# The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity

*Authors*

- Efendiev, Messoud A.
- Gajewski, Herbert
- Zelik, Sergei

*2010 Mathematics Subject Classification*

- 35B40 35B45

*Keywords*

- Cahn-Hilliard system, global attractors, fractal dimension

*DOI*

*Abstract*

The paper is devoted to study the long-time behaviour of solutions of the following 4th order parabolic system in a bounded smooth domain Ω ⊂ ⊂ ℝ^{n}:

(1) b∂_{t}u = - Δ_{x}u(aΔ_{x}u - α∂_{t}u - ƒ(u) + g̃),

where u = (u^{1},...u^{k}) is an unknown vector-valued function, a and b are given constant matrices such that a + a* > 0, b = b* > 0, α > 0 is a positive number, and ƒ and g are given functions. Note that the nonlinearity ƒ is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for the system (1) is proved under some natural assumptions on the nonlinear term ƒ.

*Appeared in*

- Adv. Differential Equations, 7 (2002) pp. 1073--1100.

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# Numerical study of the statistical characteristics of the mixing processes in rivers

*Authors*

- Engelhardt, Christian
- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl K.
- Sukhodolov, Alexander

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Langevin type stochastic models, Random Displacement Models, Mixing process, ejection and sweep statistics

*DOI*

*Abstract*

A detailed analysis of statistical characteristics of the vertical mixing process in a horizontally homogeneous and stationary river flow is given. Stochastic models of Langevin type and random displacement models are developed to calculate the statistical characteristics of the vertical mixing. For validation, Langevin type models and random displacement models conventionally applied in this field are compared. All the methods show a good qualitative agreement. However the random diplacement model with constant coefficients is shown to perform with considerable deviations.

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# On a nonlocal model of non-isothermal phase separation

*Authors*

- Gajewski, Herbert

*2010 Mathematics Subject Classification*

- 35K45 35K57 35B40 80A20 80A22

*Keywords*

- Coupled Cahn-Hilliard equations, binary alloys, segregation model, nonlocal interaction, free energy, entropy, Onsager relations, initial boundary value problem, global existence and uniqueness, Lyapunov function, asymptotic behaviour

*DOI*

*Abstract*

A nonlocal model of non-isothermal phase separation in binary alloys is presented. The model is deduced from a free energy with a nonconvex part taking into account nonlocal particle interaction. The model consists of a system of second order parabolic evolution equations for heat and mass, coupled by nonlinear drift terms and a state equation which involves a nonlocal interaction potential. The negative entropy turns out to be Lyapunov functional of the system and yields the key estimate for proving global existence and uniqueness results and for analyzing the asymptotic behaviour as time goes to infinity.

*Appeared in*

- Adv. Math. Sci. Appl., 12 (2002) pp. 569--586.

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# A special reaction-diffusion system --- The pseudo-steady-state case

*Authors*

- Zacharias, Klaus

*2010 Mathematics Subject Classification*

- 35K45 35K57 80A30 92E20

*Keywords*

- Initial boundary value problems, reaction-diffusion equations, chemical reactions, chemical kinetics

*DOI*

*Abstract*

A system of reaction-diffusion equations modelling the diffusion of iodine and its reaction with radicals in a thin layer of radiation-activated polyethylene is considered. A reduced model, the pseudo-steady-state case, is investigated.

*Appeared in*

- ZAMM Z. Angew. Math. Mech. 78 (1998)

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# A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Verdi, Claudio

*2010 Mathematics Subject Classification*

- 65M15 35K60 80A22 35K45

*Keywords*

- phase-field model, semidiscretization, a posteriori error estimates

*DOI*

*Abstract*

A time discrete scheme is used to approximate the solution to a phase field system of Penrose-Fife type with a non-conserved order parameter. An a posteriori error estimate is presented that allows to estimate the difference between continuous and semidiscrete solutions by quantities that can be calculated from the approximation and given data.

*Appeared in*

- IMA Journal of Numerical Analysis, 23 (2003), pp. 55-80

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# Mutually catalytic branching in the plane: Infinite measure states

*Authors*

- Dawson, Donald A.
- Etheridge, Alison M.
- Fleischmann, Klaus
- Mytnik, Leonid
- Perkins, Edwin A.
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Catalyst, reactant, measure-valued branching, interactive branching, state-dependent branching, two-dimensional process, absolute continuity, self-similarity, collision measure, collision local time, martingale problem, moment equations, segregation of types, coexistence of types, self-duality, long-term behavior, scaling, Feynman integral

*DOI*

*Abstract*

A two-type infinite-measure-valued population in R^{2} is constructed which undergoes diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a collision rate sufficiently small compared with the diffusion rate, the model is constructed as a pair of infinite-measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit (in law), local extinction of one type is shown. The process constructed is a rescaled limit of the corresponding Z^{2} lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model under critical scaling. This part of a trilogy extends results from the finite-measure-valued case, whereas uniqueness questions are again deferred to the third part.

*Appeared in*

- Electron. J. Probab. 7 (2002), No. 15, 61 pp.

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# Strong clumping of super-Brownian motion in a stable catalytic medium

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus
- Mörters, Peter

*2010 Mathematics Subject Classification*

- 60K37 60K35 60J80 60G57 60F05

*Keywords*

- catalytic super-Brownian motion, stable catalysts, critical branching, measure-valued branching, random medium, clumping, functional limit law, historical superprocess, Brownian snake in a random medium, subordination, exit measures, good and bad paths, stopped measures, collision local time, heavy tails, Feynman-Kac formula, annealed and quenched random medium approach

*DOI*

*Abstract*

A typical feature of the long time behaviour of continuous super-Brownian motion in a stable catalytic medium is the development of large mass clumps or clusters at spatially rare sites. We describe this phenomenon by means of a functional limit law under renormalisation. The limiting process is a Poisson point field of mass clumps with no spatial motion component and with infinite variance. The mass of each cluster evolves independently according to a continuous process trapped at mass zero, which we describe explicitly by means of a Brownian snake construction in a random medium. We also determine the survival probability and asymptotic size of the clumps.

*Appeared in*

- Ann. Probab. 30(4) (2002), pp. 1990-2045

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# A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H99 60H10 34F05 34E15

*Keywords*

- Stochastic resonance, noise-induced synchronization, double-well potential, additive noise, random dynamical systems, non-autonomous stochastic differential equations, singular perturbations, pathwise description, concentration of measure

*DOI*

*Abstract*

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period 1/ε where ε is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity σ crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to 1/2 in the symmetric case, and exponentially close to 1 in the asymmetric case. The transition zones are localised in time near the points of minimal barrier height. We give a mathematically rigorous description of the behaviour of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on ε and on the minimal barrier height, as well as the asymptotics of the transition and non-transition probabilities.

*Appeared in*

- Ann. Appl. Probab. 12, 1419-1470 (2002)

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# On a nonlocal phase separation model

*Authors*

- Gajewski, Herbert
- Zacharias, Klaus

*2010 Mathematics Subject Classification*

- 35K45 35K57 35B40 80A22 92C15 92D25

*Keywords*

- Cahn-Hilliard equation, initial boundary value problem, reaction-diffusion equations, a priori estimates, Lyapunov function, equilibria, asymptotic behaviour, classical thermodynamics, phase changes, chemotaxis

*DOI*

*Abstract*

An alternative to the Cahn-Hilliard model of phase separation for two-phase systems in a simplified isothermal case is given. It introduces nonlocal terms and allows reasonable bounds for the concentrations. Using the free energy as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.

*Appeared in*

- J. Math. Anal. Appl., 286 (2003) pp. 11--31.

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# Parameter estimation and geometrical optimal design for Bingham measurement devices

*Authors*

- Logashenko, Dmitriy
- Maar, Bernd
- Schulz, Volker

ORCID: 0000-0001-7665-130X - Wittum, Gabriel

*2010 Mathematics Subject Classification*

- 65K05 65N22 76A05 90C52

*Keywords*

- Parameter estimation, optimal experimental design, Bingham fluids.

*DOI*

*Abstract*

Bingham models are frequently used for describing the flow of pastes. Usually, Bingham material parameters have to be determined in a rather cumbersome and time consuming manner. In this paper we develop a parameter estimation method for the automatic numerical determination of certain model parameters. Theresult is a tool for the simultaneous determination of all model parameters by using data from a single experiment sweep. Additionally, a method is presented to compute optimal shapes of corresponding measurement devices which lead to a high reliability of the resulting parameter estimation.

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# Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics

*Authors*

- Sprekels, Jürgen
- Krejčí, Pavel

*2010 Mathematics Subject Classification*

- 34C55 35K60 47J40 74N30 80A22

*Keywords*

- Phase-field systems, phase transitions, hysteresis operators, parabolic systems, Prandtl-Ishlinskii operators

*DOI*

*Abstract*

Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase-field models in which hysteresis nonlinearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so-called Prandtl-Ishlinskii operators. For these operators, the corresponding phase-field systems are in the multi-dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi-dimensional Prandtl-Ishlinskii operators having non-polyhedral convex characteristic sets.

*Appeared in*

- Math. Methods Appl. Sci.,25 (2002), pp. 309--325

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# Aging in the random energy model

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Gayrard, Véronique

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

- aging, Glauber dynamics, random energy model, trap models, metastability, extreme values

*DOI*

*Abstract*

In this letter we announce rigorous results on the phenomenon of aging in the Glauber dynamics of the random energy model and their relation to Bouchaud's 'REM-like' trap model. We show that, below the critical temperature, if we consider a time-scale that diverges with the system size in such a way that equilibrium is almost, but not quite reached on that scale, a suitably defined autocorrelation function has the same asymptotic behaviour than its analog in the trap model.

