# Heteroclinic orbits between rotating waves of semilinear parabolic equation on the circle

*Authors*

- Fiedler, Bernold
- Rocha, Carlos
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 35B41 34C29

*Keywords*

- global attractor, heteroclinic orbit, zero number

*Appeared in*

- J. Differential Equations, 201 (2004), pp. 99-138

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# On some mathematical topics in classic synchronization

*Authors*

- Shilnikov, Andrey
- Shilnikov, Leonid
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37G15 37E15 37C27 34C26

*Keywords*

- synchronization, saddle-node, global bifurcations, stability boundaries, blue sky bifurcation

*Abstract*

A few mathematical problems arising in the classical synchronization theory are discussed, especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive descriptiob of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori.

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# Kinetic flux-vector splitting schemes for the hyperbolic heat conduction

*Authors*

- Dreyer, Wolfgang
- Qamar, Shamsul

*2010 Mathematics Subject Classification*

- 65M99 76Y05 80A99 76M12 35L15

*Keywords*

- Heat transfer, hyperbolic moment system, initial and boundary value problems, Bose-gas, phonons, high order accuracy

*Abstract*

A kinetic solver is developed for the initial and boundary value problems (IBVP) of the symmetric hyperbolic moment system. This nonlinear system of equations is related to the heat conduction in solids at low temperatures. The system consists of a conservation equation for the energy density e and a balance equation for the heat flux 𝘘^{𝑖}, where 𝑒 and 𝘘^{𝑖} are the four basic fields of the theory. We use kinetic flux vector splitting (KFVS) scheme to solve these equations in one and two space dimensions. The flux vectors of the equations are splitted on the basis of the local equilibrium distribution of phonons. The resulting computational procedure is efficient and straightforward to implement. The second order accuracy of the scheme is acheived by using MUSCL-type reconstruction and min-mod nonlinear limitters. The solutions exhibit second order accuracy, and satisfactory resolution of gradients with no spurious oscillations. The secheme is extended to the two-dimensional case in a usual dimensionally split manner. In order to prescribe the boundary data we need the knowledge of the 𝑒 and 𝘘^{𝑖}. However, in experiments only one of the quantities can be controlled at the boundary. This problem is removed by using a continuity condition. It turned out that after some short time energy and heat flux are related to each other according to Rankine Hugoniot jump relations. To illustrate the performance of the KFVS scheme, we perform several one- and two-dimensional test computations. For the comparison of our results we use high order central schemes. The present study demonstrates that this kinetic method is effective in handling such problems.

*Appeared in*

- J. Comput. Phys. 198 (2004), No. 2, pp. 403--423

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# Fourier inversion algorithms for generalized CreditRisk+ models and an extension to incorporate market risk

*Authors*

- Reiß, Oliver

*2010 Mathematics Subject Classification*

- 91B28 91B30 60E10

*Keywords*

- Credit Risk, Generalization of CreditRisk+, Market Risk, Fourier inversion, Characteristic function

*DOI*

*Abstract*

A popular model to describe credit risk in practice is CreditRisk+ and in this paper a Fourier inversion to obtain the distribution of the credit loss is proposed. A deeper analysis of the Fourier transformation showed that there are at least two methods to obtain the distribution although the corresponding characteristic function is not integrable. The CreditRisk+ model will be extended such, that general dependent sector variables can be taken into consideration, for example dependent lognormal sector variables. Then the transfer to a continuous time model will be performed and the sector variables become processes, more precisely geometric Brownian motions. To have a time continuous credit risk model is an important step to combine this model with market risk. Additionally a portfolio model will be presented where the changes of the spreads are driven by the sector variables. Using a linear expansion of the market risk, the distribution of this portfolio can be determined. In the special case that there is no credit risk, this model yields the well known Delta normal approach for market risk, hence a link between credit risk and market risk has been established.

