intersection graphs

*Authors*

- Yao, Xin
- Chen, Jinwen
- Zhang, Changshui
- Li, Yanda

*2010 Mathematics Subject Classification*

- 05C80 05C75 05C40

*Keywords*

- random intersection graphs, clustering coefficient, phase transition

*DOI*

*Abstract*

A random intersection graph mtlmcalG_V,W,p is induced from a random bipartite graph mtlmcalG^*_V,W,p with vertices classes mtlV, mtlW and the edges incident between mtlv in V and mtlw in W with probability mtlp. Two vertices in mtlV are considered to be connected with each other if both of them connect with some common vertices in mtlW. The clustering properties of the random intersection graph are investigated completely in this article. Suppose that the vertices number be mtlN = mabsV and mtlM=mabsW and mtlM = N^alpha, p=N^-beta, where mtlalpha > 0,, beta > 0, we derive the exact expressions of the clustering coefficient mtlC_v of vertex mtlv in mtlmcalG_V,W,p. The results show that if mtlalpha < 2beta and mtlalpha neq beta, mtlC_v decreases with the increasing of the graph size; if mtlalpha = beta or mtlalpha geq 2beta, the graph has the constant clustering coefficients, in addition, if mtlalpha > 2beta, the graph connecChangshui Zhangts almost completely. Therefore, we illustrate the phase transition for the clustering property in the random intersection graphs and give the condition that mtlriG being high clustering graph.

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# Elastostatics of a half-plane under random boundary excitations

*Authors*

- Shalimova, Irina
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*Keywords*

- Elastostatics of a half-plane, Random boundary excitations, Karhunen-Loeve expansion, partially homogeneous spectral tensor

*DOI*

*Abstract*

A stochastic analysis of an elastostatics problem for a half-plane under random white noise excitations of the displacement vector prescribed on the boundary is given. Solutions of the problem are inhomogeneous random fields homogeneous in the longitudinal direction. This is used to model the displacements and represent their correlation tensor via spectral expansion. This approach makes it possible to derive exact representations for other functionals of interest, in particular, the vorticity, the strain tensor, and the elastic energy.

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# Stochastic flow simulation and particle transport in a 2D layer of random porous medium

*Authors*

- Kurbanmuradov, Orazgeldi
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 76S05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb, 02.70.Lq

*Keywords*

- Darcy equation, random hydraulic conductivity, Randomized spectral models, diffusion regime, long-range correlations

*DOI*

*Abstract*

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.

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# Stochastic analysis of an elastic 3D half-space respond to random boundary displacements: Exact results and Karhunen--Loéve expansion

*Authors*

- Shalimova, Irina
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 76S05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 02.70.Lq

*Keywords*

- Boundary white noise, Karhunen-Loève expansion, Poisson integral formula, boundary random excitations, 3D Lamé equation

*DOI*

*Abstract*

A stochastic response of an elastic 3D half-space to random displacement excitations on the boundary plane is studied. We derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary. Solutions to this elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction. This enables us to construct explicitly the Karhunen-Loève (K-L) series expansion by solving the eigen-value problem for the correlation operator. Simulation results are presented and compared with the exact representations derived for the displacement correlation tensor. This paper is a complete 3D generalization of the 2D case study we presented in J. Stat. Physics, v.132 (2008), N6, 1071-1095.

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# Stokes flows under random boundary velocity excitations

*Authors*

- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*Keywords*

- Stokes flow, Random boundary excitations, Karhunen-Loève expansion, velocity correlation tensor, stress, vorticity, and pressure

*DOI*

*Abstract*

A viscous Stokes flow over a disc under random fluctuations of the velocity on the boundary is studied. We give exact Karhunen-Loève (K-L) expansions for the velocity components, pressure, stress, and vorticity, and the series representations for the corresponding correlation tensors. Both the white noise fluctuations, and general homogeneous random excitations of the velocities prescribed on the boundary are studied. We analyze the decay of correlation functions in angular and radial directions, both for exterior and interior Stokes problems. Numerical experiments show the fast convergence of the K-L expansions. The results indicate that ignoring the boundary condition uncertainty dramatically underestimates the variance of the velocity and pressure in the interior/exterior of the domain.

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# Constrained Delaunay tetrahedral mesh generation and refinement

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 52B55 65D18

*Keywords*

- constrained Delaunay tetrahedralization, mesh generation, boundary recovery, mesh refinement

*DOI*

*Abstract*

A it constrained Delaunay tetrahedralization of a domain in $mathbbR^3$ is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis. This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is for the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions.

*Appeared in*

- Finite Elem. Anal. Des., 46 pp. 33--46.

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# A Kohn--Sham system at zero temperature

*Authors*

- Cornean, Horia
- Hoke, Kurt
- Neidhardt, Hagen
- Racec, Paul N.
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 34L40 34L30 47H05 81V70

*Keywords*

- Kohn-Sham systems, Schrödinger-Poisson systems, non-linear operators, density operator, zero temperature, Fermi-Dirac distribution

*DOI*

*Abstract*

An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.

*Appeared in*

- J. Phys. A, 41 (2008) pp. 385304/1--385304/21.

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# Finite element error analysis for state-constrained optimal control of the Stokes equations

*Authors*

- de los Reyes, Juan Carlos
- Meyer, Christian
- Vexler, Boris

*2010 Mathematics Subject Classification*

- 49K20 49M25 65N30

*Keywords*

- Linear-quadratic optimal control problems, Stokes equations, state constraints, numerical approximation, finite elements

*DOI*

*Abstract*

An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerened with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the $L^infty$-error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one of the interpolation error. The assumptions of the $L^infty$-finite-element-error can be verified for different numerical settings. The theoretical results are confirmed by numerical examples.

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# Optimal control of static plasticity with linear kinematic hardening

*Authors*

- Griesse, Roland
- Meyer, Christian

*2010 Mathematics Subject Classification*

- 49J40 49M30 35J85 35Q72

*Keywords*

- Optimal control of variational inequalities, static plasticity, Yosida approximation

*DOI*

*Abstract*

An optimal control problem for the static problem of infinitesimal elastoplasticity with linear kinematic hardening is considered. The variational inequality arising on the lower-level is regularized using a Yosida-type approach, and an optimal control problem for the so-called viscoplastic model is obtained. Existence of a global optimizer is proved for both the regularized and original problems, and strong convergence of the solutions is established.

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# Exact artificial boundary conditions for problems with periodic structures

*Authors*

- Ehrhardt, Matthias
- Zheng, Chunxiong

*2010 Mathematics Subject Classification*

- 65M99 81-08

*Keywords*

- artificial boundary conditions, periodic potential, Schrödinger equation, hyperbolic equation, unbounded domain

*DOI*

*Abstract*

Based on the work of Zheng on the artificial boundary condition for the Schrödinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second order ODE problems with periodic coefficients and its validity is shown to be strongly supported by numerical evidences. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schrödinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered.

*Appeared in*

- J. Comput. Phys., 227 (2008) pp. 6877--6894.

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# Sharp-optimal adjustment for multiple testing in the multivariate two-sample problem

*Authors*

- Rohde, Angelika

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Combinatorial process, exponential concentration bound, coupling, decoupling inequality, exact multiple test, nearest-neighbors, sharp asymptotic adaptivity

*DOI*

*Abstract*

Based on two independent samples $X_1, ...,X_m$ and $X_m+1, ...,X_n$ drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearestneighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The adjustment for multiple testing is sharp-optimal for typical arrangements of the observation values which appear with probability close to one, and it relies on a new coupling Bernstein type exponential inequality, reflecting the nonsubgaussian tail behavior of the combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis ''$p = q$'' to a simple one with the multivariate mixed density $(m/n)p + (1 ? m/n)q$ as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes.

*Appeared in*

- Probab. Theory Relat. Fields (2010) under new title: Optimal calibration for multiple testing against local inhomogeneity in higher dimension

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# Bifurcations in a model of monolithic passively mode-locked semiconductor laser

*Authors*

- Vladimirov, Andrei G.
- Pimenov, Alexander
- Rachinskii, Dmitrii

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

- semiconductor laser, bifurcations, mode-locking

*DOI*

*Abstract*

Bifurcation mechanisms of the development and break up of different operation regimes in a passively mode-locked monolithic semiconductor laser are studied by solving numerically partial differential equations for amplitudes of two counterpropagating waves and carrier densities in gain and absorber sections. It is shown that harmonic mode-locking regime with two pulses in the cavity can exhibit a period-doubling bifurcation leading to different amplitudes and separations of the pulses. The effect of linewidth enhancement factors in gain and absorber sections on the laser dynamics is discussed.

*Appeared in*

- IEEE J. Quantum Electron., 45 (2009) pp. 462--468.

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# Method for computing the nonlinear refractive index via Keldysh theory

*Authors*

- Brée, Carsten
- Demircan, Ayhan
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A10 78A60

*Keywords*

- Nonlinear Optics, Laser-matter interaction

*DOI*

*Abstract*

By making use of the multiphoton limit of Keldysh theory, we show that for the case of two-photon absorption a Kramers-Kronig expansion can be used to calculate the nonlinear refractive index for different wavelenghts. We apply this method to various inert gases and compare the obtained numerical values to different experimental and theoretical results for the dispersion of the Kerr nonlinearity.