*Appeared in*

- Phys.Rev.Letts. 88, 087201

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# Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions

*Authors*

- Sprekels, Jürgen
- Zheng, Songmu

*2010 Mathematics Subject Classification*

- 35B40 35K50 45J05 45K05

*Keywords*

- Phase transitions, nonlocal models, initial-boundary value problems, a priori estimates, asymptotic behaviour, well-posedness, integrodifferential equations

*DOI*

*Abstract*

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krejčí and the authors for phase-field systems involving hysteresis operators.

*Appeared in*

- J. Math. Anal. Appl., 279 (2003), pp. 97-110

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# An analytic approach to a generalized Naghdi shell model

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 74K25 35Q72

*Keywords*

- Shells with little regularity, simplified approach

*DOI*

*Abstract*

In this paper a shell model of generalized Naghdi type is studied which requires only low regularity conditions. It is shown that the corresponding system of linear variational equations (representing a boundary value problem for a linear system of six partial differential equations on the shell) admits a unique solution. The main step in the proof is to show the coercivity of the corresponding bilinear form which is equivalent to a Korn inequality in curvilinear coordinates. In this paper, a direct approximation argument is used for the proof of coercivity.

*Appeared in*

- Adv. Math. Sci. Appl., 12/1 (2002), 175--190

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# Relaxation properties of a 1D flow through a porous material without and with adsorption

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152 - Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 76S05 76E20 74J05

*Keywords*

- flows in porous media, stability of geophysical flows, linear waves

*DOI*

*Abstract*

In this paper we investigate relaxation properties of a 1D steady state flow in a porous medium which is linearly perturbed in flow direction. We consider two cases of relaxation: without adsorption and with adsorption. The fields are assumed to be a superposition of a stationary (nonuniform) solution and of infinitesimal disturbances in the form of a linear wave ansatz. We show that such flows are absolutely stable with respect to longitudinal disturbances. It means that a smaller real part of the exponent in this ansatz yields a faster relaxation of the perturbation and the flow recovers faster the equilibrium. We solve numerically the eigenvalue problem for the first step field equations using a finite difference scheme and compare the results for the perturbation without mass exchange with the analytical solution. Calculations demonstrate the range of permeability coefficients with the fastest relaxation and the fastest convergence of numerical solutions.

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# An efficient stochastic chemistry approximation for the PDF transport equation

*Authors*

- Kraft, Markus

ORCID: 0000-0002-4293-8924 - Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C35 60K40

*Keywords*

- PDF transport equation, particle method, stochastic chemistry approximation, convergence, efficiency

*DOI*

*Abstract*

In this paper we present an efficient algorithm for the numerical treatment of the PDF transport equation. Using the partially stirred plug flow model in conjunction with the IEM mixing model we construct a numerical scheme that is based on a time splitting technique and a stochastic chemistry approximation. For this purpose a particle/sub-particle system is introduced. The dynamics of this particle system is determined by a mixing step and a chemistry step. The chemistry step is solved by a jump process where forward and reverse reactions are combined. Various numerical experiments are carried out to study convergence with respect to particle number and sub-particle number. In case of a linear reaction, the comparison between analytical solution and numerical approximation of the third moment reveals that the systematic error is inversely proportional to the number of particles and sub-particles, respectively. The performance of the algorithm is evaluated by studying the combustion of a premixed stoichiometric mixture of n-heptane and air. The stochastic chemistry algorithm is compared with a deterministic approach using the ODE solver DASSL and it is found, for the examples studied, that the stochastic algorithm is more efficient than the deterministic approach.

*Appeared in*

- Monte Carlo Methods Appl. 8 (2002), pp. 371-394

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# Inverse problem of diffractive optics: Conditional stability

*Authors*

- Bruckner, Gottfried
- Cheng, Jin
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35J05 78A47

*Keywords*

- inverse scattering, perfectly reflecting periodic grating, profile reconstruction, far field data, conditional stability, Helmholtz equation

*DOI*

*Abstract*

In this paper we prove conditional stability for an inverse problem in diffractive optics of determining a periodic curve from far field observations on a segment, in the case of perfect reflection. Our proof is based on a Carleman estimate for the Laplace operator.

*Appeared in*

- Inverse Problems 18 (2002), pp. 415-433

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# Stochastic, analytic and numerical aspects of coagulation processes

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C35 60K40

*Keywords*

- Coagulation, gelation effect, stochastic algorithms

*DOI*

*Abstract*

In this paper we review recent results concerning stochastic models for coagulation processes and their relationship to deterministic equations. Open problems related to the gelation effect are discussed. Finally we present some new conjectures based on numerical experiments performed with stochastic algorithms.

*Appeared in*

- Math. Comput. Simulation, 62 (2003), pp.265-275

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# Modellierung und Simulation von Bauelementen der Nano- und Optoelektronik

*Authors*

- Gajewski, Herbert
- Glitzky, Annegret

ORCID: 0000-0003-1995-5491 - Griepentrog, Jens André
- Hünlich, Rolf
- Kaiser, Hans-Christoph
- Rehberg, Joachim
- Röpke, Wilfried
- Stephan, Holger
- Wenzel, Hans

*DOI*

*Abstract*

In vielen Zweigen der modernen Technik spielen nano- und optoelektronische Bauelemente eine wichtige Rolle. Zu ihrer Entwicklung sind mathematische Modellierung und numerische Simulation unverzichtbare Hilfsmittel. Am Weierstraß-Institut wurde in den letzten Jahren intensiv an der mathematischen Modellierung von Technologieschritten zur Herstellung von Bauelementen und der in ihnen ablaufenden Ladungstransportprozesse gearbeitet. Auf der Grundlage der Analyse der beschreibenden Systeme nichtlinearer partieller Differentialgleichungen entstanden die weithin akzeptierten Simulationsprogramme DIOS (Diffusion, Implantation, Oxidation in Semiconductors) und ToSCA (Two-dimensional Semiconductor Analysis Package).

Im folgenden wird die Nutzung unserer Ergebnisse zur Entwicklung von Silizium-Germanium-Heterobipolartransistoren bzw. Quantum-Well-Halbleiterlasern im Rahmen zweier, durch das BMBF gefoerderter Projekte beschrieben. Beide Projekte leben von enger interdisziplinaerer Kooperation mit unseren Partnern vom Institut für Halbleiterphysik Frankfurt(Oder) bzw. vom Ferdinand-Braun-Institut für Höchstfrequenztechnik Berlin.

*Appeared in*

- Mathematik-Schluesseltechnologie fuer die Zukunft, Springer, 1997, pp. 303-313

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# Suboptimal control of laser surface hardening using proper orthogonal decomposition

*Authors*

- Hömberg, Dietmar
- Volkwein, Stefan

*2010 Mathematics Subject Classification*

- 35Kxx 49J20 49K20 65Nxx

*Keywords*

- Laser hardening, optimality conditions, properorthogonal decomposition, error estimates, suboptimal control

*DOI*

*Abstract*

Laser surface hardening of steel is formulated in terms of an optimal control problem, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the high temperature phase. The optimal control problem is analyzed and first-order necessary optimality conditions are derived. An error estimate for POD (proper orthogonal decomposition) Galerkin methods for the state system is proved. Finally a strategy to obtain suboptimal controls using POD is developed and validated by computing some numerical examples.

*Appeared in*

- Math. Comput. Modelling, 37 (2003), pp. 1003-1028 under new title: Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition.

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# The moment Lyapunov exponent for conservative systems with small periodic and random perturbations

*Authors*

- Imkeller, Peter
- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H10 93E15

*Keywords*

- Linear stochastic systems with periodic coefficients, stochastic stability, moment Lyapunov exponent, stability index, Hill and Mathieu equations with random excitations

*DOI*

*Abstract*

Much effort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic differential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coefficients. Most extensively we study these problems for second order conservative systems with small random and periodic excitations. With respect to relations between the intrinsic period of the system and the period of perturbations we consider the incommensurable and commensurable cases. In the first case we obtain an asymptotic expansion of the moment Lyapunov exponent. In the second case we obtain a finite expansion except in situations of resonance. As an application we consider the Hill and Mathieu equations with random excitations.

*Appeared in*

- Stochastics and Dynamics, vo. 2 (2002), no. 1, pp. 25-48

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# Numerical parameter identification in multiphase flow through porous media

*Authors*

- Hazra, Subhendu Bikash
- Schulz, Volker

ORCID: 0000-0001-7665-130X

*2010 Mathematics Subject Classification*

- 65K05 76T10 65L10 93E24

*Keywords*

- Numerical parameter identification, multiphase flow, reduced Gauss-Newton technique, multiple shooting

*DOI*

*Abstract*

Multiphase flow is of high interest for the investigation of the behavior of waste in groundwater. The high nonlinearity of the model equations pose special problems. Here, a new parameter identification technique in this context is proposed which takes advantage of recently developed highly efficient numerical simulation techniques. It is based on a reduced Gauss-Newton technique in combination with an efficient gradient computation. Numerical experiments are performed for the McWhorter model problem.