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# Compact interface property for symbiotic branching

*Authors*

- Etheridge, Alison M.
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Symbiotic branching, mutually catalytic branching, stepping stone model, Anderson model, interacting superprocess, stochastic equation, collision localtime, self-dual, moment dual, moment equations, correlated noise, colourednoise, compact interface property, at most linear speed of propagation

*Abstract*

A process which we call symbiotic branching, is suggested covering three well-known interacting models: mutually catalytic branching, the stepping stone model, and the Anderson model. Basic tools such as self-duality, particle system moment duality, measure case moment duality, and moment equations are still available in this generalized context. As an application, we show that in the setting of the one-dimensional continuum the compact interface property holds: starting from complementary Heaviside states, the interface is finite at all times almost surely.

*Appeared in*

- Stochastic Process. Appl., 114 (2004), pp. 127--160

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# Stability and bifurcations in neural fields with axonal delay and general connectivity

*Authors*

- Atay, Fatihcan M.
- Hutt, Axel

*2010 Mathematics Subject Classification*

- 92C20

*Keywords*

- synaptic networks, non-local interaction, delay, bifurcations, spatial patterns, travelling waves

*Abstract*

A stability analysis is presented for neural field equations in the presence of axonal delays and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the delays play a crucial role in non-stationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the kernel. Bounds are determined for the frequencies of bifurcating periodic solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillatory solutions, and traveling waves. For high transmission speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the connectivity kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible second order systems with combinations of Gaussian excitatory and inhibitory connections.

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# Strong uniqueness for cyclically symbiotic branching diffusions

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60J60 60J80

*Keywords*

- Catalyst, reactant, superprocess, cyclically catalytic branching, stochastic equation, cyclic reaction, growth of moments, moment equation system, moment dual

*Abstract*

A uniqueness problem raised in 2001 for critical cyclically catalytic super-Brownian motions is solved in the simplified space-less case, that is, for cyclically catalytic branching diffusions X. More precisely, X is characterized as the unique strong solution of a singular stochastic equation.

*Appeared in*

- Statist. Probab. Lett., 73 (2005) pp. 251-257

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# Element method for epitaxial growth with attachment-detachment kinetics

*Authors*

- Bänsch, Eberhard
- Haußer, Frank
- Lakkis, Omar
- Li, Bo
- Voigt, Axel

*2010 Mathematics Subject Classification*

- 35Q99 35R35 65N30 65Z05 74S05

*Keywords*

- epitaxial growth, island dynamics, free or moving boundary problem, adatom diffusion, attachment-detachment kinetics, surface diffusion, mean curvature flow, finite elements, adaptivity, front tracking

*DOI*

*Abstract*

An adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth. Such problems consist of an adatom (adsorbed atom) diffusion equation on terraces of different height, boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional "surface" diffusion. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. The diffusion equation is discretized by the first-order implicit scheme in time and the linear finite element method in space. A technique of extension is used to avoid the complexity in the spatial discretization near boundaries. All the elements are marked, and the marking is updated in each time step, to trace the terrace height. The evolution of the terrace boundaries includes both the mean curvature flow and the surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements. Simple adaptive techniques are employed in solving the adatom diffusion as well as the boundary motion problem. Numerical tests on pure geometrical motion, mass balance, and the stability of a growing circular island demonstrate that the method is stable, efficient, and accurate enough to simulate the growing of epitaxial islands over a sufficiently long time period.

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# Numerical analysis of Monte Carlo finite difference evaluation of Greeks

*Authors*

- Milstein, Grigori N.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 60H30 65C30 91B28

*Keywords*

- Derivative pricing and hedging, probabilistic representations, weak approximation of solutions of stochastic differential equations, Monte Carlo simulation

*Abstract*

An error analysis of approximation of derivatives of the solution to the Cauchy problem for parabolic equations by finite differences is given taking into account that the solution itself is evaluated using weak-sense numerical integration of the corresponding system of stochastic differential equations together with the Monte Carlo technique. It is shown that finite differences are effective when the method of dependent realizations is exploited in the Monte Carlo simulations. This technique is applicable to evaluation of Greeks. In particular, it turns out that it is possible to evaluate both the option price and deltas by a single simulation run that reduces the computational costs. Results of some numerical experiments are presented.