*Appeared in*

- IEEE J. Quantum Electron., 46 (2010) pp. 433--437.

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# Self-pinching of pulsed laser beams in a plasma filament

*Authors*

- Brée, Carsten
- Demircan, Ayhan
- Skupin, Stefan
- Bergé, Luc
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60 81V80 35Q55 37K40

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 42.65.-k 52.38.Hb 42.68.Ay

*Keywords*

- Nonlinear Schrödinger Equations, Optical self-focusing, Ultrashort pulse propagation

*DOI*

*Abstract*

Competing nonlinear optical effects that act on femtosecond laser pulses propagating in a self-generated plasma filament may give rise to a pronounced radial deformation of the beam, similar to the z-pinch contraction of pulsed high-current discharges. This self-pinching locally increases the photon density. The process is further identified as the first stage in the recently observed self-compression of femtosecond laser pulses propagating in filaments. Self-pinching also explains the complicated spatio-temporal shapes generally observed in filament compression experiments.

*Appeared in*

- Optics Express, 17 (2009) pp. 16429-16435.

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# Structural adaptive smoothing in diffusion tensor imaging: The R package dti

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 62P10 92C55 62G05

*Keywords*

- Structural adaptive smoothing, diffusion weighted imaging, diffusion tensor model, Rician bias, software

*DOI*

*Abstract*

Diffusion Weighted Imaging has become and will certainly continue to be an important tool in medical research and diagnostics. Data obtained with Diffusion Weighted Imaging are characterized by a high noise level. Thus, estimation of quantities like anisotropy indices or the main diffusion direction may be significantly compromised by noise in clinical or neuroscience applications. Here, we present a new package dti for R, which provides functions for the analysis of diffusion weighted data within the diffusion tensor model. This includes smoothing by a recently proposed structural adaptive smoothing procedure based on the Propagation-Separation approach in the context of the widely used Diffusion Tensor Model. We extend the procedure and show, how a correction for Rician bias can be incorporated. We use a heteroscedastic nonlinear regression model to estimate the diffusion tensor. The smoothing procedure naturally adapts to different structures of different size and thus avoids oversmoothing edges and fine structures. We illustrate the usage and capabilities of the package through some examples.

*Appeared in*

- J. Statist. Software, 31 (2009) pp. 1--24.

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# Fixed domain transformations and split-step finite difference schemes for nonlinear Black--Scholes equations for American options

*Authors*

- Ankudinova, Julia
- Ehrhardt, Matthias

*2010 Mathematics Subject Classification*

- 91B26 35A35 65N99

*Keywords*

- nonlinear Black-Scholes models, fixed domain transformation, split-step methods, American options

*DOI*

*Abstract*

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection equations, where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself.

In this chapter we will be concerned with several models from the most relevant class of nonlinear Black-Scholes equations for American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives.

We will analytically approach the option price by following the ideas proposed by Ševčovič and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. Finally, we will present the results of a split-step finite difference schemes for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.

*Appeared in*

- Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science, 2008, ISBN: 978-1-60456-931-5.

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# Optimal Hölder index for density states of superprocesses with (1 + $beta$)-branching mechanism

*Authors*

- Fleischmann, Klaus
- Mytnik, Leonid
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60G75

*Keywords*

- Dichotomy for density of superprocess states, Hölder continuity, optimal exponent, critical index, local unboundedness, multifractal spectrum, Hausdorff dimension

*DOI*

*Abstract*

For 0 < alpha leq 2, a super-alpha-stable motion X in R^d with branching of index 1 + beta in (1,2) is considered. If d < alpha / beta, a dichotomy for the density of states X_t at fixed times t > 0 holds: the density function is locally Hölder continuous if d = 1 and alpha > 1 + beta, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal Hölder index.

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# Analyticity for some operator functions from statistical quantum mechanics

*Authors*

- Hoke, Kurt
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 81Q10 35J10 35P20

*2008 Physics and Astronomy Classification Scheme*

- 31.15.bt

*Keywords*

- Schrödinger operator, analyticity of operator functions, statistical ensemble of quantum systems, quantum mechanical particle density in many particle systems

*DOI*

*Abstract*

For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.

*Appeared in*

- Ann. Henri Poincare, 10 (2009) pp. 749--771.

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# Slow decorrelations in KPZ growth

*Authors*

- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35

*Keywords*

- KPZ class, correlations, Airy processes

*DOI*

*Abstract*

For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in $1+1$ dimensions, fluctuations grow as $t^1/3$ during time $t$ and the correlation length at a fixed time scales as $t^2/3$. In this note we discuss the scale of time correlations. For a representant of the KPZ class, the polynuclear growth model, we show that the space-time is non-trivially fibred, having slow directions with decorrelation exponent equal to $1$ instead of the usual $2/3$. These directions are the characteristic curves of the PDE associated to the surface's slope. As a consequence, previously proven results for space-like paths will hold in the whole space-time except along the slow curves.

*Appeared in*

- J. Stat. Mech. Theory Exp., (2008) pp. P07022/1-P07022/18.

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# Parallel simulation of high power semiconductor lasers

*Authors*

- Lichtner, Mark
- Spreemann, Martin

*2010 Mathematics Subject Classification*

- 65Y05 37M05 37L65 35L45 35Q60 65M70 65M06

*Keywords*

- parallel computation, numerical simulation of optoelectronic devices, laser dynamics, initial boundary value problem of hyperbolic type, comparison with experimental data

*DOI*

*Abstract*

High power tapered semiconductor lasers are characterized by a huge amount of structural and geometrical design parameters and are subject to time-space instabilities like pulsations, self-focussing, filamentation and thermal lensing which yield restrictions to output power, beam quality and wavelength stability. Numerical simulations are an important tool for finding optimal design parameters, understanding the complicated dynamical behavior and for predicting new laser designs. We present fast dynamic high performance parallel simulation results based on traveling wave equations which are suitable for model calibration and parameter scanning of the long time dynamics in reasonable time. Simulation results are compared to experimental data.

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# Mathematical modeling of channel-porous layer interfaces in PEM fuel cells

*Authors*

- Ehrhardt, Matthias
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Linke, Alexander

ORCID: 0000-0002-0165-2698 - Holzbecher, Ekkehard

*2010 Mathematics Subject Classification*

- 76S05 35Q35 76D05

*2008 Physics and Astronomy Classification Scheme*

- 47.56.+r

*Keywords*

- fluid-porous interface, porous media, PEM fuel cell, incompressible flow, Stokes equation, Darcy equation, Brinkman extension, Beavers-Joseph-Saffman interface conditions

*DOI*

*Abstract*

In *proton exchange membrane (PEM) fuel cells*, the transport of the fuel to the active zones, and the removal of the reaction products are realized using a combination of channels and porous diffusion layers. In order to improve existing mathematical and numerical models of PEM fuel cells, a deeper understanding of the coupling of the flow processes in the channels and diffusion layers is necessary.

After discussing different mathematical models for PEM fuel cells, the work will focus on the description of the coupling of the free flow in the channel region with the filtration velocity in the porous diffusion layer as well as interface conditions between them.

The difficulty in finding effective coupling conditions at the interface between the channel flow and the membrane lies in the fact that often the orders of the corresponding differential operators are different, e.g., when using stationary (Navier-)Stokes and Darcy's equation. Alternatively, using the Brinkman model for the porous media this difficulty does not occur.

We will review different interface conditions, including the well-known Beavers-Joseph-Saffman boundary condition and its recent improvement by Le Bars and Worster.

*Appeared in*

- M. Ehrhardt, J. Fuhrmann, A. Linke, E. Holzbecher, Proceedings of FDFC2008 --- Fundamentals and Developments of Fuel Cell Conference 2008, Nancy, France, December 10--12 (CD), 2008, pp. 8 pages

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# Holomorphic transforms with application to affine processes

*Authors*

- Belomestny, Denis
- Kampen, Jörg
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60J25 91B28

*Keywords*

- Itô-Lévy processes, holomorphic transforms, affine processes

*DOI*

*Abstract*

In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

*Appeared in*

- Journal of Functional Analysis, Vol. 257, 4, (2009) pp. 1222-1250

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# Global regularity and probabilistic schemes for free boundary surfaces of multivariate American derivatives and their Greeks

*Authors*

- Kampen, Jörg

*2010 Mathematics Subject Classification*

- 35R35 60G46

*Keywords*

- multivariate American derivatives, regularity, free boundary surfaces, probabilistic schemes, front fixing

*DOI*

*Abstract*

In a rather general setting of multivariate stochastic volatility market models we derive global iterative probabilistic schemes for computing the free boundary and its Greeks for a generic class of American derivative models using front-fixing methods. Establishment of convergence is closely linked to a proof of global regularity of the free boundary surface.

*Appeared in*

- SIAM J. Appl. Math., 71 (2011) pp. 288--308.

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# Confidence sets for the optimal approximating model --- Bridging a gap between adaptive point estimation and confidence regions

*Authors*

- Rohde, Angelika
- Dümbgen, Lutz

*2010 Mathematics Subject Classification*

- 62G15 62G20

*Keywords*

- Adaptivity, confidence sets, coupling, exponential inequality, model selection, multiscale inference, risk optimality

*DOI*

*Abstract*

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. Our construction is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral $chi^2$--distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.