*Appeared in*

- Computing and Visualization in Science, 5 (2002), pp. 1107-113

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# Error bounds and their application

*Authors*

- Bosch, Paul
- Jourani, Abderrahim
- Henrion, René

*2010 Mathematics Subject Classification*

- 90C31 49J52

*Keywords*

- Error Bounds, Approximate Subdifferential, Sensitivity Analysis, Local Controllability Error Bounds, Local Controllability

*DOI*

*Abstract*

Our aim in this paper is to present sufficient conditions for error bounds in terms of Frechet and limiting Frechet subdifferentials outside of Asplund spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (𝑥, 𝑦) ∈ 𝐶 × 𝐷, 𝑔(𝑥, 𝑦, 𝑢) = 0, where 𝑔 takes values in an infinite dimensional space and 𝑢 plays the role of a parameter. This symmetric structure offers us the choice to impose condtions either on 𝐶 or 𝐷. We use these results to prove nonemptyness and weak-star compactness of Fritz-John and Karuch-Kuhn-Tucker multiplier sets, to establish Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.

*Appeared in*

- Applied Mathematics and Optimization 50 (2004), pp. 161-181 under the new title: Sufficient Conditions for Error Bounds and Applications.

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# Singular limit in parabolic differential inclusions and the stop operator

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K85 35B25 47J40

*Keywords*

- Parabolic differential inclusion, singular limit, hysteresis operators, penalty approximation, phase transitions

*DOI*

*Abstract*

Parabolic differential inclusions with convex constraints in a finite-dimensional space are considered with a small "diffusion" coefficient ε in the elliptic term. This problem arises for instance in multicomponent phase-field systems. We prove the strong convergence of solutions as ε → 0 to the solution of the singular limit equation and show the connection to elementary hysteresis operators.

*Appeared in*

- Interfaces and Free Boundaries, 4 (2002), pp. 423-435

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# Random walk on spheres methods for iterative solution of elasticity problems

*Authors*

- Sabelfeld, Karl K.
- Shalimova, Irina

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Random Walk on Spheres Process, Schwarz iterations, Lame equation, elastic plates

*DOI*

*Abstract*

Random Walk on Spheres method for solving some 2D and 3D boundary value problems of elasticity theory are developed. The boundary value problems studied include the elastic thin plate problems with simply supported boundary, rigid fixing of the boundary, and general 2D and 3D problems for the Lamé equation. Unbiased estimators for some special classes of domains based on the generalized Mean Value Theorem which relates the solution at an arbitrary point inside the sphere with the integral of the solution over the sphere. We study a variance reduction technique based on the explicit simulation of the first passage of a sphere for the Wiener process starting at an arbitary point inside this sphere. Along with the conventional random walk methods we apply another type of iteration method, the Schwarz iterative procedure whose convergence for the Lamé equation was proved in 1936 by S.L. Sobolev. We construct also different types of iterative procedures which combine the probabilistic and conventional deterministic methods of solutions.

*Appeared in*

- Monte Carlo Methods Appl., 8 (2002) pp. 171--202.

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# Endogenous interest rate dynamics in asset markets

*Authors*

- Reiß, Oliver
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Schweizer, Martin

*2010 Mathematics Subject Classification*

- 91B28 60G35 91B70

*Keywords*

- asset market, interest rates, market structure, asset indices, endogenous dynamic relations

*DOI*

*Abstract*

Starting from a general Ito process model with more assets than driving Brownian motions, we study the term structure model endogenously induced by this complete market. In the Markovian diffusion case, we provide the resulting HJM description and point out a link to finite factor models. But the main contribution is the conceptual approach of considering assets and interest rates within one model which is completely specified by the assets alone. This allows endogenous derivations of dynamic relations between assets and interest rates from global structural assumptions (homogeneity and some spherical symmetry) on the market. Related issues in financial market modelling have been studied by E. Platen.

*Appeared in*

- Journal of Economic Dynamics & Control, 31, 2007, 593-612, under new title: From Structual Assumptions to a Link between Assets and Interest Rates.

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# Mean-square symplectic methods for Hamiltonian systems with multiplicative noise

*Authors*

- Milstein, Grigori N.
- Repin, Yuri M.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 60H10 65U05

*Keywords*

- Stochastic Hamiltonian systems, symplectic integration, implicit methods, mean-square convergence

*DOI*

*Abstract*

Stochastic systems with multiplicative noise, phase flows of which have integral invariants, are considered. For such systems, numerical methods preserving the integral invariants are constructed using full implicit schemes of a new type for stochastic differential equations. In these full implicit schemes increments of Wiener processes are substituted by some truncated random variables. They are important for both theory and practice of numerical integration of stochastic differential equations. A special attention is paid to systems with separable Hamiltonians and to Hamiltonian systems with small noise. Liouvillian methods for stochastic systems preserving phase volume are also proposed. Some results of numerical experiments are presented.

*Appeared in*

- SIAM J. on Numerical Analysis, vol. 40 (2003), no. 4, pp. 1583-1604, under new title: Numerical methods for stochastic systems preserving symplectic structure.

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# Symplectic methods for Hamiltonian systems with additive noise

*Authors*

- Milstein, Grigori N.
- Repin, Yuri M.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 60H10 65C30 65P10

*Keywords*

- Hamiltonian systems with additive noise, symplectic integration, mean-square methods for stochastic differential equations

*DOI*

*Abstract*

Stochastic systems, phase flows of which have integral invariants, are considered. Hamiltonian systems with additive noise being a wide class of such systems possess the property of preserving symplectic structure. For them, numerical methods preserving the symplectic structure are constructed. A special attention is paid to systems with separable Hamiltonians, to second order differential equations with additive noise, and to Hamiltonian systems with small additive noise.

*Appeared in*

- SIAM J. on Numerical Analysis, vol. 39 (2002), no. 6, pp. 2066-2088, under new title: Symplectic integration of Hamiltonian systems with additive noise.

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# Asymptotic analysis of surface waves at vacuum/porous medium and liquid/porous medium interfaces

*Authors*

- Edelman, Inna
- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 35C20 35L50 74J15

*Keywords*

- asymptotic expansions, waves in porous media

*DOI*

*Abstract*

Surface waves at a free interface of a saturated porous medium and at an interface between a porous medium and a liquid are investigated. Existence and peculiarities of such surface waves are revealed. At a free interface two types of surface waves are proved to be possible: the true Stoneley wave, propagating almost without attenuation, and the leaky generalized Rayleigh wave, which reradiates a part of its energy into interior of a medium. At a porous medium/liquid interface three types of surface waves are expected. These are the true Stoneley wave, the pseudo Stoneley wave, and the generalized Rayleigh wave.

*Appeared in*

- Contin. Mech. Thermodyn. 14 (2002), pp. 25--44

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# The time-varying stabilization of linear discrete control systems

*Authors*

- Leonov, Gennadi A.

*2010 Mathematics Subject Classification*

- 93D15 93C55

*Keywords*

- stabilization, linear control, discrete system, feedback, transfer function

*DOI*

*Abstract*

The Brockett stabilization problem for linear discrete control systems is considered. The method of synthesis of time-varying feedback for stabilization is described.

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# The asymptotic behavior of semi-invariants for linear stochastic systems

*Authors*

- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H10 93E15

*Keywords*

- Stochastic stability, moment Lyapunov exponent, analytic characteristic function, semi-invariants

*DOI*

*Abstract*

The asymptotic behavior of semi-invariants of the random variable ln ｜X(t, 𝑥)｜, where X(t,𝑥) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(𝑝). Namely, it is obtained that the 𝑛-th semi-invariant is asymptotically proportional to the time 𝗍 with the coefficient of proportionallity g^{(n)}(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln ｜X(t, 𝑥)｜ in a neighbourhood of the origin on the complex plane is controlled by the extension g(𝑖𝓏) of g(𝑝).

*Appeared in*

- Stochatics and Dynamics, vol. 2 (2002), no.2, pp.281-294

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# On estimation of the linearized drift for nonlinear stochastic differential equations

*Authors*

- Khasminskii, Rafail
- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H10 62F 93E15

*Keywords*

- asymptotic efficiency, maximum likelihood estimator, local asymptotic normality, stochastic stability

*DOI*

*Abstract*

The estimation of linearized drift for stochastic differential equations with equilibrium points is considered. It is proved that the linearized drift matrix can be estimated efficiently if the initial condition for the system is chosen close enough to the equilibrium point. Some bounds for initial conditions providing the asymptotical efficiency of estimators are found.

*Appeared in*

- Stochastics and Dynamics, vol. 1 (2001), no. 1, pp. 23-43

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# Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael
- Kunik, Matthias

*2010 Mathematics Subject Classification*

- 82C40 82C70

*Keywords*

- Kinetic theory of phonons, Maximum Entropy Principle

*DOI*

*Abstract*

The evolution of heat in crystalline solids is described at low temperatures by the Boltzmann-Peierls-Equation which is a kinetic equation for the phase density of phonons. In this study we solve initial value problems for the Boltzmann-Peierls-Equation with respect to the following questionings: In thermodynamics, a given kinetic equation is usually replaced by its truncated moment systems which in turn is supplemented by a closure principle so that there results a system of PDE's for some moments as thermodynamic variables. A very popular closure principle is the Maximum Entropy Principle yielding a symmetric hyperbolic system. In recent times this strategy has lead to serious studies on two problems that might arise. 1. Do solutions of the Maximum Entropy Principle exist? 2. Is the physics which is embodied in the kinetic equation more or less equivalently displayed by the truncated moment system? It was Junk who proved for the Boltzmann equation of gases that Maximum Entropy solutions do not exist. The same failure appears for the Fokker-Planck-Equation, which was proved by means of explicit solutions by Dreyer/Junk/Kunik. The current study yields a positive existence result. We prove for the Boltzmann-Peierls-Equation hat the Maximum Entropy Principle is well posed and that it can be used to establish a closed hyperbolic moment system of PDE's. Regarding the second question on the equivalence of moments that are calculated by solutions of the Boltzmann-Peierls-Equation and moments that result from the Maximum Entropy system we develop a numerical method that allows a comparison of both solutions. In particular, we introduce a numerical kinetic scheme that consists of free flight periods and two classes of update rules. The first class of rules are the same for the kinetic equation as well as for the Maximum Entropy system, while the second class of update rules contain additional rules for the Maximum Entropy system. It is illustrated that if sufficient many moments are taken into account, both solutions converge to each other. However, it is additionally illustrated, that the numerical effort to solve the kinetic equation is less than the effort to solve the Maximum Entropy system.