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# Nonlinear dynamics of semiconductor lasers with active optical feedback

*Authors*

- Bauer, Stefan
- Brox, Olaf
- Kreissl, Jochen
- Sartorius, Bernd
- Radziunas, Mindaugas
- Sieber, Jan
- Wünsche, Hans-Jürgen
- Henneberger, Fritz

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 5.45.-a

*Keywords*

- semiconductor laser, nonlinear dynamics, optical feedback, bifurcations

*DOI*

*Abstract*

An in-depth theoretical as well as experimental analysis of the nonlinear dynamics in semiconductor lasers with active optical feedback is presented. Use of a monolithically integrated multi-section device of sub-mm total length provides access to the short-cavity regime. By introducing an amplifier section as novel feature, phase and strength of the feedback can be separately tuned. In this way, the number of modes involved in the laser action can be adjusted. We predict and observe specific dynamical scenarios. Bifurcations mediate various transitions in the device output, from single-mode steady-state to self-pulsation and between different kinds of self-pulsations, reaching eventually chaotic behavior in the multi-mode limit.

*Appeared in*

- Phys. Rev. E., 69 (2004), pp. 016206-1 - 016206-10

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# An efficient dual Monte Carlo upper bound for Bermudan style derivatives

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C05 91B28

*Keywords*

- Bermudan options, Monte Carlo, duality approach, LIBOR models

*Abstract*

Based on a duality approach for Monte Carlo construction of upper bounds for American/Bermudan derivatives (Rogers, Haugh & Kogan), we present a new algorithm for computing dual upper bounds in an efficient way. The method is applied to Bermudan swaptions in the context of a LIBOR market model, where the dual upper bound is constructed from the maximum of still alive swaptions. We give a numerical comparison with Andersen's lower bound method and its dual considered by Andersen & Broadie.

*Appeared in*

- Monte Carlo Methods and Applications Vol.10 (3-4), 331-343 under the title ''Upper Bounds for Bermudan Style Derivatives''

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# Multi-scale clustering for a non-Markovian spatial branching process

*Authors*

- Fleischmann, Klaus
- Vatutin, Vladimir A.

*2010 Mathematics Subject Classification*

- 60J80 60G70 60J15

*Keywords*

- branching particle system, Bellman-Harris process, age-dependent process, continuous-state branching, critical dimension, scaling limit theorem, superprocess

*Abstract*

Consider a system of particles which move in R^d according to a symmetric alpha-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index 1+beta. In case of the critical dimension d=alpha/beta, the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index 1+beta. Our result generalizes the case alpha=2 of Brownian particles of Klenke (1998), where pde methods had been used which are not available in the present setting.

*Appeared in*

- J. Theoret. Probab., 18, (2005), pp. 719-755

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# Two- and three-dimensional transient melt-flow simulation in vapour-pressure-controlled Czochralski crystal growth

*Authors*

- Bänsch, Eberhard
- Davis, Dominic
- Langmach, Hartmut
- Reinhardt, Gerd
- Uhle, Manfred

*2010 Mathematics Subject Classification*

- 35Q30 65M60 76-04 76D05

*Keywords*

- axisymmetric, buoyancy, Czochralski, crystal growth, finite-element, GaAs, Navier-Stokes equations, vapour-pressure controlled