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# Interface conditions for limits of the Navier--Stokes--Korteweg model

*Authors*

- Hermsdörfer, Katharina
- Kraus, Christiane
- Kröner, Dietmar

*2010 Mathematics Subject Classification*

- 35Q30, 35R35, 76T10, 82B26

*Keywords*

- Phase boundary, phase transition, Navier-Stokes-Korteweg model, compressible flow, interface condition

*DOI*

*Abstract*

In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

*Appeared in*

- Interfaces Free Bound., 13 (2011) pp. 239--254.

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# Locally time homogeneous time series modelling

*Authors*

- Elagin, Mstislav
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62M10

*Keywords*

- Adaptive estimation, local homogeneity, model selection, stagewise aggregation, volatility model, Poisson model, exponential model, Bernoulli model, propagation, oracle

*DOI*

*Abstract*

In this paper three locally adaptive estimation methods are applied to the problems of variance forecasting, value-at-risk analysis and volatility estimation within the context of nonstationary financial time series. A general procedure for the computation of critical values is given. Numerical results exhibit a very reasonable performance of the methods.

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# Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

*Authors*

- Bovier, Anton
- den Hollander, Frank
- Spitoni, Cristian

*2010 Mathematics Subject Classification*

- 60K35 82C26

*Keywords*

- Glauber dynamics, Kawasaki dynamics, critical droplet, metastable transition time, last-exit biased distribution, Dirichlet principle, Berman-Konsowa principle, capacity, flow, cluster expansion

*DOI*

*Abstract*

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $b$ denote the inverse temperature and let $L_b subset Z^2$ be a square box with periodic boundary conditions such that $lim_btoinfty L_b =infty$. We run the dynamics on $L_b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles) are small. For large $b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single emphcritical droplet somewhere in $L_b$. Using potential-theoretic methods, we compute the emphaverage nucleation time (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $btoinfty$. It turns out that this time grows as $Ke^Gammab/ L_b $ for Glauber dynamics and $Kb e^Gammab/ L_b $ for Kawasaki dynamics, where $Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $ L_b $). The fact that the average nucleation time is inversely proportional to $ L_b $ is referred to as emphhomogeneous nucleation, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $L_b$.

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# Sharp asymptotics for metastability in the random field Curie--Weiss model

*Authors*

- Bianchi, Alessandra

ORCID: 0000-0003-1566-6000 - Bovier, Anton
- Ioffe, Dmitry

*2010 Mathematics Subject Classification*

- 82C44 60K35 60G70

*Keywords*

- Disordered system, random field Curie-Weiss model, Glauber dynamics, metastability, potential theory, Dirichlet form, capacity

*DOI*

*Abstract*

In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.

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# A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations

*Authors*

- Antoine, Xavier
- Arnold, Anton
- Besse, Christophe
- Ehrhardt, Matthias
- Schädle, Achim

*2010 Mathematics Subject Classification*

- 65M12 35Q40 45K05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Bf 31.15.Fx

*Keywords*

- Schrödinger equation, transparent boundary conditions, discrete convolution, unbounded domain, finite difference schemes, finite elements

*DOI*

*Abstract*

In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

*Appeared in*

- Commun. Comput. Phys., 4 (2008) pp. 729--796.

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# Fast, stable and accurate numerical method for the Black--Scholes equation of American options

*Authors*

- Ehrhardt, Matthias
- Mickens, Ronald

*2010 Mathematics Subject Classification*

- 35A35 65N99 91B26

*Keywords*

- Black-Scholes equation, computational finance, option pricing, finite difference method, artificial boundary condition, free boundary problem, American option

*DOI*

*Abstract*

In this work we improve the algorithm of Han and Wu (SIAM J. Numer. Anal. 41 (2003), 2081-2095) for American Options with respect to stability, accuracy and order of computational effort.

We derive an exact discrete artificial boundary condition (ABC) for the Crank-Nicolson scheme for solving the Black-Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.

Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.

Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs o f Mayfield and Han and Wu.

*Appeared in*

- Int. J. Theor. Appl. Finance, 11 (2008) pp. 471--501.

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# Numerical simulation of waves in periodic structures

*Authors*

- Ehrhardt, Matthias
- Han, Houde
- Zheng, Chunxiong

*2010 Mathematics Subject Classification*

- 35B27 65M99 35Q60 35J05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Bf 31.15.-p 42.82.Et 85.35.-p 85.35.Be

*Keywords*

- periodic media, Helmholtz equation, Schrödinger equation, Dirichlet-to-Neumann maps, Robin-to-Robin maps, band structure, Floquet-Bloch theory, high-order finite elements

*DOI*

*Abstract*

In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu.

In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.

*Appeared in*

- Commun. Comput. Phys., 5 (2009) pp. 849--872.

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# A threestepped coordinated level set segmentation method for identifying atherosclerotic plaques on MR-images

*Authors*

- Gloger, Oliver
- Ehrhardt, Matthias
- Dietrich, Thore
- Hellwich, Olaf
- Graf, Kristof
- Nagel, Eike

*2010 Mathematics Subject Classification*

- 92B05 65M06

*Keywords*

- level set segmentation, active contours, medical image segmentation, anisotropic diffusion, atherosclerotic plaques, canny edges

*DOI*

*Abstract*

In this work we propose an adapted level set segmentation technique for the recognition of atherosclerotic plaque tissue on magnetic resonance images. The images are 2dimensional crosssectional images and show different profiles from ex-vivo human vessels with high variability in vessel shape. We used a curvature based anisotropic diffusion technique to denoise the magnetic resonance images.

The segmentation technique is subdivided into three level set steps. Hence, the result of every phase serves as constructive knowledge for the next level set step. By analyzing and combining carefully all available channel information during the first and second step we are capable to delineate exactly the vessel walls by using and adapting two well-known level set segmentation techniques.

The third step controls an enclosing level set which separates the plaque patterns from healthy media tissue. In this step we introduce a local weighting concept to consider intensity information for conspicuous plaque patterns. Furthermore, we propose the introduction of a maximal shrinking distance for the third level set in the vessel wall and compare the results of the local weighting algorithm with and without the concept of the maximal shrinking distance.

The incorporation of locally weighted intensity information into the level set method allows the algorithm to automatically distinguish plaque from healthy media tissue. The knowledge of the maximal shrinking distance can improve the segmentation results and enables to delineate tissue areas where plaque is most likely.

*Appeared in*

- Comm. Numer. Methods Engrg., 25 (2009) pp. 615--638.

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# High resolution fMRI: Overcoming the signal-to-noise problem

*Authors*

- Tabelow, Karsten

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

ORCID: 0000-0001-7471-2658 - Voss, Henning U.

*2010 Mathematics Subject Classification*

- 62P10 92C55

*Keywords*

- functional MR, structure adaptive smoothing, high-resolution fMRI

*DOI*

*Abstract*

Increasing the spatial resolution in functional Magnetic Resonance Imaging (fMRI) inherently lowers the signal-to-noise ratio (SNR). In order to still detect functionally significant activations in high-resolution images, spatial smoothing of the data is required. However, conventional non-adaptive smoothing comes with a reduced effective resolution, foiling the benefit of the higher acquisition resolution. We show how our recently proposed structural adaptive smoothing procedure for functional MRI data can improve signal detection of high-resolution fMRI experiments regardless of the lower SNR. The procedure is evaluated on human visual and sensory-motor mapping experiments. In these applications, the higher resolution could be fully utilized and high-resolution experiments were outperforming normal resolution experiments by means of both statistical significance and information content.

*Appeared in*

- J. Neurosci. Meth., 178 (2009) pp. 357--365.

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# Regularity and uniqueness in quasilinear parabolic systems

*Authors*

- Krejčí, Pavel
- Panizzi, Lucia

*2010 Mathematics Subject Classification*

- 35K50 35K60

*Keywords*

- Parabolic system, regularity, uniqueness

*DOI*

*Abstract*

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

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# The existence of triangulations of non-convex polyhedra without new vertices

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 52B55 65D18

*Keywords*

- non-convex polyhedron, regular subdivision, triangulation, Steiner points

*DOI*

*Abstract*

It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called it Steiner points). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra.

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# A modified lookdown construction for the Xi--Fleming--Viot process with mutation and populations with recurrent bottlenecks

*Authors*

- Birkner, Matthias
- Blath, Jochen
- Möhle, Martin
- Steinrücken, Matthias
- Tams, Johanna

*2010 Mathematics Subject Classification*

- 60K35 60G09 92D10 60C05 92D15

*Keywords*

- Coalescent, duality, Fleming-Viot process, measure-valued process, modified lookdown construction

*DOI*

*Abstract*

Let $Lambda$ be a finite measure on the unit interval. A $Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($Lambda$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $Lambda$ is the Dirac measure in $0$. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($Xi$-coalescent) with mutation, the $Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a populations is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $Xi$-Fleming-Viot processes naturally arise as limiting models.