*Appeared in*

- Contin. Mech. Thermodyn. 16 (2004), pp. 453--469

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# Stochastic Lagrangian model for spatially inhomogeneous Smoluchowski equation governing coagulating and diffusing particles

*Authors*

- Kolodko, Anastasia
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 76F99

*Keywords*

- Smoluchowski coagulation equation, turbulent mixing, Lagrangian stochastic model

*DOI*

*Abstract*

The following generally unsolved yet problem is studied: construct the solution of a spatially inhomogeneous Smoluchowski equation governing coagulating and diffusing particles in a host gas, on the basis of solutions to homogeneous Smoluchowski equation. In citeksw we solved this problem in the case when there is no diffusion. The non-zero diffusion term drastically complicates the situation. Under some general assumptions we give the interrelations between the homogeneous and inhomogeneous cases. This provides an effective numerical scheme especially when the host gas is incompressible. New Lagrangian scheme leads to a new model governing by a Smoluchowski type equation with an additional effective source. We give a numerical comparison of these two models. The computer time of the new algorithm is so dramatically decreased, compared to the conventional deterministic algorithm (tens of hours drop down to several minutes) that many practical problems like the formation of soot particles in flames or chemical reactions coupled to formation of a new phase can be solved in a reasonable computer time. However this method works only if the diffusion coefficient of all particles is the same which can be a reasonable approximation only for special systems. The problem of generalisation of the method presented to the case when the diffusion coefficient depends on the particle's size is open.

*Appeared in*

- Monte Carlo Methods Appl., 7 (2001) pp. 223--228.

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# Transition density estimation for stochastic differential equations via forward-reverse representations

*Authors*

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

ORCID: 0000-0002-4389-8266 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G07 60H10 65C05

*Keywords*

- transition density, forward and reverse diffusion, statistical estimation, Monte Carlo simulation

*DOI*

*Abstract*

The general reverse diffusion equations are derived. They are applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem it is shown that density estimation based on forward-reverse representations allows for achieving essentially better results in comparison with usual kernel or projection estimation based on forward representations only.

*Appeared in*

- Bernoulli, vol. 10(2), 2004, pp. 281--312

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# Phase-field models with hysteresis in one-dimensional thermo-visco-plasticity

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen
- Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 34C55 35K60 47J40 74K05 74N30 80A22

*Keywords*

- Phase-field systems, phase transitions, hysteresis operators, thermo-visco-plasticity, thermodynamic consistency

*DOI*

*Abstract*

The mathematical modelling of nonlinear thermo-visco-plastic developments and of phase transitions in solids have drawn much attention in past years. On the one hand, one is interested how phase transformations on the micro- and/or mesoscales (for instance, between different geometric configurations of the crystal lattice) influence the global thermo-visco-plastic behaviour, on the other hand, the global evolution of solid-solid phase transformations is strongly affected by the presence of micro- and/or mesoscopic stresses. In such situations, a typical macroscopic phenomenon is the occurrence of hysteresis effects, and it is therefore important to model these effects. This paper is a contribution towards this direction. A new one-dimensional model is considered that incorporates both the occurrence of hysteresis effects and of phase transitions. In this connection, the phase transition is described by the evolution of a phase-field (which is usually closely related to an order parameter of the phase transition), while the hysteresis effects are accounted for using the mathematical theory of hysteresis operators developed in the past thirty years. The model extends recent works of the first two authors on phase-field models with hysteresis to the case when mechanical effects can no longer be ignored or even prevail. It leads to a strongly nonlinear coupled system of partial differential equations in which hysteresis nonlinearities occur at several places, even under time and space derivatives. We show the thermodynamic consistency of the model, and we prove its well-posedness.

*Appeared in*

- SIAM J. Math. Anal., Vol.34, 2 (2002), pp. 409-434

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# Mass-time-space scaling of a super-Brownian catalyst reactant pair

*Authors*

- Fleischmann, Klaus
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Catalyst, reactant, superprocess, martingale problem, stochastic equation, density field, collision measure, colision local time, extinction, critical scaling

*DOI*

*Abstract*

The one-dimensional super-Brownian reactant X^{ϱ} with a super-Brownian catalyst ϱ has a jointly continuous density field satisfying a stochastic partial differential equation. Consider any expectation preserving mass-time-space scaling of X^{ϱ}. Using the density field, one can pass to an fdd scaling limit of the measure-valued process, which degenerates also under the critical scaling of ϱ. For some of the scaling indexes, convergence on path spaces holds, too.

*Appeared in*

- J. Theoret. Probab., 19 (2006) pp. 557-588.

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# Flow instabilities in granular media due to porosity inhomogeneities

*Authors*

- Wilhelm, Theo
- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 86-99 76E20 73Q05 76S05 76T05

*Keywords*

- Piping, channeling, flow instability in granular materials (fluidization), multicomponent model of saturated soils

*DOI*

*Abstract*

The paper concerns a theoretical description of the piping phenomenon appearing in saturated sands at high filtration velocities. Motivated by own experiments we propose a thermodynamical two component model which accounts for a threshold effect at a critical value of the relative velocity of components. This property is incorporated in the source term of momentum balance equations by means of a nonlinear contribution accounting for spatial variations of the porosity. We prove the thermodynamical admissibility of such a model. By means of a linear stability analysis we show the existence of the onse of instability for realistic values of material parameters gained from experiments.

*Appeared in*

- Internat. J. Multiphase Flows 28 (2002), pp. 1929--1944

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# Mass exchange, diffusion and large deformations of poroelastic materials

*Authors*

- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 80A15 76S05 73S10 73D40 76R50

*Keywords*

- Flows in porous media, micromechanics of solids, interfaces in porous media

*DOI*

*Abstract*

The paper contains a review of fundamental equations of the two component thermoporoelastic materials with the balance equation of porosity. By means of the exploitation of the second law of thermodynamics restricted to small deviations from thermodynamical equilibrium it is proven that there exists no thermodiffusional coupling of components through intrinsic parts of fluxes. Certainly such a coupling is still present due to convective contributions. Simultaneously we show that classical partial dynamical compatibility conditions on material interfaces cannot hold. For boundary conditions on permeable boundaries to hold true it must be required that global balance equations contain at least surface sources of momentum, entropy, and porosity. We show as well that the requirement of the local thermodynamical equilibrium on permeable interfaces yields the continuity of absolute temperature. It means that temperature becomes a measurable physical field in porous materials undergoing processes with small deviations from thermodynamical equilibria. This result allows to extend models of mass exchange in poroelastic materials from adsorption isothermal processes to chemical reactions, and phase transformations. Details of the latter problems are not discussed in this paper.

*Appeared in*

- Modeling and Mechanics of Granular and Porous Materials, G. Capriz, V. N. Ghionna, P. Giovine (eds.) pp. 211--242, Birkhaeuser, 2002

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# On calmness of a class of multifunctions

*Authors*

- Henrion, René
- Jourani, Abderrahim
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 90C31 49J52

*Keywords*

- calmness, multifunctions, constraint qualifications, nonsmooth calculus, solution stability, equilibrium problems, weak sharp minima

*DOI*

*Abstract*

The paper deals with calmness of a class of multifunctions in finite dimensions. Its first part is devoted to various calmness criteria which are derived in terms of coderivatives and subdifferentials. The second part demonstrates the importance of calmness in several areas of nonsmoooth analysis. In particular, we focus on nonsmooth calculus and solution stability in mathematical programming and in equilibrium problems. The derived conditions find a number of applications there.

*Appeared in*

- SIAM Journal on Optimization, 13 (2002) pp. 603-618

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# On a class of compactly epi-Lipschitzian sets

*Authors*

- Jourani, Abderrahim

*2010 Mathematics Subject Classification*

- 49J52

*Keywords*

- nonsmooth analysis, compactly epi-lipschitzian sets, locally compact cones

*DOI*

*Abstract*

The paper is devoted to the study of the so-called compactly epi-Lipschitzian sets. These sets are needed for many aspects of generalized differentiation, particulary for necessary optimality conditions, stability of mathematical programming problems and calculus rules for subdifferentials and normal cones. We present general conditions under which sets defined by general constraints are compactly epi-Lipschitzian. This allows us to show how the compact epi-Lipschitzness properties behave under set intersections.

*Appeared in*

- Nonlinear Anal. 54 (2003), no. 3, pp. 471-483

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# Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter

*2010 Mathematics Subject Classification*

- 80A20 65M99 65Z05 35K55 78A55 78M25

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 81.10.Bk 84.32.Hh 44.30.+v 44.90.+c

*Keywords*

- numerical simulation, sublimation growth, physical vapor transport, modified Lely method, SiC single crystal, induction heating, heat transfer

*DOI*

*Abstract*

This article presents transient numerical simulations of the temperature evolution during sublimation growth of SiC single crystals via physical vapor transport (also called the modified Lely method) including diffusion and radiation, investigating the influence of induction heating. Using the imposed voltage as input data, the heat sources are computed via an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic PDE. The presented results include stationary simulations of magnetic potential distributions and resulting heat sources as well as transient simulations of the temperature evolution during the heating process. We examine the effects of imposed voltage (i.e. heating power), of different coil positions, and of a moving induction coil on the evolution of the global temperature field and on the temperature at the source, at the seed, and at the blind holes. All material data used are either included or referenced.