*Abstract*

Flow and thermal properties associated with semiconductor melt flow in an axisymmetric crucible container are studied numerically. Axisymmetric and three-dimensional computational solutions are obtained using a standard-Galerkin, finite-element solver. The crucible and crystal are optionally rotated, and the influence of gravity (through buoyancy) is accounted for via a Boussinesq approximation in the controlling Navier-Stokes equations. The results indicate a strong dependence of the flow on both rotation and buoyancy. Results for axisymmetric flows, computed in both flat and curved geometries, are presented first, and strongly suggest that rotation of crystal and crucible in the same direction (iso-rotation) is most favourable for producing a desired convexity for the crystal/melt interface. Three-dimensional results are then presented for higher Reynolds numbers, and, in particular, reveal that for iso-rotation under moderate buoyancy, the flow undergoes a switch from a steady, 2D state to an unsteady 3D state, and that the temperature becomes non-trivially advected by the flow beneath the crystal. Further evidence reveals however, that on a time scale more appropriate to the crystal growth process, the (time-averaged) flow has a weaker three-dimensionality, in relation to its axisymmetric mode, and there is only slight distortion to the temperature field beneath the crystal.

*Appeared in*

- Comput. Fluids, 25 (2006) pp. 1400--1419.

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# Snake representation of a super-Brownian reactant in the catalytic region

*Authors*

- Fleischmann, Klaus
- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Markov branching process, admissible catalyst, part of reactant, superprocess, collision measure, collision local time, catalytic Brownian snake, modified hitting measure

*Abstract*

For a continuous super-Brownian reactant X in R^{d} with general catalyst ρ a Brownian snake representation is derived for the part X^{c} of X in the catalyst region. This extends results of Dawson et al. (2002) and Klenke (2003) in that it allows the collision local time of an intrinsic reactant particle with the catalyst to have flat pieces, caused by catalyst-free regions.

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# Solution of linear systems with sparse matrices

*Authors*

- Grund, Friedrich

*2010 Mathematics Subject Classification*

- 65F05 65F50 65Y05

*Keywords*

- Direct Linear Solver, Sparse-Matrix-Techniques, Parallel Computation, Circuit Simulation, Chemical Process Simulation

*Abstract*

For large scale problems in electric circuit simulation as well as in chemical process simulation, the linear solver often needs about 50 - 80 % of the total amount of computing time. For that purpose, we consider direct methods for the numerical solution of linear systems of equations with unsymmetric sparse coefficient matrices. The Gaussian elimination method is applied to solve the linear system. Here, the row permutation is used to provide numerical stability and the column permutation is chosen to control sparsity. In a new approach, implemented in the solver GSPAR2, the determination of the pivot columns is done with a modified algorithm, which has only a complexity of O(n). A partial pivoting technique is used to maintain numerical stability. For solving several linear systems with the same pattern structure of the coefficient matrix efficiently, we generate a list of pseudo code instructions for the factorization of the matrices. With it, the solver GSPAR2 has been proven successful within the simulation of several real life problems. For a number of linear systems arising from different technical problems, the computing times of GSPAR2 are compared to that of some recently released linear solvers.

*Appeared in*

- K. Antreich, R. Bulirsch, A. Gilg, and P. Rentrop, editors: Modeling, Simulation and Optimization of Integrated Circuits, volume 146 of International Series of Numerical Mathematics, pages 333-347. Birkhäuser Verlag Basel/Switzerland, 2003

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# Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height

*Authors*

- Kurbanmuradov, Orazgeldi
- Rannik, Ullar
- Levykin, Alexander I.
- Sabelfeld, Karl
- Vesala, Timo

*2010 Mathematics Subject Classification*

- 65C05 76S05

*Keywords*

- Forest canopy, Flux footprint functions, surface roughness change, closure model, backward Lagrangian trajectories

*Abstract*

Forward and backward stochastic Lagrangian trajectory simulation methods are developed to calculate the footprint and cumulative footprint functions of concentration and fluxes in the case when the ground surface has an abrupt change of the roughness height. The statistical characteristics to the stochastic model are extracted numerically from a closure model we developed for the atmospheric boundary layer. The flux footprint function is perturbed in comparison with the footprint function for surface without change in properties. The perturbation depends on the observation level as well as roughness change and distance from the observation point. It is concluded that the footprint function for horizontally homogeneous surface, widely used in estimation of sufficient fetch for measurements, can be seriously biased in many cases of practical importance.