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# Infinite harmonic chain with heavy mass

*Authors*

- Herrmann, Michael
- Segatti, Antonio

*2010 Mathematics Subject Classification*

- 37K60 70F45

*Keywords*

- Harmonic lattice, atomic chain, macroscopic continuum limit

*DOI*

*Abstract*

Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to $1$, except for one heavy particle which has mass $M$. We investigate the macroscopic behaviour of such a system when $M$ is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle.

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# Nonlinear models in option pricing --- An introduction

*Authors*

- Ehrhardt, Matthias

*2010 Mathematics Subject Classification*

- 91B26

*Keywords*

- nonlinear Black-Scholes equation, computational finance, free boundary problem, European options, American options

*DOI*

*Abstract*

Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option price itself.

This book consists of a collection of contributed chapters of well-known outstanding scientists working successfully in this challenging research area. It discusses concisely several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will present in this book both analytical techniques and numerical methods to solve adequately the arising nonlinear equations.

The purpose of this book is to give an overview on the current state-of-the-art research on nonlinear option pricing. The intended audience is on the one hand graduate and Ph.D. students of (mathematical) finance and on the other hand lecturer of mathematical finance and and people working in banks and stock markets that are interested in new tools for option pricing.

*Appeared in*

- Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science, 2008, ISBN: 978-1-60456-931-5.

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# Exponential decay of the free energy for discretized electro-reaction-diffusion systems

*Authors*

- Glitzky, Annegret

*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*

Our focus are 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 statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system 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. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.

*Appeared in*

- Nonlinearity, 21 (2008) pp. 1989--2009.

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# Evaluation of exact boundary mappings for one-dimensional semi-infinite periodic arrays

*Authors*

- Ehrhardt, Matthias
- Sun, Jiguang
- Zheng, Chunxiong

*2010 Mathematics Subject Classification*

- 65M99 35B27 35Q60, 35J05

*Keywords*

- periodic arrays, Helmholtz equation, Sommerfeld-to-Sommerfeld mapping, dispersion diagram, Floquet-Bloch theory

*DOI*

*Abstract*

Periodic arrays are structures consisting of geometrically identical subdomains, usually called periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of the exact boundary mappings for general one-dimensional semi-infinite periodic arrays for any real wavenumber. The well-posedness of the Helmholtz equation is established via the *limiting absorption principle*.

An algorithm based on the doubling procedure and extrapolation technique is proposed to derive the exact Sommerfeld-to-Sommerfeld boundary mapping. The advantages of this algorithm are the robustness and simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers.

To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of two generalized eigenvalue problems, but still the overall computational cost is significantly reduced.

Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given.

*Appeared in*

- Comm. Math. Sci., 7 (2009) pp. 347--364.

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# Modeling solutions with jumps for rate-independent systems on metric spaces

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 35K55 49Q20 58E99

*Keywords*

- Metric flow, rate-independent systems, quasistatic evolution, vanishing-viscosity limit, parametrized metric solution, approximable solutions

*DOI*

*Abstract*

Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our *parametrized metric solutions* of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of *BV solutions* is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.

*Appeared in*

- Discrete Contin. Dyn. Syst., 25 (2009) pp. 585--615.

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# Asymptotic pulse shapes in filamentary propagation of intense femtosecond pulses

*Authors*

- Krüger, Carsten
- Demircan, Ayhan
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60 35Q55 37K40 81V80

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 42.65.-k 52.38.Hb 42.68Ay

*Keywords*

- Nonlinear Schroedinger equations, Ultrashort pulse propagation

*DOI*

*Abstract*

Self-compression of intense ultrashort laser pulses inside a self-guided filament is discussed. The filament self-guiding mechanism requires a balance between diffraction, plasma self-defocusing and Kerr-type self-focusing, which gives rise to asymptotic intensity profiles on axis of the filament. The asymptotic solutions appear as the dominant pulse shaping mechanism in the leading part of the pulse, causing a pinch of the photon density close to zero delay, which substantiates as pulse compression. The simple analytical model is backed up by numerical simulations, confirming the prevalence of spatial coupling mechanisms and explaining the emerging inhomogeneous spatial structure. Numerical simulations confirm that only spatial effects alone may already give rise to filament formation. Consequently, self-compression is explained by a dynamic balance between two optical nonlinearities, giving rise to soliton-like pulse formation inside the filament.

*Appeared in*

- Laser Physics, 19 (2009) pp. 330-335.

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# Diffraction of stochastic point sets: Exactly solvable examples

*Authors*

- Baake, Michael
- Birkner, Matthias
- Moody, Robert V.

*2010 Mathematics Subject Classification*

- 60G55, 60G57, 78A45

*Keywords*

- Diffraction, stochastic point sets

*DOI*

*Abstract*

Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.

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# Lagrangian and Hamiltonian two-scale reduction

*Authors*

- Giannoulis, Johannes
- Herrmann, Michael
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2008 Physics and Astronomy Classification Scheme*

- 02.30.Mv, 45.20.Jj, 52.35.-g

*Keywords*

- Reduced dynamics, multi-scale problems, Hamiltonian structures, modulation equations

*DOI*

*Abstract*

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

*Appeared in*

- J. Math. Phys., 49 (2008) pp. 103505/1--103505/42.

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# Bernstein--Walsh type theorems for real analytic functions in several variables

*Authors*

- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 41A17, 41A10, 41A60, 41A63, 41A25, 32U35, 32U05, 32E30, 32D20

*Keywords*

- Polynomial approximation in higher dimensions, Bernstein-Walsh's type theorems, real-analytic functions in $mathbb(R)^N$, maximal convergence, plurisubharmonicity, pluricomplex Green functions

*DOI*

*Abstract*

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in R^N. In particular, we investigate the polynomial approximation behavior for functions $F: L to C, L= (Re z, Im z ) : z in K$, of the type $F= g overline h$, where g and h are holomorphic in a neighborhood of a compact set $K subset C^N$. To this end the maximal convergence number $rho(S_c,f)$ for continuous functions f defined on a compact set $S_c subset C^N$ is connected to a maximal convergence number $rho(S_r,F)$ for continuous functions F defined on a compact set $S_r subset R^N$.

*Appeared in*

- Constr. Approx., 33 (2011) pp. 191--217.

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# On the evaluation of dilatometer experiments

*Authors*

- Hömberg, Dietmar
- Togobytska, Nataliya
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 74F05 74N99

*Keywords*

- dilatometer, phase transitions, inverse problem

*DOI*

*Abstract*

The goal of this paper is a mathematical investigation of dilatometer experiments to measure the kinetics of solid-solid phase transitions in steel upon cooling from the high temperature phase. Usually, the data are only used for measuring the start and end temperature of the phase transition. In the case of several coexisting product phases, lavish microscopic investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, we show that the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data and present some numerical identification results.

*Appeared in*

- Appl. Anal., 88 (2009) pp. 669-681.

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# Phase transformation modeling and parameter identification from dilatometric investigations

*Authors*

- Suwanpinij, Piyada
- Togobytska, Nataliya
- Keul, Christoph
- Weiss, Wolf
- Prahl, Ulrich
- Hömberg, Dietmar
- Bleck, Wolfgang

*2010 Mathematics Subject Classification*

- 74N15 35K05 35R30 35B30 35K20

*Keywords*

- Phase transformation modeling, dilatometry, parameter identification

*DOI*

*Abstract*

The goal of this paper is to propose a new approach towards the evaluation of dilatometric results, which are often employed to analyse the phase transformation kinetics in steel, especially in terms of continuous cooling transformation (CCT) diagram. A simple task of dilatometry is deriving the start and end temperatures of the phase transformation. It can yield phase transformation kinetics provided that plenty metallographic investigations are performed, whose analysis is complicated especially in case of several coexisting product phases. The new method is based on the numerical solution of a thermomechanical identification problem. It is expected that the phase transformation kinetics can be derived by this approach with less metallographic tasks. The first results are remarkably promising although further investigations are required for the numerical simulations.

*Appeared in*

- Steel Res. Int., 79 (2008), pp. 793-799

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# Uniform boundedness of norms of convex and nonconvex processes

*Authors*

- Henrion, René
- Seeger, Alberto

*2010 Mathematics Subject Classification*

- 34A60 47H04 52A20

*Keywords*

- Convex processes, positively homogeneous maps, controllability, Painleve-Kuratowski limits, graph-convergence

*DOI*

*Abstract*

The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.

*Appeared in*

- Numer. Funct. Anal. Optim., 29 (2008) pp. 551--573.

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# A model equation for ultrashort optical pulses

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Vladimirov, Andrei G.
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 78A60 35Q60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.81.Dp 42.65.Tg 05.45.Yv

*Keywords*

- ultrashort pulses, few-cycle pulses

*DOI*

*Abstract*

The nonlinear Schrödinger equation based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function such that the resonances are naturally accounted for. This approach allows us to derive a simple non-envelope model for short pulses propagating in one spatial dimension. This model is further investigated numerically and analytically.

*Appeared in*

- Eur. Phys. J. D, 58 (2010) pp. 219--226.