*Appeared in*

- Journal of Crystal Growth, 247 (2003), pp. 219-235

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# Phase-field systems with vectorial order parameters including diffusional hysteresis effects

*Authors*

- Kenmochi, Nobuyuki
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K45 35K50 47J40 80A20 80A22

*Keywords*

- Parabolic systems, phase-field models, hysteresis, a priori estimates, existence, uniqueness, phase transitions

*DOI*

*Abstract*

This paper is concerned with phase-field systems of Penrose-Fife type which model the dynamics of a phase transition with non-conserved vectorial order parameter. The main novelty of the model is that the evolution of the order parameter vector is governed by a system consisting of one partial differential equation and one partial differential inclusion, which in the simplest case may be viewed as a diffusive approximation of the so-called multi-dimensional stop operator, which is one of the fundamental hysteresis operators. Results concerning existence, uniqueness and continuous dependence on data are presented which can be viewed as generalizations of recent results by the authors to cases where a diffusive hysteresis occurs.

*Appeared in*

- Comm. Pure Appl. Anal. 4 (2002), pp. 495-511

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# Excitability of lasers with integrated dispersive reflector

*Authors*

- Tronciu, Vasile Z.
- Wünsche, Hans-Jürgen
- Schneider, Klaus R.
- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 78A60 34C60

*Keywords*

- semiconductor laser, excitability, dispersive reflector, optical feedback

*DOI*

*Abstract*

This paper is concerned with the phenomenon of excitability in semiconductor lasers consisting of a DFB section and a passive dispersive reflector (PDR). We assume that the PDR section contains a Bragg grating and (or) a passive Fabry Perot filter guaranteeing a dispersive reflection of the optical field. We investigate a single mode model for PDR lasers and derive conditions under which excitable behavior can be demonstrated. Especially, we show the existence of a threshold, that is, only perturbations above the threshold imply a large excursion from the steady state, and where the response is almost independent of the strength of the perturbation, moreover we establish the existence of a refractory period, i.e., if a second perturbation is applied before the refractory time has passed, then the system does not respond. Finally, we discuss the importance of excitability for the transmission of signals in communication networks.

*Appeared in*

- SPIE Proceedings Series, vol 4283, pp. 347-354, (2001)

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# Finite element solution of conical diffraction problems

*Authors*

- Elschner, Johannes
- Hinder, Rainer
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 78M10 65N30

*Keywords*

- Conical diffraction, system of Helmholtz equations, transmission problem, strongly elliptic variational formulation, finite element solution

*DOI*

*Abstract*

This paper is devoted to the numerical study of diffraction by periodic structures of plane waves under oblique incidence. For this situation Maxwell's equations can be reduced to a system of two Helmholtz equations in ℝ^{2} coupled via quasiperiodic transmission conditions on the piecewise smooth interfaces between different materials. The numerical analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for discrete solutions and provide the corresponding error analysis.

*Appeared in*

- Advances in Computational Mathematics 16 (2002), pp. 139-156

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# Modeling diffusional coarsening in microelectronic solders

*Authors*

- Dreyer, Wolfgang
- Müller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 76R50 82B26 82B24 35K30 35K35

*Keywords*

- Diffusion, phase transisions, interface problems, initial value problems for higher order parabolic equations, boundary value problems for higher order parabolic equations, solders, lead, tin, lead-free, coarsening, aging

*DOI*

*Abstract*

This paper presents a detailed numerical simulation of the coarsening phenomenon observed in microelectronic solder materials that are subjected to high homologeous temperatures in combination with thermo-mechanical stresses. The simulations are based on a phase field model which, for simplicity, is explicitly formulated for a binary alloy. To this end, the thermomechanical stresses originating within a Representative Volume Element (RVE) of the solder material are calculated first. This is achieved by means of a closed-form solution of the Navier equations resulting in explicit expressions for the displacements of an anisotropic, heterogeneous, thermally stressed elastic medium in discrete Fourier space. Inverse discrete Fourier transforms are then applied to these expressions in order to obtain the local stresses in real space. These in turn are inserted into an extended expression for the diffusion flux, which, in addition to the classical driving force of a concentration gradient takes the influence of different surface tensions between the solder phases as well as the local strain energy into account. The equations are evaluated numerically for the exemplary case of eutectic SnPb solder, for which all material constants are known explicitly. A comparison with aging experiments is performed.

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# On a model for phase separation in binary alloys driven by mechanical effects

*Authors*

- Bonetti, Elena
- Colli, Pierluigi
- Dreyer, Wolfgang
- Gilardi, Gianni
- Schimperna, Giulio
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K45 35K50 74N15 74N20 74N25 74M25

*Keywords*

- Free boundary value problems, phase-field models, phase transitions in mechanical stress fields, a priori estimates, existence, uniqueness

*DOI*

*Abstract*

This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary) balance of linear momentum with a fourth order evolution equation of Cahn_Hilliard type for the phase separation, and it is highly nonlinearly coupled. Existence and uniqueness results are shown.

*Appeared in*

- Phys. D, 165 (2002), pp. 48--65

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# On adaptive inverse estimating linear functionals of unknown smoothness in Hilbert scales

*Authors*

- Goldenshluger, Alexander
- Pereverzev, Sergei V.

*2010 Mathematics Subject Classification*

- 62G05 62G20 65R30

*Keywords*

- adaptive estimation, Hilbert scales, inverse problems, linear functionals, minimax risk, regularization

*DOI*

*Abstract*

We address the problem of estimating the value of a linear functional ⟨ ƒ,𝑥 ⟩ from random noisy observations of 𝑦 = A 𝑥 in Hilbert scales. Both the white noise and density observation models are considered. We develop an inverse estimator of ⟨ ƒ,𝑥 ⟩ which automatically adapts to unknown smoothness of 𝑥 and ƒ. It is shown that accuracy of this adaptive estimator is only by a logarithmic factor worse than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.

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# Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics

*Authors*

- Bovier, Anton
- Manzo, Francesco

*2010 Mathematics Subject Classification*

- 60K35 82C20

*Keywords*

- Metastability, Markow chains, Glauber dynamics, kinetic Ising model

*DOI*

*Abstract*

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as T ↓ 0. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann [NS] and Ben Arous-Cerf [BC]. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein in [BEGK1,BEGK2].

*Appeared in*

- J. Statist. Phys. 107 (2002), pp. 757-779

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# One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen
- Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 74N30 47J40 34C55 35K60

*Keywords*

- Phase-field systems, phase transitions, hysteresis operators, thermo-visco-plasticity

*DOI*

*Abstract*

We consider a strongly coupled system of partial differential equations as a model for the dynamics of a thermo-visco-elasto-plastic solid under phase transitions. It consists of the momentum balance equation for the displacement, the energy balance equation for the absolute temperature, and an order parameter equation describing the dynamics of the phase transition. Both the phase transition and the strain-stress law involve hysteresis dependence represented by hysteresis operators. We show the thermodynamic consistency of the model, and prove its well-posedness.

*Appeared in*

- Adv. Math. Sci. Appl., Volume 13, No.2 (2003), 695-712

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# Universal bounds on the selfaveraging of random diffraction measures

*Authors*

- Külske, Christof

*2010 Mathematics Subject Classification*

- 78A45 82B44 60F10 82B20

*2008 Physics and Astronomy Classification Scheme*

- 61.10.Dp 05.50.+q 02.50.Cw

*Keywords*

- Diffraction theory, random scatterers, random point sets, quasicrystals, large deviations, cluster expansions

*DOI*

*Abstract*

We consider diffraction at random point scatterers on general discrete point sets in ℝ^{ν}, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence of the random scattering measures applied to an observable towards its mean, when the finite volume tends to infinity. We give an explicit universal large deviation upper bound that is exponential in the number of scatterers. The rate is given in terms of a universal function that depends on the point set only through the minimal distance between points, and on the observable only through a suitable Sobolev-norm. Our proof uses a cluster expansion and also provides a central limit theorem.

*Appeared in*

- Probab. Theory Related Fields, 126 (2003), pp. 29-50

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# Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H20 60H10 34E15 82C31

*Keywords*

- Stochastic resonance, dynamical hysteresis, bifurcation delay, double-well potential, first-exit time, scaling laws, Lorenzmodel, thermohaline circulation, white noise, coloured noise

*DOI*

*Abstract*

We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial amounts of time in metastable equilibrium, instead of adiabatically following the stationary distribution of the frozen system. This behaviour can be characterized by describing the location of typical paths, and bounding the probability of atypical paths. We illustrate this approach by giving a quantitative description of phenomena associated with bistability, for three famous examples of simple climate models: Stochastic resonance in an energy balance model describing Ice Ages, hysteresis in a box model for the Atlantic thermohaline circulation, and bifurcation delay in the case of the Lorenz model for Rayleigh-Bénard convection.

*Appeared in*

- Stoch. Dyn. 2:327-356 (2002), DOI 10.1142/S0219493702000455 WorldSciNet

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# An optimization method for grating profile reconstruction

*Authors*

- Bruckner, Gottfried
- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35J05 78A46 78M50

*Keywords*

- diffraction grating, profile reconstruction, Tikhonov regularization, optimization method, Levenberg-Marquardt algorithm.