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# Stability of gyroscopic systems under small random excitations

*Authors*

- Khasminskii, Rafail
- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H10 93E15

*Keywords*

- Stochastic stability, Lyapunov exponents, moment Lyapunov exponents, oscillators with small random excitations, gyroscopic forces, stochastic averaging principle

*Abstract*

Gyroscopic systems with two degrees of freedom under small random perturbations are investigated by use of the stochastic averaging principle. It is proved that the principal term of the Lyapunov exponent for the original system coincides with the Lyapunov exponent for the averaged system. An explicit formula for the averaged Lyapunov exponent is derived. The averaged moment Lyapunov exponent is considered as well. An example is given in which an unstable gyroscopical system is stabilized by noise of the Sratonovich type.

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# A super-stable motion with infinite mean branching

*Authors*

- Fleischmann, Klaus
- Sturm, Anja

*2010 Mathematics Subject Classification*

- 60J80 60K35 60G57 60F15

*Keywords*

- Neveu's continuous state branching process, superprocess, branching processwith infinite mean, non-Lipschitz non-linearity, immortal process, instantaneous mass propagation, locally countably infinite biodiversity

*DOI*

*Abstract*

Impressed by Neveu's (1992) continuous-state branching process we learned about from Bertoin and Le Gall (2000), a class of finite measure-valued càdlàg superprocesses X with Neveu's branching mechanism is constructed. To this end, we start from certain supercritical (α,d,β)-superprocesses X^{(β)} with symmetric α-stable motion and (1+β)-branching and let β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function 𝑢 log 𝑢 as non-linear term (instead of 𝑢^{1+β} in the case of X^{(β)}) and is thus interesting in its own. X has infinite expectations, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity.

*Appeared in*

- Ann. Inst. H. Poincare Probab. Statist., 40 (2004), pp. 513--537

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# On a micro-macro transition for poroelastic Biot's model and corresponding Gassmann-type relations

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 74F10 74L10 74Q15 74E30

*Keywords*

- Micro-macro transitions, mechanics of poroelastic materials, Biot's model

*Abstract*

In the paper we consider a micro-macro transition for a linear thermodynamical model of poroelastic media which yields the Biot's model. We investigate a two-component poroelastic linear model in which a constitutive dependence on the porosity gradient is incorporated and this is compared with the classical Biot's model without added mass effects. We analyze three Gedankenexperiments: jacketed undrained, jacketed drained and unjacketed and derive a generalization of classical Gassmann relations between macroscopic material parameters and microscopic compressibility moduli of the solid, and of the fluid. Dependence on the porosity is particularly exposed due to its importance in acoustic applications of the model.

In particular we show that Gassmann relations follow as one of two physically justified solutions of the full set of micro-macro compatibility relations. In this solution the coupling to the porosity gradient is absent. Simultaneously, we demonstrate the second solution which lies near the Gassmann results but admits the coupling. In both models couplings are weak enough to admit, within the class of problems of acoustic wave analysis, an approximation by a "simple mixture" model in which coupling of stresses is fully neglected.

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# Direct simulation of the uniformly heated granular Boltzmann equation

*Authors*

- Gamba, Irene M.
- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82C40 82C80 65R20

*Keywords*

- inelastic Boltzmann equation, granular flow, Monte Carlo algorithm

*DOI*

*Abstract*

In the present paper we give an overview of the analytical properties of the steady state solution of the spatially homogeneous uniformly heated granular Boltzmann equation. The asymptotic properties of this distribution (so called tails) are formulated for different models of interaction. A new stochastic numerical algorithm for this problem is presented and tested using analytical relaxation of the temperature. The "tails" of the steady state distribution are computed using this algorithm and the results are compared with the available analytical information.