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# On the co-derivative of normal cone mappings to inequality systems

*Authors*

- Henrion, René
- Outrata, Jiří
- Surowiec, Thomas

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 90C30 49J53

*Keywords*

- Mordukhovich co-derivative, normal cone mapping, calmness

*DOI*

*Abstract*

The paper deals with co-derivative formulae for normal cone mappings to smooth inequality systems. Both, the regular (Linear Independence Constraint Qualification satisfied) and nonregular (Mangasarian-Fromovitz Constraint Qualification satisfied) case are considered. A major part of the results relies on general transformation formulae previously obtained by Mordukhovich and Outrata. This allows to derive exact formulae for general smooth, regular and polyhedral, possibly nonregular systems. In the nonregular, nonpolyhedral case a generalized transformation formula by Mordukhovich and Outrata applies, however a major difficulty consists in checking a calmness condition of a certain multivalued mapping. The paper provides a translation of this condition in terms of much easier to verify constraint qualifications. A series of examples illustrates the use and comparison of the presented formulae.

*Appeared in*

- Nonlinear Anal., 71 (2009) pp. 1213-1226

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# Deviational particle Monte Carlo for the Boltzmann equation

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65R20 82C40

*2008 Physics and Astronomy Classification Scheme*

- 05.10.Ln

*Keywords*

- Monte Carlo method, Boltzmann equation, reference Maxwellian, deviational particles

*DOI*

*Abstract*

The paper describes the deviational particle Monte Carlo method for the Boltzmann equation. The approach is an application of the general ``control variates'' variance reduction technique to the problem of solving a nonlinear equation. The deviation of the solution from a reference Maxwellian is approximated by a system of positive and negative particles. Previous results from the literature are modified and extended. New algorithms are proposed that cover the nonlinear Boltzmann equation (instead of a linearized version) with a general interaction model (instead of hard spheres). The algorithms are obtained as procedures for generating trajectories of Markov jump processes. This provides the framework for deriving the limiting equations, when the number of particles tends to infinity. These equations reflect the influence of various numerical approximation parameters. Detailed simulation schemes are provided for the variable hard sphere interaction model.

*Appeared in*

- Monte Carlo Methods Appl., 14 (2008) pp. 191--268.

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# Integral equations for conical diffraction by coated gratings

*Authors*

- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 35Q60 31A10, 35J05, 35K50, 45F15, 78A45

*Keywords*

- Diffraction, periodic structure, integral equation method, oblique incidence, system of singular integral equations

*DOI*

*Abstract*

The paper is devoted to integral formulations for the scattering of plane waves by diffraction gratings under oblique incidence. For the case of coated gratings Maxwell's equations can be reduced to a system of four singular integral equations on the piecewise smooth interfaces between different materials. We study analytic properties of the integral operators for periodic diffraction problems and obtain existence and uniqueness results for solutions of the systems corresponding to electromagnetic fields with locally finite energy.

*Appeared in*

- J. Integral Equations Appl., 23 (2011) pp. 71--112.

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# Locally adaptive estimation methods with application to univariate time series

*Authors*

- Elagin, Mstislav

*2010 Mathematics Subject Classification*

- 62M10 62F10, 62P20

*Keywords*

- Adaptive estimation, local homogeneity, model selection, stagewise aggregation, volatility model, Poisson model, exponential model, Bernoulli model, propagation, oracle

*DOI*

*Abstract*

The paper offers a unified approach to the study of three locally adaptive estimation methods in the context of univariate time series from both theoretical and empirical points of view. A general procedure for the computation of critical values is given. The underlying model encompasses all distributions from the exponential family providing for great flexibility. The procedures are applied to simulated and real financial data distributed according to the Gaussian, volatility, Poisson, exponential and Bernoulli models. Numerical results exhibit a very reasonable performance of the methods.

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# Numerical study of the systematic error in Monte Carlo schemes for semiconductors

*Authors*

- Muscato, Orazio
- Di Stefano, Vincenza
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82D37 65C05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Ss

*Keywords*

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

*DOI*

*Abstract*

The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., M2AN 44 (2010) pp. 1049-1068.

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# Hölder continuity for second order elliptic problems with nonsmooth data

*Authors*

- Haller-Dintelmann, Robert
- Meyer, Christian
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35J25 35R05 49K20

*Keywords*

- Elliptic problems, mixed boundary value problems, Hölder continuity, optimal control

*DOI*

*Abstract*

The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.

*Appeared in*

- Appl. Math. Optim., 60 (2009) pp. 397--428.

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# Mode transitions in distributed-feedback tapered master-oscillator power-amplifier

*Authors*

- Radziunas, Mindaugas
- Tronciu, Vasile Z.
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Lichtner, Mark
- Spreemann, Martin
- Wenzel, Hans

*2010 Mathematics Subject Classification*

- 35Q60 37L15

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.Sf

*Keywords*

- master oscillator, power amplifier, taper, longitudinal mode analysis

*DOI*

*Abstract*

Theoretical and experimental investigations have been carried out to study the spectral and spatial behavior of monolithically integrated distributed-feedback tapered master-oscillators power-amplifiers emitting around 973 nm. Introduction of self and cross heating effects and the analysis of longitudinal optical modes allows us to explain experimental results. The results show a good qualitative agreement between measured and calculated characteristics.

*Appeared in*

- Opt. Quantum Electron., 40 (2008) pp. 1103-1109.

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# Numerical simulation of quantum waveguides

*Authors*

- Arnold, Anton
- Ehrhardt, Matthias
- Schulte, Maike

*2010 Mathematics Subject Classification*

- 65M12 35Q40 45K05

*Keywords*

- Quantum waveguide, Schrödinger equation, transparent boundary condition, finite difference scheme

*DOI*

*Abstract*

This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss *transparent boundary conditions* (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain.

First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive *discrete transparent boundary conditions* (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary.

Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct *approximate* DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion.

In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.

*Appeared in*

- Kenzo Watanabe (ed): VLSI and Computer Architecture, Nova Sciece Publishers, 2008, ISBN: 978-1-60692-075-6

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# On unwanted nucleation phenomena at the wall of a VGF chamber

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank
- Eichler, Stefan
- Naldzhieva, Margarita

*2010 Mathematics Subject Classification*

- 74N05

*2008 Physics and Astronomy Classification Scheme*

- 61.50.-f 61.82.Fk 62.10.+s 62.20.-x 64.60.Qb 64.70.Ja 68.03.Cd

*Keywords*

- crystal growth, homogeneous and heterogenous nucleation, undercooled melt, phase transition, gallium arsenide

*DOI*

*Abstract*

This is preliminary study on a phenomenon that happens during crystal growth of GaAs in a vertical gradient freeze (VGF) device. Here unwanted polycrystals nucleate at the chamber wall and move into the interior of the crystal. This happens within an undercooled region in the vicinity of the triple point, where the liquid-solid interface meets the chamber wall. The size and shape of that region is modelled by the Gibbs-Thomson law, which will be rederived in this paper. Hereafter we identify the crucial parameter, whose proper adjustment may minimize the undercooled region. Finally we give a simple estimate to calculate and evaluate the energy barrier for homogeneous and heterogeneous nucleation of a solid nucleus in the undercooled melt

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# On a thermomechanical milling model

*Authors*

- Chełminski, Krzysztof
- Hömberg, Dietmar
- Rott, Oliver

*2010 Mathematics Subject Classification*

- 74F05 35Q80 35L15

*Keywords*

- Thermo-elasticity, milling, process-structure interaction, work piece effects, stability, delay-differential equations

*DOI*

*Abstract*

This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions.

*Appeared in*

- Nonlinear Anal. Real World Appl., 12 (2011) pp. 615--632.

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# On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 49J40 74C05 74F05

*Keywords*

- Shape-memory materials, rate-independent energetic formulation, temperature-induced phase transformation, differential inclusion, convergence for space-time discretization

*DOI*

*Abstract*

This paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.

*Appeared in*

- SIAM J. Math. Anal., 41 (2009) pp. 1388--1414.

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# A milling model with thermal effects including the dynamics of machine and work piece

*Authors*

- Rott, Oliver
- Rasper, Patrick
- Hömberg, Dietmar
- Uhlmann, Eckart

*2010 Mathematics Subject Classification*

- 74H99 93A30

*Keywords*

- process machine interaction, stability, time domain simulation

*DOI*

*Abstract*

This paper deals with the development of a new mathematical model that characterizes the structure-process interaction for a complex milling system. The structure is divided into a work piece and a machine part, which are represented by different models. While the machine dynamics is characterized by a standard multi-body system, the work piece is described as a linear thermo-elastic continuum. The coupling of both parts is carried out by an empirical process model permitting an estimate of heat and coupling forces occurring during milling. This work reports the derivation of the governing equations emphasizing the coupling and summarizes the numerical algorithms being applied to solve the coupled equation system. The results of numerical simulations that show the dynamics of the complex thermo-mechanical system are presented at the end.

*Appeared in*

- B. Denkena, ed., Proceedings, 1st International Conference on Process Machine Interactions, Hannover, September 3-4, 2008, PZH Produktionstechnisches Zentrum GmbH, Garbsen, 2008, pp. 369--378

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# Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Petrov, Adrien
- Martins, Joao A. C.

*2010 Mathematics Subject Classification*

- 34A60, 47H06, 73E50, 73E99, 74C05

*Keywords*

- Rate-independent processes, quasi-static problems, differential inclusions,, elastoplasticity, hardening, variational formulations, slow time scale.

*DOI*

*Abstract*

This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.