*DOI*

*Abstract*

We consider the inverse diffraction problem to recover a two-dimensional periodic structure from scattered waves measured above the structure. Following an approach by Kirsch and Kress, the inverse problem is reformulated as a nonlinear optimization problem. The resulting Tikhonov regularized least squares problem is then solved iteratively by the Levenberg-Marquardt algorithm. Numerical results for synthetic data demonstrate the practicability of the inversion algorithm. We also present some convergence results for the Tikhonov regularization of the reconstruction problem and for the optimization method.

*Appeared in*

- Progress in Analysis, Proceed. 3rd ISAAC congress, World Scientific, Singapore, 2003, pp. 1391-1404

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# Correct voltage distribution for axisymmetric sinusoidal modeling of induction heating with prescribed current, voltage, or power

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter

*2010 Mathematics Subject Classification*

- 78A25 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 84.32.Hh, 02.60.Cb

*Keywords*

- Sinusoidal induction heating, axisymmetric modeling, voltage distribution, numerical simulation

*DOI*

*Abstract*

We consider the problem of determining the voltage in coil rings, which arise as an axisymmetric approximation of a single connected induction coil during modeling of induction heating. Assuming axisymmetric electromagnetic fields with sinusoidal time dependence, the voltages are computed from the condition that the total current must be equal in each ring. Depending on which of the quantities total current, total voltage, or total power is to be prescribed, the ring voltages are given by different linear systems of complex equations. In two sets of numerical simulations, varying the number of coil rings, we compare results using the correct voltage distribution to the corresponding results using the simple homogeneous voltage distribution.

*Appeared in*

- IEEE transactions on Magnetics, 38 (2002), no. 3, pp. 1519-1523

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# On a class of singularly perturbed partly dissipative reaction-diffusion systems

*Authors*

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

*2010 Mathematics Subject Classification*

- 35B25 35K57

*Keywords*

- Initial boundary value problem, singularly perturbed partly dissipative reaction-diffusion system, exchange of stabilities, asymptotic lower and upper solutions

*DOI*

*Abstract*

We consider the singularly perturbed partly dissipative reaction-diffusion system $ve^2 left(fracpartial upartial t - fracpartial^2 upartial x^2right) =g(u,v,x,t,ve), fracpartial vpartial t = f(u,v,x,t,ve)$ under the condition that the degenerate equation $g(u,v,t,0)=0$ has two solutions $u= varphi_i(v,x,t), i=1,2,$ that intersect (exchange of stabilities) and that $v$ is a vector. Our main result concerns existence and asymptotic behavior in $ve$ of the solution of the initial boundary value problem under consideration. The proof is based on the method of asymptotic lower and upper solutions.

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# Inverse scattering for periodic structures: Stability of polygonal interfaces

*Authors*

- Elschner, Johannes
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A46 35R30 35Q60

*Keywords*

- Inverse problem, diffraction grating, local stability, material derivative

*DOI*

*Abstract*

We consider the two-dimensional TE and TM diffraction problems for a time harmonic plane wave incident on a periodic grating structure. An inverse diffraction problem is to determine the grating profile from measured reflected and transmitted waves away from the structure. We present a new approach to this problem which is based on the material derivative with respect to the variation of the dielectric coefficient. This leads to local stability estimates in the case of interfaces with corner points.

*Appeared in*

- Inverse Problems 17 (2001), pp. 1817-1829

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# A quadrature algorithm for wavelet Galerkin methods

*Authors*

- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 65N38 65T60 65R20 65D30

*Keywords*

- wavelet Galerkin methods, first kind integral operator, quadrature algorithm

*DOI*

*Abstract*

We consider the wavelet Galerkin method for the solution of boundary integral equations of the first and second kind including integral operators of order r less than zero. This is supposed to be based on an abstract wavelet basis which spans piecewise polynomials of order d_{T}. For example, the bases can be chosen as the basis of tensor product interval wavelets defined over a set of parametrization patches. We define and analyze a quadrature algorithm for the wavelet Galerkin method which utilizes Smolyak quadrature rules of finite order. In particular, we prove that quadrature rules of an order larger than 2d_{T} - r are sufficient to compose a quadrature algorithm for the wavelet Galerkin scheme such that the compressed and quadrature approximated method converges with the maximal order 2d_{T} - r and such that the number of necessary arithmetic operations is less than 𝒪(N log N) with N the number of degrees of freedom. For the estimates, a degree of smoothness greater or equal to 2[2d_{T} - r]+1 is needed.

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# Macroscopic current induced boundary conditions for Schrödinger-type operators

*Authors*

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

*2010 Mathematics Subject Classification*

- 35P10 47A55 47B44 81Q15

*2008 Physics and Astronomy Classification Scheme*

- 03.65.Yz 03.65.Db

*Keywords*

- non-selfadjoint Schrödinger-type operator, spectral asymptotics, Abel basis of root vectors, dissipative operators, open quantum systems.

*DOI*

*Abstract*

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.

*Appeared in*

- Integral Equations and Operator Theory, 2003, 45, pp. 39-63

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# Stationary measures and phase transition for a class of Probabilistic Cellular Automata

*Authors*

- Dai Pra, Paolo
- Louis, Pierre-Yves
- Rœlly, Sylvie

*2010 Mathematics Subject Classification*

- 60G60 60J10 60K35 82C20 82B26

*Keywords*

- Probabilistic Cellular Automata, stationary measure, Gibbs measure

*DOI*

*Abstract*

We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.

*Appeared in*

- ESAIM probab. Statist. 6 (2002), pp. 89-104

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# Gibbs measures of disordered spin systems

*Authors*

- Külske, Christof

*2010 Mathematics Subject Classification*

- 82B44 82B20 82B28

*2008 Physics and Astronomy Classification Scheme*

- 05.50.+q

*Keywords*

- Disordered systems, Gibbs measures, random field model, interfaces, continuous spins, metastates, non-Gibbsian measures

*DOI*

*Abstract*

We give a brief introduction to some aspects of the field of Gibbs measures of disordered (lattice) spin systems. We present a summary of some of the main results of our own contributions to the subject.

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# Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H20 60H10 34E15 82C31

*2008 Physics and Astronomy Classification Scheme*

- 02.50.-r, 05.10.Gg, 75.60.-d, 92.40.Cy

*Keywords*

- Langevin equation, Fokker-Planck equation, double-well potential, first-exit time, scaling laws, stochastic resonance, dynamicalhysteresis, bifurcation delay, white noise, coloured noise

*DOI*

*Abstract*

We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.

*Appeared in*

- J. Phys. A: Math. Gen. 35 (2002), 2057-2091

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# An adaptive Uzawa FEM for Stokes: Convergence without the Inf-Sup

*Authors*

- Bänsch, Eberhard

ORCID: 0000-0003-2743-1612 - Morin, Pedro
- Nochetto, Ricardo H.

*2010 Mathematics Subject Classification*

- 65N12 65N15 65N30 65N50 65Y20

*Keywords*

- A posteriori error estimators, adaptive mesh refinement, convergence, data oscillation, performance, quasi-optimal meshes

*DOI*

*Abstract*

We introduce and study an adaptive finite element method for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree 𝑘 for velocity whereas for pressure the elements can be either discontinuous of degree 𝑘-1 or continuous of degree 𝑘-1 and 𝑘. The popular Taylor-Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver, and provide consistent computational evidence that the resulting meshes are quasi-optimal.

*Appeared in*

- SIAM J. Num. Anal. 40, no. 4, 1207-1229 (2002)

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# Fundamental obstacles to self-pulsations in low-intensity lasers

*Authors*

- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 78A60 34E15 37D10 34C29

*Keywords*

- distributed feedback semiconductor laser, singular perturbation, averaging, invariant manifold, normal form, rate equations

*DOI*

*Abstract*

We investigate most general properties of possible laser equations in the case where optics is linear. Exploiting the presence of a natural small parameter (the ratio of the photon lifetime in the laser device to the relaxation time of the population density) we establish the existence of an exponentially attracting invariant manifold which contains all bounded orbits, and show that only a small number of electromagnetic modes is sufficient to describe accurately the dynamics of the system. We give a general form of the reduced few-mode systems. We analyze the behavior of single-mode models and a double-mode model with a single optical frequency. We show that in the case where only one electromagnetic mode is excited, the rate equations are close to integrable ones, so the dynamics in this case can be understood by analytic means (by averaging method). In particular, it is shown that a non-stationary (periodic) output is possible only in relatively small (of order of some fractional powers of the small parameter) regions in the space of parameters of the system near some specially chosen parameter constellations. Estimates on the size of these regions and on the frequency of periodic self-pulsations are given for different situations.

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# Numerical bifurcation analysis for multi-section semiconductor lasers

*Authors*

- Sieber, Jan

*2010 Mathematics Subject Classification*

- 34C60 78A60

*Keywords*

- semiconductor lasers, delayed optical feedback, numerical bifurcation analysis

*DOI*

*Abstract*

We investigate the dynamics of a multi-section laser resembling a delayed feedback experiment where the length of the cavity is comparable to the length of the laser. Firstly, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of partial differential equations) to a system of ordinary differential equations (ODEs) describing the semiflow on a local center manifold. Then, we analyse the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters feedback phase and feedback strength to obtain a complete bifurcation diagram for small and moderate feedback strength. This diagram allows to understand the roots of a variety of nonlinear phenomena like, e. g., self-pulsations, excitability, hysteresis or chaos, and to locate them in the parameter plane.