*Appeared in*

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

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# Second order accurate explicit finite volume schemes for the solution of Boltzmann-Peierls equation

*Authors*

- Dreyer, Wolfgang
- Qamar, Shamsul

*2010 Mathematics Subject Classification*

- 65M99 76Y05 80A99 76M12 35L15

*Keywords*

- Boltzmann-Peierls equation, heat transfer, hyperbolic moment system, Bose-gas, phonons, upwind schemes, central schemes, high order accuracy

*Abstract*

In this article we present the first and second order numerical schemes for the solution of initial value problems of the Boltzmann-Peierls equation (BPE). We also modify the numerical schemes for the solution of initial and boundary value problems (IBVP) of its derived hyperbolic moment system. BPE is an integro-differential equation which describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE, while it still retain all the properties of the original BPE. Using special coordinates, we get a further reduction of the kinetic equation in one space dimension. We introduce the discrete-velocity model of the reduce BPE in one space dimension. This discrete-velocity model can be discretized in space and time by using finite volume schemes. We derive both first and second order explicit upwind and central schemes for the discrete-velocity kinetic equation as well as for the derived moment system. We use the kinetic approach in order to prescribe boundary conditions for the IBVP of the moment system. Several numerical test cases are considered in order to validate the theory.

*Appeared in*

- Z. Angew. Math. Mech. 85 (2005), no. 1, pp. 4-22.

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# Nonlocal phase-field models for non-isothermal phase transitions and hysteresis

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35B40 35K50 45J05 45K05

*Keywords*

- Phase transitions, nonlocal models, integrodifferential equations, hysteresis operators

*Abstract*

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper complements recent investigations by S. Zheng and the second author and treats the case when the part of the free energy density forcing the order parameter to attain values within the physically meaningful range [0,1] is not given by a logarithmic expression but by the indicator function of [0,1]. The resulting field equations form a system of integro-partial differential inclusions that are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved by first transforming the system of inclusions into an equivalent system of equations in which hysteresis operators occur, and then employing techniques similar to those recently developed by the authors for phase-field systems involving hysteresis operators.

*Appeared in*

- Adv. Math. Sci. Appl., Volume 14, No. 2 (2004), pp. 593-612

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# An L^1-stability and uniqueness result for balance laws with multifunctions: A model from the theory of granular media

*Authors*

- Gwiazda, Piotr
- Świerczewska, Agnieszka

*2010 Mathematics Subject Classification*

- 35L65 35L45 35B35

*Keywords*

- system of hyperbolic conservation law, multifunction, weak entropy solutions, L^1-stability, uniqueness, well-posedness

*Abstract*

In this paper it is studied uniqueness and L^{1}-stability for 2×2 system coming out from the theory of granular media. The investigations are done in a class of weak entropy solutions. The appearance of multifunction in a source term, given by Coulomb-Mohr friction law, requires a modification of definition of the solution.

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# Controlled topological transitions in thin film phase separation

*Authors*

- Hennessy, Mathew G.
- Burlakov, Victor M.
- Münch, Andreas
- Wagner, Barbara
- Goriely, Alain

*2010 Mathematics Subject Classification*

- 76M45 76E17 35B40 35C20

*2008 Physics and Astronomy Classification Scheme*

- 64.60.an

*Keywords*

- phase separation and segregation in thin films, metastable phases

*Abstract*

In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

*Appeared in*

- SIAM J. Appl. Math., 75 (2015) pp. 38--60.

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# Stochastic models and Monte Carlo algorithms for Boltzmann type equations

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- Boltzmann equation, stochastic models, Monte Carlo algorithms

*Abstract*

In this paper we are concerned with three typical aspects of the Monte Carlo approach. First there is a certain field of application, namely physical systems described by the Boltzmann equation. Then some class of stochastic models is introduced and its relation to the equation is studied using probability theory. Finally Monte Carlo algorithms based on those models are constructed. Here numerical issues like efficiency and error estimates are taken into account. In Section 1 we recall some basic facts from the kinetic theory of gases, introduce the Boltzmann equation and discuss some applications. Section 2 is devoted to the study of stochastic particle systems related to the Boltzmann equation. The main interest is in the convergence of the system (when the number of particles increases) to the solution of the equation in an appropriate sense. In Section 3 we introduce a modification of the standard "direct simulation Monte Carlo" method, which allows us to tackle the problem of variance reduction. Results of some numerical experiments are presented.