*Appeared in*

- J. Math. Anal. Appl., 348 (2008) pp. 1012--1020.

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# Optimal control for the thermistor problem

*Authors*

- Hömberg, Dietmar
- Meyer, Christian
- Rehberg, Joachim
- Ring, Wolfgang

*2010 Mathematics Subject Classification*

- 35K55 35M10 49J20 49K20

*Keywords*

- Partial differential equations, optimal control problems, state constraints

*DOI*

*Abstract*

This paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.

*Appeared in*

- SIAM J. Control Optim., 48 (2010) pp. 3449--3481.

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# Coarsening dynamics of slipping droplets

*Authors*

- Kitavtsev, Georgy
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 35G25 35K55

*2008 Physics and Astronomy Classification Scheme*

- 47.20Ma

*Keywords*

- lubrication models, coarsening, wall slippage, fluid dynamics

*DOI*

*Abstract*

This paper studies the late phase dewetting process of nanoscopic thin polymer films on hydrophobized substrates using some recently derived lubrication models that take account of large slippage at the polymer-substrate interface. The late phase of this process is characterized by the slow-time coarsening dynamics of arrays of droplets that remain after rupture and the initial dewetting phases. For this situation a reduced system of ordinary differential equations is derived from the lubrication model for large slippage using asymptotic analysis. This extends known results for the no-slip case. On the basis of the reduced model, the role of the slippage as a control parameter for droplet migration is analysed and several new qualitative effects for the coarsening process are identified.

*Appeared in*

- Journal of Engineering Mathematics, 66 (2010) pp. 271--292.

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# The Airy$_1$ process is not the limit of the largest eigenvalue in GOE matrix diffusion

*Authors*

- Bornemann, Folkmar
- Ferrari, Patrik
- Prähofer, Michael

*2010 Mathematics Subject Classification*

- 82C22 60K35

*Keywords*

- KPZ class, Random Matrices, Airy processes

*DOI*

*Abstract*

Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy$_1$-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion

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# Spinodal dewetting of thin films with large interfacial slip: Implications from the dispersion relation

*Authors*

- Rauscher, Markus
- Blossey, Ralf
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 35G25 35K55

*2008 Physics and Astronomy Classification Scheme*

- 68.55.-a

*Keywords*

- thin liquid films, stability, fluid dynamics

*DOI*

*Abstract*

We compare the dispersion relations for spinodally dewetting thin liquid films for increasing magnitude of interfacial slip length in the lubrication limit. While the shape of the dispersion relation, in particular the position of the maximum, are equal for no-slip up to moderate slip lengths, the position of the maximum shifts to much larger wavelengths for large slip lengths. Here, we discuss the implications of this fact for recently developed methods to assess the disjoining pressure in spinodally unstable thin films by measuring the shape of the roughness power spectrum. For PS films on OTS covered Si wafers (with slip length $bapprox 1,mu$m) we predict a 20% shift of the position of the maximum of the power spectrum which should be detectable in experiments.

*Appeared in*

- Langmuir, 24 (21) (2008), pp. 12290--12294 .

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# How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms

*Authors*

- Kampen, Jörg

*2010 Mathematics Subject Classification*

- 65N99 35F20 35K10

*Keywords*

- length of geodesic, regular polynomial interpolation

*DOI*

*Abstract*

We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_ijdx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximations of solutions of linear parabolic partial differential equations leading to obvious applications in finance, physics and other sciences.

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# Stability results for a soil model with singular hysteretic hydrology

*Authors*

- Krejčí, Pavel
- O'Kane, J. Philip
- Pokrovskii, Alexei
- Rachinskii, Dmitrii

*2010 Mathematics Subject Classification*

- 34C55 34C25 76S05

*Keywords*

- Preisach operator, singular differential equation, periodic solution

*DOI*

*Abstract*

We consider a differential equation describing the mass balance in a soil hydrology model with noninvertible Preisach-type hysteresis. We approximate the singular Preisach operator by regular ones and show, as main result, that the solutions of the regularized problem converge to a solution of the original one as the regularization parameter tends to zero. For monotone right hand sides, we prove that the solution is unique. If in addition the external water sources are time periodic, then we find sufficient conditions for the existence, uniqueness, and asymptotic stability of periodic solutions.

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# Magnetohydrodynamic flow with hysteresis

*Authors*

- Eleuteri, Michela
- Kopfová, Jana
- Krejčí, Pavel

*2010 Mathematics Subject Classification*

- 76W05 47J40 35K60

*Keywords*

- Preisach hysteresis operator, magnetohydrodynamics

*DOI*

*Abstract*

We consider a model system describing the 2D flow of a conducting fluid surrounded by a ferromagnetic solid under the influence of the hysteretic response of the surrounding medium. We assume that this influence can be represented by the Preisach hysteresis operator. Existence and uniqueness of solutions for the resulting system of PDEs with hysteresis nonlinearities is established in the convexity domain of the Preisach operator.

*Appeared in*

- SIAM J. Math. Anal., 41 (2009) pp. 435--464.

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# A note on a parabolic equation with nonlinear dynamical boundary condition

*Authors*

- Sprekels, Jürgen
- Wu, Hao

*2010 Mathematics Subject Classification*

- 35B40 35B41 35B45

*Keywords*

- Parabolic equation, dynamical boundary condition, global attractor, convergence to equilibrium, Lojasiewicz-Simon inequality

*DOI*

*Abstract*

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Łojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f, g are real analytic. Moreover, we provide an estimate for the convergence rate.

*Appeared in*

- Nonlinear Anal., 72 (2010) pp. 3028--3048.

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# Global higher integrability of minimizers of variational problems with mixed boundary conditions

*Authors*

- Fiaschi, Alice
- Knees, Dorothee
- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 74C05 49N60 49S05 35B65

*Keywords*

- Higher integrability of gradients of minimizers, p-growth, mixed boundary conditions, damage, uniform Caccioppoli-like inequality

*DOI*

*Abstract*

We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.

*Appeared in*

- J. Math. Anal. Appl., 401 (2013) pp. 269--288.

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# Exponential asymptotic stability via Krein--Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems

*Authors*

- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 35B25 35B10 35B35 35K10 35K90

*Keywords*

- singularly perturbed parabolic Dirichlet problems, exponential asymptotic stability, Krein-Rutman theorem, lower and upper solutions

*DOI*

*Abstract*

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.

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# Dependence on the dimension for complexity of approximation of random fields

*Authors*

- Serdyukova, Nora

*2010 Mathematics Subject Classification*

- 41A63 60G60 60G15, 41A58

*Keywords*

- Gaussian processes, random fields, Karhunen-Loève expansion, linear approximation error, information-based complexity, tractability, curse of dimensionality, multivariate linear problems

*DOI*

*Abstract*

We consider the $e $-approximation by $n$-term partial sums of the Karhunen-Loève expansion to $d$-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as $dto infty$, of the information complexity $n(e,d)$ of approximation with error not exceeding a given level $e$. It was recently shown by M. A. Lifshits and E. V. Tulyakova that for this problem one observes the curse of dimensionality (intractability) phenomenon. The aim of this paper is to give the exact asymptotic expression for $n(e,d)$.

*Appeared in*

- Theory Probab. Appl., 54 (2010) pp. 272--284.

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# Global weak solutions of the Navier--Stokes--Vlasov--Poisson system

*Authors*

- Anoschenko, Olga
- Khruslov, Evgeni
- Stephan, Holger

*2010 Mathematics Subject Classification*

- 76D05 35J05 76T20

*2008 Physics and Astronomy Classification Scheme*

- 82.70.Kj 52.65.Ff

*Keywords*

- Navier-Stokes equation, Vlasov-Poisson equation, suspensions, global weak solution, modified Galerkin method, compactness of approximations

*DOI*

*Abstract*

We consider the Navier-Stokes-Vlasov-Poisson system of partial differential equations, describing the motion of a viscous incompressible fluid with small solid charged particles therein. We prove the existence of a weak global solution of the initial boundary value problem for this system.

*Appeared in*

- J. Mathematical Physics Analysis Geometry (MAG), 6 (2010) pp. 143--182.

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# An optimisation method in inverse acoustic scattering by an elastic obstacle

*Authors*

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

*2010 Mathematics Subject Classification*

- 35R30 76Q05 35J05

*Keywords*

- acoustic and elastic waves, inverse scattering, potential representation, Kirsch-Kress method

*DOI*

*Abstract*

We consider the interaction between an elastic body and a compressible inviscid fluid, which occupies the unbounded exterior domain. The inverse problem of determining the shape of such an elastic scatterer from the measured far field pattern of the scattered fluid pressure field is of central importance in detecting and identifying submerged objects. Following a method proposed by Kirsch and Kress, we approximate the acoustic and elastodynamic wave by potentials over auxiliary surfaces, and we reformulate the inverse problem as an optimisation problem. The objective function to be minimised is the sum of three terms. The first is the deviation of the approximate far field pattern from the measured one, the second is a regularisation term, and the last a control term for the transmission condition. We prove that the optimisation problem has a solution and that, for the regularisation parameter tending to zero, the minimisers tend to a solution of the inverse problem. In contrast to a numerical method from a previous paper, the presented method does require neither a direct solution method nor an additional treatment of possible Jones modes.