*Appeared in*

- SIAM J. Appl. Dyn. Syst., 1 (2002), pp. 248-270

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# Quasi-stability of the primary flow in a cone and plate viscometer

*Authors*

- Azerad, Pascal
- Bänsch, Eberhard

ORCID: 0000-0003-2743-1612

*2010 Mathematics Subject Classification*

- 35Q30 35B40 65M60 76D05

*Keywords*

- Navier-Stokes equations, shallow domains, rotating fluids, nonlinear stability, asymptotic analysis, haemodynamics, flow chamber, haemostasis, rheometry, CFD, finite elements

*DOI*

*Abstract*

We investigate the flow between a shallow rotating cone and a stationary plate. This cone and plate device is used in rheometry, haemostasis as well as in food industry to study the properties of the flow w.r.t. shear stress. Physical experiments and formal computations show that close to the apex the flow is approximately azimuthal and the shear-stress is constant within the device, the quality of the approximation being controlled essentially by the single parameter Re ε^{2}, where Re is the Reynolds number and ε the thinness of the cone-plate gap. We establish this fact by means of rigorous energy estimates and numerical simulations. Surprisingly enough, this approximation is valid though the primary flow is not itself a solution of the Navier-Stokes equations, and it does not even fulfill the correct boundary conditions, which are in this particular case discontinuous along a line, thus not allowing for a usual Leray solution. To overcome this difficulty we construct a suitable corrector.

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# Glauber dynamics of the random energy model II. Aging below the critical temperature

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Gayrard, Véronique

*2010 Mathematics Subject Classification*

- 82C44 60K35

*Keywords*

- aging, Glauber dynamics, random energy model, trap models, metastability, extreme values

*DOI*

*Abstract*

We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We establish that for a suitably chosen timescale that diverges with the size of the system, one can prove that a natural autocorrelation function exhibits aging. Moreover, we show that the long-time asymptotics of this function coincide with those of the so-called "REM-like trap model" proposed by Bouchaud and Dean. Our results rely on very precise estimates on the distribution of transition times of the process between different states of extremely low energy.

*Appeared in*

- Commun. Math. Phys. 236 (2003) 1, pp. 1-54. DOI 10.107/s00220-003-799-3 .

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# Glauber dynamics of the random energy model I. Metastable motion on the extreme states

*Authors*

- Ben Arous, Gérard
- Bovier, Anton
- Gayrard, Véronique

*2010 Mathematics Subject Classification*

- 82C44 60K35

*Keywords*

- aging, Glauber dynamics, random energy model, trap models, metastability, extreme values

*DOI*

*Abstract*

We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T . The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail.

*Appeared in*

- Commun. Math. Phys. 235 (2003) 3, pp. 379-425. DOI 10.1007/s00220-0033-07984 .

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# Cluster expansions and Pirogov Sinai theory for long range spin systems

*Authors*

- Bovier, Anton
- Zahradník, Miloš

*2010 Mathematics Subject Classification*

- 82A25

*Keywords*

- Low temperature Gibbs states, discrete spin lattice models of Kac Ising typerestricted ensembles with low density constraints, cluster expansion, contours, Pirogov-Sinai theory

*DOI*

*Abstract*

We investigate the low temperature phases of lattice spin systems with interactions of Kac type, that is interactions that are weak but long range in such a way that the total interaction of one spin with all the others is of order unity. In particular we develop a systematic approach to convergent low temperature expansions in situations where interactions are weak but long range. This leads to a reformulation of the model in in terms of a generalized abstract Pirogov-Sinai model, that is a representation in terms of contours interacting through cluster fields. The main point of our approach is that all quantities in the contour representation satisfy estimates that are uniform in the range of the interaction and depend only on the overall interaction strength. The extension of the Pirogov-Sinai theory to such models developed in [Z3] allows then the investigation of the low-temperature phase diagram of these models.

*Appeared in*

- Markov Proc. Rel. fields 8 (2002), pp. 443-478

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# The effect of additive noise on dynamical hysteresis

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H20 60H10 34C55 34E15 82C31

*Keywords*

- dynamical systems, singular perturbations, hysteresis cycles, scaling laws, non-autonomous stochastic differential equations, double-well potential, pathwise description, concentration of measure

*DOI*

*Abstract*

We investigate the properties of hysteresis cycles produced by a one-dimensional, periodically forced Langevin equation. We show that depending on amplitude and frequency of the forcing and on noise intensity, there are three qualitatively different types of hysteresis cycles. Below a critical noise intensity, the random area enclosed by hysteresis cycles is concentrated near the deterministic area, which is different for small and large driving amplitude. Above this threshold, the area of typical hysteresis cycles depends, to leading order, only on the noise intensity. In all three regimes, we derive mathematically rigorous estimates for expectation, variance, and the probability of deviations of the hysteresis area from its typical value.

*Appeared in*

- Nonlinearity 15 (2002), 605-632

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# A mathematical model for impulse resistance welding

*Authors*

- Hömberg, Dietmar
- Duderstadt, Frank
- Khludnev, Alexander M.

*2010 Mathematics Subject Classification*

- 35D05 35Q72 74F15

*Keywords*

- Joule heating, resistance welding, thermo-viscoelasticity

*DOI*

*Abstract*

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are nonlinearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non-elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research.

*Appeared in*

- Math. Methods Appl. Sci. 26 (2003), pp. 717--737

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# Discrete Fourier transforms and their application to stress-strain problems in composite mechanics: A convergence study

*Authors*

- Brown, Calum M.
- Dreyer, Wolfgang
- Müller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 42B10 45A05 15A09 74E30

*Keywords*

- Discrete Fourier transforms, Neumann iteration, fixpoint theorem, composites

*DOI*

*Abstract*

We present in this paper a method for determining the convergence characteristics of the Neumann iterative solution of a discrete version of a second-type Fredholm equation. Implemented as the so-called "equivalent inclusion problem" within the context of mechanical stress/strain analysis, it allows the modeling of elastically highly heterogeneous bodies with the aid of Discrete Fourier Transforms (DFT). A method is developed with which we can quantify pre-analysis (i.e., at iteration zero) the convergence behavior of the Neumann scheme depending on the choice of an auxiliary stiffness tensor, specifically for the linear-elastic case. It is shown that a careful choice of this tensor results in both guaranteed convergence and a smaller convergence radius for the solution. Furthermore, there is some indication that as the convergence radius decreases, the scheme may converge to a solution at a faster rate translating into an increase in computational efficiency.

*Appeared in*

- Proc. R. Soc. A, 458 (2002), pp. 1-21

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# Generalized necessary scaling condition and stability of chemical reactors with several educts

*Authors*

- Efendiev, Messoud A.
- Hebermehl, Georg
- Lasser, Rupert

*2010 Mathematics Subject Classification*

- 35Q60 65F15 65N22

*Keywords*

- Reaction-Diffusion equation, stability, chemical radical reactions

*DOI*

*Abstract*

We present, for a class of industrially relevant chemical reactions with two educts the dependence of stability on important chemical parameters, such as coolant, dilution and diffusion rates. The main analytical tools are generalized upscaling balance condition for the equilibria concentrations and spectral properties of corresponding operators. Although we illustrate the stability analysis for a model reactor (2 educts, E_{1} and E_{2}), it should be emphasized that our approach is applicable to more complex reaction mechanisms.

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# To the uniqueness problem for nonlinear parabolic equations

*Authors*

- Gajewski, Herbert
- Skrypnik, Igor V.

*2010 Mathematics Subject Classification*

- 35B45 35K15 35K20 35K65

*Keywords*

- Nonlinear parabolic equations, bounded solutions, uniqueness, nonstandard assumptions, degenerate type

*DOI*

*Abstract*

We prove a priori estimates in $L^2(0,T,W^1,2(Omega)) cap L^infty(Q)$, existence and uniqueness of solutions to Cauchy-Dirichlet problems for parabolic equations $$ fracpartial sigma(u)partial t - sum_i=1^n fracpartialpartial x_i Big rho(u) b_i Big (t,x,fracpartial upartial x Big ) Big + a Big (t,x,u,fracpartial upartial x Big ) = 0, $$ $(t,x) in Q = (0,T) times Omega$, where $rho(u) = fracddusigma(u)$. We consider solutions $u$ such that $rho^frac12(u) left fracpartial upartial x right in L^2 ( 0,T,L^2 (Omega)), fracpartialpartial tsigma(u) in L^2 (0,T, [ cirWhspace*-0.1mm^1,2 (Omega) ]^ast )$. Our nonstandard assumption is that $log rho (u)$ is concave. Such assumption is natural in view of drift diffusion processes for example in semiconductors and binary alloys, where $u$ has to be interpreted as chemical potential and $sigma$ is a distribution function like $sigma=e^u$ or $sigma=frac11+e^u$.

*Appeared in*

- Discrete Contin. Dyn. Syst., 9 (2003)

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# Equilibrium figures of viscous fluids governed by external forces and surface tension

*Authors*

- Guckel, Ralf

*2010 Mathematics Subject Classification*

- 35R35 76D05 76D45 76U05

*Keywords*

- Equilibrium Figures, Viscous Fluids, External Forces, Surface Tension, Rotating Drops, Free Boundary Problem, Navier-Stokes Equations, Perturbation Problem, Implicit Function Theorem

*DOI*

*Abstract*

We reconsider the problem of determining equilibrium figures of an isolated drop of an incompressible viscous liquid. The fluid body is subject to an external force density, and, along the free boundary, to surface tension. Here the term "equilibrium figure" means that the whole configuration is assumed to be stationary with respect to a uniformly rotating reference frame. Moreover, the pressure outside the fluid body is assumed to be close to a constant, and the fluid body itself is assumed to be close to the unit ball. The existence of such configurations is proved by applying successive approximations, under certain smallness and symmetry conditions on the external and inertial forces. The smallness assumptions are in some sense stronger, while the symmetry assumptions are weaker compared to previous results.

In case surface tension is no longer present and is (or is not) replaced by self-gravitation, the perturbation problem degenerates. The mathematical difficulties are sketched, along with a proposal of how to overcome these difficulties. Details will be presented in forthcoming papers.

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# Mathematical modeling and numerical simulation of semiconductor detectors

*Authors*

- Gajewski, Herbert
- Kaiser, Hans-Christoph
- Langmach, Hartmut
- Nürnberg, Reiner
- Richter, Rainer H.

*2010 Mathematics Subject Classification*

- 65M60 35K57 35B40 65M12

*2008 Physics and Astronomy Classification Scheme*

- 85.60.Gz 07.85.Fv 95.55.Ka

*Keywords*

- semiconductor device simulation, reaction-diffusion systems, asymptotic behavior, time and 3d-space discretization, software design, x-ray pixel detector, DEPFET, DEPMOS

*DOI*

*Abstract*

We report on a system of nonlinear partial differential equations describing signal conversion and amplification in semiconductor detectors. We explain the main ideas governing the numerical treatment of this system as they are implemented in our code WIAS-TeSCA. This software package has been used by the MPI Semiconductor Laboratory for numerical simulation of innovative radiation detectors. We present some simulation results focussing on three-dimensional effects in X-ray detectors for satellite missions.

*Appeared in*

- Mathematics - Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jaeger, H.-J. Krebs, eds., Springer-Verlag Berlin Heidelberg, 2003, pp. 355-364

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# Propagation of sound and surface waves in porous materials

*Authors*

- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 35C20 35L50 74J15

*Keywords*

- asymptotic expansions, waves in porous media

*DOI*

*Abstract*

We review main properties of acoustic waves which are described by continuous models of saturated porous materials. Bulk waves are only mentioned in the introduction, and we concentrate on the propagation of surface waves. We demonstrate differences between a one-component approach to the problem typical for seismology, and a two-component approach. In the latter case we rely on a model proposed by K. Wilmanski in the recent ten years, and we indicate similarities, and differences of results obtained within this model and that of the classical Biot's model of porous materials. The main conclusion is that both models predict the same modes of propagation, and differences are only quantitative in spite of flaws of the Biot's model which contains contributions violating the second law of thermodynamics, and the principle of material objectivity.

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# Mutually catalytic branching in the plane: Uniqueness

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus
- Mytnik, Leonid
- Perkins, Edwin A.
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Catalytic super-Brownian motion, collision local time, martingale problem, duality, uniqueness, Markov property

*DOI*

*Abstract*

We study a pair of populations in ℝ^{2} which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

*Appeared in*

- Ann. Inst. H. Poincare Probab. Statist. 39(1) (2003), pp. 135-191

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# Experimental and numerical investigation of edge tones

*Authors*

- Bamberger, Andreas
- Bänsch, Eberhard

ORCID: 0000-0003-2743-1612 - Siebert, Kunibert G.

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- Edge tones, experimental investigation, numerical methods, Navier-Stokes equations, adaptive finite elements

*DOI*

*Abstract*

We study both, by experimental and numerical means the fluid dynamical phenomenon of so-called edge tones. Of particular interest is the clarification of certain scaling laws relating the frequency ƒ to geometrical quantities, namely 𝑑, the height of the jet, 𝑤, the stand-off distance and the velocity of the jet. We conclude that the Strouhal number S_{𝑑} is given by S_{𝑑} = C · (𝑑/𝑤)^{𝑛} with 𝑛 ≈ 1 in our case. Moreover, the constant C of the experiment agrees within 10-15% with the result of the simulation. As for the frequency dependence on the geometry and on the jet velocity there is a very good agreement of experimental and numerical results.

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# On one dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions

*Authors*

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

*2010 Mathematics Subject Classification*

- 47B44 47E05

*Keywords*

- dissipative Schrödinger-type operators, Sturm-Liouville operators, characteristic function, minimal dilation, eigenfunction expansion

*DOI*

*Abstract*

We study in detail Schrödinger-type operators on a bounded interval of the real axis with dissipative boundary conditions. The characteristic function of such operators is computed, its minimal self-adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.

*Appeared in*

- Mathematische Nachrichten, 2003, Vol. 252, pp.51-69

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# Some new properties of the kinetic equation for the consistent Boltzmann algorithm

*Authors*

- Garcia, Alejandro L.
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 82C40 76P05

*Keywords*

- Kinetic theory, direct simulation Monte Carlo, consistent Boltzmann algorithm, dense gases, H-theorem

*DOI*

*Abstract*

We study properties of the consistent Boltzmann algorithm for dense gases, using its limiting kinetic equation. First we derive an H-theorem for this equation. Then, following the classical derivation by Chapman and Cowling, we find approximations to the equations of continuity, momentum and energy. The first order correction terms with respect to the particle diameter turn out to be the same as for the Enskog equation. These results confirm previous derivations, based on the virial, of the corresponding equation of state.

*Appeared in*

- Transport Theory Statist. Phys. 31, (2002), pp. 579-594

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# Boundary-oriented subdifferential characterization of calmness for convex systems

*Authors*

- Henrion, René
- Jourani, Abderrahim

*2010 Mathematics Subject Classification*

- 90C31 26E25 49J52

*Keywords*

- calmness, multifunctions, convex systems, constraint qualifications

*DOI*

*Abstract*

We study subdifferential characterizations of the calmness property for multifunctions representing convex constraint systems in a Banach space. Extending earlier work in finite dimensions, we show that - in contrast to the stronger Aubin property of a multifunction (or metric regularity of its inverse), calmness can be ensured by corresponding weaker constraint qualifications which are based on boundaries of subdifferentials and normal cones only rather than on the full objects.

*Appeared in*

- SIAM Journal on Optimization, 13 (2002) 520-534, under new title: Subdifferential Conditions for Calmness of Convex Constraints.

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# The spin-glass phase-transition in the Hopfield model with p-spin interactions

*Authors*

- Bovier, Anton
- Niederhauser, Beat

*2010 Mathematics Subject Classification*

- 82A87 60K35

*Keywords*

- spin glasses, Hopfield models, phase transition, overlap distribution

*DOI*

*Abstract*

We study the Hopfield model with pure 𝑝-spin interactions with even 𝑝 ≥ 4, and a number of patterns, M(N) growing with the system size, N, as M(N) = αN^{𝑝-1}. We prove the existence of a critical temperature β_{𝑝} characterized as the first time quenched and annealed free energy differ. We prove that as 𝑝 ↑ ∞, β_{𝑝} → √α2ln2. Moreover, we show that for any α > 0 and for all inverse temperatures β, the free energy converges to that of the REM at inverse temperature β / √α. Moreover, above the critical temperature the distribution of the replica overlap is concentrated at zero. We show that for large enough α, there exists a non-empty interval of in the low temperature regime where the distribution has mass both near zero and near ±1. As was first shown by M. Talagrand in the case of the 𝑝-spin SK model, this implies the the Gibbs measure at low temperatures is concentrated, asymptotically for large N, on a countable union of disjoint sets, no finite subset of which has full mass. Finally, we show that there is α_{𝑝} ∼ 1 / 𝑝! such that for α > α_{𝑝} the set carrying almost all mass does not contain the original patterns. In this sense we describe a genuine spin glass transition. Our approach follows that of Talagrand's analysis of the 𝑝-spin SK-model. The more complex structure of the random interactions necessitates, however, considerable technical modifications. In particular, various results that follow easily in the Gaussian case from integration by parts formulas have to be derived by expansion techniques.

*Appeared in*

- Adv. Theor. Math. Phys. 5 (2001), pp. 1001-1046

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# A new type of travelling wave solutions

*Authors*

- Schneider, Klaus R.
- Shchepakina, Elena
- Sobolev, Vladimir

*2010 Mathematics Subject Classification*

- 34E15 35B25 35K55 80A25

*Keywords*

- travelling wave, singular perturbations, canards, combustion

*DOI*

*Abstract*

We study the existence of combustion waves for an autocatalytic reaction in the non-adiabatic case. Basing on the fact that the reaction system has canard solutions separating the slow combustion regime from the explosive one, we prove by applying the geometric theory of singularly perturbed differential equations the existence of a new type of travelling waves solutions, the so-called canard travelling waves.

*Appeared in*

- Math. Methods Appl. Sci., 26 (2003), 1349-1361

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# Longtime dynamics in adaptive gain control systems

*Authors*

- Leonov, Gennady A.
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 93C40 34D23 34D05 34D45 93C80

*Keywords*

- adaptive control systems, frequency domain methods, global attractor, Hausdorff dimension

*DOI*

*Abstract*

We study the longtime dynamics of a nonlinear adaptive control system introduced by Mareels et al. [10] to control the behavior of a plant which can be described by a finite dimensional SISO linear time invariant system stabilizable by a high gain output feedback. We apply frequency domain methods to derive conditions for global stability, to approximate the region containing the global attractor and to estimate its Hausdorff dimension.

*Appeared in*

- Lecture Notes in Control and Inform. Sci., 273 (2002), pp. 241-254

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