*Appeared in*

- SIAM J. Appl. Math., 70 (2009) pp. 168--187.

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# Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave

*Authors*

- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35B60

*Keywords*

- Inverse scattering problem, uniqueness, sound-hard, polyhedral obstacle

*DOI*

*Abstract*

We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far field measurements. It is proved that a polyhedral scatterer in $R^n, nge 2$, consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.

*Appeared in*

- Inverse Problems 24 (2008) 035004

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# Existence and stability of solutions with periodically moving weak internal layers

*Authors*

- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 35B25 35B10 35K57 35K20

*Keywords*

- singularly perturbed reaction diffusion equation, periodic boundary value, problem, boundary layers

*DOI*

*Abstract*

We consider the periodic parabolic differential equation $ep^2 Big( fracpartial^2 upartial x^2 -fracpartial upartial t Big)=f(u,x,t,ep)$ under the assumption that $ve$ is a small positive parameter and that the degenerate equation $f(u,x,t,0) =0$ has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution $u_p(x,t,ep)$ which is $T$-periodic in $t$, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as $eprightarrow 0$.

*Appeared in*

- J. Math. Anal. Appl., 348 (2008) pp. 508-515.

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# Anisotropic growth of random surfaces in $2+1$ dimensions

*Authors*

- Borodin, Alexei
- Ferrari, Patrik

*2010 Mathematics Subject Classification*

- 82C22 60K35 60G55 60G1

*Keywords*

- Anisotropic KPZ, Gaussian free field

*DOI*

*Abstract*

We construct a family of stochastic growth models in $2+1$ dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield $1+1$ dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order $ln(t)$ for time $tgg 1$. (3) There is a map of the $(2+1)$-dimensional space-time to the upper half-plane $H$ such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on $H$.

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# Experimental and numerical model study of the limiting current in a channel flow cell with a circular electrode

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Zhao, Hong
- Holzbecher, Ekkehard
- Langmach, Hartmut
- Chojak, Malgorzata
- Halseid, Rune
- Jusys, Zenonas
- Behm, Rolf Jürgen

*2010 Mathematics Subject Classification*

- 35K20 65N99

*Keywords*

- limiting current, Finite Volume Method, boundary layer, parameter estimation

*DOI*

*Abstract*

We describe first measurement in a novel thin-layer channel flow cell designed for the investigation of heterogeneous electrocatalysis on porous catalysts. For the interpretation of the measurements, a macroscopic model for coupled species transport and reaction, which can be solved numerically, is feasible. In this paper, we focus on the limiting current. We compare numerical solutions of a macroscopic model to a generalization of a Leveque-type asymptotic estimate for circular electrodes, and to measurements obtained in the aforementioned flow cell. We establish, that on properly aligned meshes, the numerical method reproduces the asymptotic estimate. Furthermore, we demonstrate, that the measurements are partially performed in the sub-asymptotic regime, in which the boundary layer thickness exceeds the cell height. Using the inlet concentration and the diffusion coefficient from literature, we overestimate the limiting current. On the other hand, the use of fitted parameters leads to perfect agreement between model and experiment.

*Appeared in*

- Phys. Chem. Chem. Phys., 10 (2008) pp. 3784--3795.

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# A discrete variational principle for rate-independent evolution

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 35K55

*Keywords*

- Variational Principle, rate-independent process, time-discretization

*DOI*

*Abstract*

We develop a global-in-time variational approach to the time-discretization of rate-independent processes. In particular, we investigate a discrete version of the variational principle based on the *weighted energy-dissipation functional* introduced by A. Mielke and M. Ortiz in ESAIM Control Optim. Calc. Var., 2008. We prove the conditional convergence of time-discrete approximate minimizers to energetic solutions of the time-continuous problem. Moreover, the convergence result is combined with approximation and relaxation. For a fixed partition the functional is shown to have an asymptotic development by Gamma convergence, cf. G. Anzellotti and S. Baldo (Appl. Math. Optim., 1993), in the limit of vanishing viscosity.

*Appeared in*

- Adv. Calculus Variations, 1 (2008) pp. 399--431.

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# Crack growth in polyconvex materials

*Authors*

- Knees, Dorothee
- Zanini, Chiara
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

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

*Keywords*

- rate-independent problems, energetic formulation, time-incremental minimization, parameterized solutions, energy-release rate, Griffith fracture criterion, finite-strain elasticity, local energetic solution

*DOI*

*Abstract*

We discuss a model for crack propagation in an elastic body, where the crack path is described a-priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed.

*Appeared in*

- Phys. D, 239 (2010) pp. 1470--1484.

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# A study on the eigenstrain problem in solid mixtures

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank
- Kimmerle, Sven-Joachim

*2010 Mathematics Subject Classification*

- 74-99 74A10 74E05 74F05 74F10 74F20 74N25

*Keywords*

- misfit, inclusions, St. Venant-Kirchhoff law, elasticity, inelastic deformation, change of reference configuration, intermediate configuration, thermal, expansion, diffusion, phase transition

*DOI*

*Abstract*

We introduce a framework that is capable to model the appearance of mechanical stresses due to inelastic deformations. Among these we consider in particular thermal expansions, diffusion and phase transitions. Among the quantities of central importance are the eigenstrain and the misfit strain. They describe the phenomenon that different material volumes of a compact body may not be compatible to each other in a stress-free reference configuration, so that here a compact body may not exist. We shall show that it is possible to find a further reference configuration, where the body is compact but not free of stress. A typical example where misfit appears concerns a body whose local parts differently transform their phase. This might be a change of the crystal lattice from the ferrite to the austenite symmetry in steel, or the formation of liquid droplets in crystalline gallium arsenide. In both cases the new interior phase has with respect to the parent phase different volume or shape in its state that is free of stress. In this study we consider the eigenstrain problem for pure substances as well as for mixtures. In the latter case subtle arguments are needed for an appropriate description. Special focus is given to the equivalence of interface boundaries with discontinues and continues displacement vectors.

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# Analysis of a spin-polarized drift-diffusion model

*Authors*

- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35K57 35R05 35B45 35B40 78A35

*Keywords*

- Reaction-diffusion systems, spin-polarized drift-diffusion processes, motion of charged particles, existence, uniqueness, energy estimates, a priori estimates, asymptotic behaviour

*DOI*

*Abstract*

We introduce a spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. We give a weak formulation of this problem and prove an existence and uniqueness result for the instationary problem. If the boundary data is compatible with thermodynamic equilibrium the free energy along the solution decays monotonously and exponentially to its equilibrium value. In other cases it may be increasing but we estimate its growth. Moreover we give upper and lower estimates for the solution.

*Appeared in*

- Adv. Math. Sci. Appl., 18 (2008) pp. 401--427.

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# Evanescent channels and scattering in cylindrical nanowire heterostructures

*Authors*

- Racec, Paul N.
- Racec, Roxana
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A40 35Q40 35P25 58J50

*2008 Physics and Astronomy Classification Scheme*

- 72.10.Bg 73.23.Ad 73.40.-c 73.63.-b

*Keywords*

- Nanowire, scattering, mesoscopic transport, resonances, evanescent states

*DOI*

*Abstract*

We investigate the scattering phenomena produced by a general finite range non-separable potential in a multi-channel two-probe cylindrical nanowire heterostructure. The multi-channel current scattering matrix is efficiently computed using the R-matrix formalism extended for cylindrical coordinates. Considering the contribution of the evanescent channels to the scattering matrix, we are able to put in evidence the specific dips in the tunneling coefficient in the case of an attractive potential. The cylindrical symmetry cancels the ''selection rules'' known for Cartesian coordinates. If the attractive potential is superposed over a non-uniform potential along the nanowire, then resonant transmission peaks appear. We can characterize them quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances (dips and peaks). Our formalism is applied to a variety of model systems like a quantum dot, a core/shell quantum ring or a double barrier, embedded into the nano-cylinder.

*Appeared in*

- Phys. Rev. B., 79 (2009) pp. 155305/1--155305/14.

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# A model for the austenite-ferrite phase transition in steel including misfit stress

*Authors*

- Dreyer, Wolfgang
- Hömberg, Dietmar
- Petzold, Thomas

*2010 Mathematics Subject Classification*

- 74-99 74N05 74N25

*Keywords*

- steel, ferrite-austenite phase transition, elasticity, diffusion, misfit, free boundaries, kinetic boundary conditions, diffusion controlled boundary conditions

*DOI*

*Abstract*

We present a thermodynamically consistent model to describe the austenite-ferrite phase transition in steel. We consider the influence of the mechanical displacement field due to eigenstrains caused by volumetric expansion. The model equations are derived in a systematical framework. They are based on the conservation laws for mass and momentum and the second law of thermodynamics. By means of numerical computations for a simplified interface controlled model, we examine the influence of the mechanical contributions to the transformation kinetics and the equilibrium states.

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# Pricing CMS spreads in the Libor market model

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G44 91B28

*Keywords*

- CMS spread option, Margrabe's formula, Libor market model

*DOI*

*Abstract*

We present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.

*Appeared in*

- Int. J. Theor. Appl. Finance, 13 (2010) pp. 45--62.

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# Escaping the Brownian stalkers

*Authors*

- Weiß, Alexander

*2010 Mathematics Subject Classification*

- 60J65 60K10

*Keywords*

- financial markets, market stability, stochastic dynamics, recurrence, transience

*DOI*

*Abstract*

We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance

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# Discrete transparent boundary conditions for the Schrödinger equation on circular domains

*Authors*

- Arnold, Anton
- Ehrhardt, Matthias
- Schulte, Maike
- Sofronov, Ivan

*2010 Mathematics Subject Classification*

- 65M12 35Q40 45K05

*Keywords*

- two--dimensional Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes

*DOI*

*Abstract*

We propose transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson finite difference scheme. The presented discrete initial boundary-value problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct *approximate* discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method.

As a by-product we also present a new formulation of discrete TBCs for the 1D Schrödinger equation, with convolution coefficients that have better decay properties than those from the literature.

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# Global existence for rate-independent gradient plasticity at finite strain

*Authors*

- Mainik, Andreas
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 49J40 49S05 74C15

*Keywords*

- Energetic rate-independent systems, energetic solution, finite-strain elastoplasticity, multiplicative decomposition of the strain, Lie group of plastic strain, dissipation distance, local theory via gradient terms

*DOI*

*Abstract*

We provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables.

*Appeared in*

- J. Nonlinear Sci., 19 (2009) pp. 221--248.

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# Maximal parabolic regularity for divergence operators including mixed boundary conditions

*Authors*

- Haller-Dintelmann, Robert
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35A05 35B65 35K15 35K20

*Keywords*

- Maximal parabolic regularity, quasilinear parabolic equations, mixed Dirichlet-Neumann conditions

*DOI*

*Abstract*

We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

*Appeared in*

- J. Differential Equations, 247 (2009) pp. 1354--1396.

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# On a mathematical model for laser-induced thermotherapy

*Authors*

- Fasano, Antonio
- Hömberg, Dietmar
- Naumov, Dmitri

*2010 Mathematics Subject Classification*

- 92C50 76Z05 35Q80

*Keywords*

- laser treatment, cancer therapy, coagulation, bio-heat equation

*DOI*

*Abstract*

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.

*Appeared in*

- Appl. Math. Modelling, 34 (2010) pp. 3831--3840.

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# Annealed vs quenched critical points for a random walk pinning model

*Authors*

- Birkner, Matthias
- Sun, Rongfeng

*2010 Mathematics Subject Classification*

- 60K35 82B44

*Keywords*

- Random walks, pinning models, annealed and quenched critical points, collision local time, disordered system

*DOI*

*Abstract*

We study a random walk pinning model, where conditioned on a simple random walk $Y$ on $Z^d$ acting as a random medium, the path measure of a second independent simple random walk $X$ up to time $t$ is Gibbs transformed with Hamiltonian $-L_t(X,Y)$, where $L_t(X,Y)$ is the collision local time between $X$ and $Y$ up to time $t$. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature $beta$ varies. We show that in dimensions $d=1,2$, the annealed and quenched critical values of $beta$ are both 0, while in dimensions $dgeq 4$, the quenched critical value of $beta$ is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with Brownian noise and the directed polymer model. For $dgeq 5$, the same result has recently been established by Birkner, Greven and den Hollander via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida, Giacomin, Lacoin and Toninelli to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case $d=3$ remains open.

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# Existence of bounded steady state solutions to spin-polarized drift-diffusion systems

*Authors*

- Glitzky, Annegret
- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 35J55 65N12 35B45 35R05

*Keywords*

- Reaction--diffusion systems, spin-polarized drift--diffusion processes, motion of charged particles, steady states, existence, a priori estimates, uniqueness, Scharfetter-Gummel scheme, boundary conforming Delaunay grid

*DOI*

*Abstract*

We study a stationary spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a Scharfetter-Gummel scheme on 3D boundary conforming Delaunay grids we show existence, boundedness and, for small applied voltages, the uniqueness of the discrete solution.

*Appeared in*

- SIAM J. Math. Anal., 41 (2010) pp. 2489--2513.

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# Elastic half-plane under random boundary excitations

*Authors*

- Shalimova, Irina
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65Z05

*Keywords*

- White noise, Karhunen-Loève expansion, Poisson integral formula, boundary random excitations, 2D Lamé equation

*DOI*

*Abstract*

We study in this paper a respond of an elastic half-plane to random boundary excitations. We treat both the white noise excitations and more generally, homogeneous random fluctuations of displacements prescribed on the boundary. Solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate. This is used to represent the displacements 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 correlation operator. The K-L expansion can be used to calculate the statistical characteristics of other functionals of interest, in particular, the strain and stress tensors and the elastic energy tensor.

*Appeared in*

- J. Statist. Phys., (2008) pp. 1--29.

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# Competing particle systems and the Ghirlanda--Guerra identities

*Authors*

- Arguin, Louis-Pierre

*2010 Mathematics Subject Classification*

- 60G55 60G10 82B44

*Keywords*

- Point processes, ultrametricity, Ghirlanda-Guerra identities

*DOI*

*Abstract*

We study point processes on the real line whose configurations $X$ can be ordered decreasingly and evolve by increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix $Q=q_ij$. Quasi-stationary systems are those for which the law of $(X,Q)$ is invariant under the evolution up to translation of $X$. It was conjectured by Aizenman and co-authors that the matrix $Q$ of robustly quasi-stationary systems must ex This was established recently, up to a natural decomposition of the system, whenever the set $S_Q$ of values assumed by $q_ij$ is finite. In this paper, we study the general case, where $S_Q$ may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda-Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case.

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# On weak solutions to the stationary MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35J55 35Q35 35Q30, 35Q60

*Keywords*

- 0:0:Nonlinear elliptic system, magnetohydrodynamics, natural interface conditions, heat equation, nonlocal radiation boudary condition

*DOI*

*Abstract*

We study the coupling of the stationary system of magnetohydrodynamics to the heat equation. Coupling occurs on the one hand from temperature-dependent coefficients and from a temperature-dependent force term in the Navier-Stokes equations. On the other hand, the heat sources are given by the dissipation of current in the electrical conductors, and of viscous stresses in the fluid. We consider a domain occupied by several different materials, and have to take into account interface conditions for the electromagnetic fields. Since we additionally want to treat high-temperatures applications, we also take into account the effect of heat radiation, which results in nonlocal boundary conditions for the heat flux. We prove the existence of weak solutions for the coupled system, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small. We prove a uniqueness result in the case of constant coefficients and small data. Finally, we discuss the regularity issue in a simplified setting.

*Appeared in*

- Nonlinear Anal. Real World Appl., 10 (2009) pp. 2914--2936.

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# Short note on global spatial regularity in elasto-plasticity with linear hardening

*Authors*

- Knees, Dorothee

*2010 Mathematics Subject Classification*

- 35B65 49N60 74C05

*Keywords*

- elasto-plasticity, linear hardening, global regularity, reflection argument

*DOI*

*Abstract*

We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L^{∞}((0,T);H^{3/2-δ}(Ω)) for the displacements and z in L^{∞}((0,T);H^{1/2-δ}(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 36 (2009) pp. 611--625.

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# Traveling wave modeling of dynamics in semiconductor ring lasers

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 78-04 78A60 35-04 78-05 35Q60 35B35

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.-v 02.30.Jr 02.60.Cb

*Keywords*

- semiconductor ring laser, traveling wave model, switching, alternate oscillations, bistability, unidirectional, optical modes, analysis

*DOI*

*Abstract*

We use the traveling wave model for simulating and analyzing nonlinear dynamics of complex semiconductor ring laser devices. This modeling allows to consider temporal-spatial distributions of the counter-propagating slowly varying optical fields and the carriers, what can be important when studying non-homogeneous ring cavities, propagation of short pulses or fast switching. By performing numerical integration of the model equations we observe several dynamic regimes as well as transitions between them. The computation of ring cavity modes explains some peculiarities of these regimes.

*Appeared in*

- M. Radziunas, Semiconductor Lasers and Laser Dynamics III, K.P. Panajotov, M. Sciamanna, A.A. Valle, R. Michalzik, eds., vol. 6997 of Proceedings of SPIE, SPIE, 2008, pp. 69971B/1--69971B/9

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# Quenched LDP for words in a letter sequence

*Authors*

- Birkner, Matthias
- Greven, Andreas
- den Hollander, Frank

*2010 Mathematics Subject Classification*

- 60F10 60G10 82D60

*Keywords*

- Letters and words, renewal process, empirical process, annealed vs. quenched, large deviation principle, rate function, specific relative entropy, collision local time

*DOI*

*Abstract*

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. We apply our LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on $Z^d$, $d geq 1$, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.

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# Adaptive goodness-of-fit tests based on signed ranks

*Authors*

- Rohde, Angelika

*2010 Mathematics Subject Classification*

- 62G10 62G20 62G35

*Keywords*

- Exact multiple testing, exponential inequality, multiscale statistic, relative asymptotic efficiency, signed ranks, sharp asymptotic adaptivity

*DOI*

*Abstract*

Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l = 0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to one in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.

*Appeared in*

- Ann. Statist. 36 (2008), pp. 1346--1374.

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