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# Derrida's Generalized Random Energy models 4: Continuous state branching and coalescents

*Authors*

- Bovier, Anton
- Kurkova, Irina

*2010 Mathematics Subject Classification*

- 82B44 60G70 60K35

*Keywords*

- Gaussian processes, generalized random energy model, continuous state branching process, subordinators, coalescent processes, genealogy, Ghirlanda-Guerra identities

*Abstract*

In this paper we conclude our analysis of Derrida's Generalized Random Energy Models (GREM) by identifying the thermodynamic limit with a one-parameter family of probability measures related to a continuous state branching process introduced by Neveu. Using a construction introduced by Bertoin and Le Gall in terms of a coherent family of subordinators related to Neveu's branching process, we show how the Gibbs geometry of the limiting Gibbs measure is given in terms of the genealogy of this process via a deterministic time-change. This construction is fully universal in that all different models (characterized by the covariance of the underlying Gaussian process) differ only through that time change, which in turn is expressed in terms of Parisi's overlap distribution. The proof uses strongly the Ghirlanda-Guerra identities that impose the structure of Neveu's process as the only possible asymptotic random mechanism.

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# Complete synchronization of symmetrically coupled autonomous systems

*Authors*

- Schneider, Klaus R.
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34C15 34K20 34C11 34K35

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt

*Keywords*

- complete synchronization, ordinary differential, differential-delay equations, Goodwin oscillator, one-sided Lipschitz condition

*Abstract*

In this paper we derive conditions for complete synchronization of two symmetrically coupled identical systems of ordinary differential equations and differential-delay equations. Using Lyapunov function approach we give an estimate of the region of attraction of the synchronized solution. We also established that complete synchronization is robust with respect to small perturbations of the identical systems.

*Appeared in*

- Applicable Analysis 82 (2003) 1127-1143.

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# On modeling acoustic waves in saturated poroelastic media

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152 - Wilmanski, Krzystof

*2010 Mathematics Subject Classification*

- 74F10 74J05 74L05

*Keywords*

- Bulk waves in poroelastic materials, Biot-Gassmann model of granular materials

*Abstract*

In this paper we present a comparison of the linear wave analysis for four models of poroelastic materials. As shown in a paper by Wilmanski (Arch. Mech. 2002) a nonlinear thermodynamical construction of a two-component model of such materials requires a dependence on the porosity gradient. In the linear version this dependence may or may not be present (WIAS-Preprint No. 868). Consequently, we may work with the model without a dependence on this gradient which is identical with Biot's model or we can use the so-called full model. In both cases we can construct simplified models without a coupling between partial stresses introduced by Biot. These simplified models have the advantage that their application to, for instance, surface wave analysis yields much simpler mathematical problems.

In the present work we show that such a simplification for granular materials leads to a good qualitative agreement of all four models in ranges of porosity and Poisson's ratio commonly appearing in geotechnical applications. Quantitative differences depend on the mode of propagation and vary between 10% and 20%. We illustrate the analysis with a numerical example corresponding to data for sands.

Simultaneously we demonstrate severe limitations of the applicability of Gassmann relations which yield an instability of models in a wide range of practically important values of parameters.

*Appeared in*

- J. Statist. Phys., 131 (2005), pp.873-996.

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# An existence result for the Leray-Lions type operators with discontinuous coefficients

*Authors*

- Gwiazda, Piotr
- Zatorska-Goldstein, Anna

*2010 Mathematics Subject Classification*

- 35J65 49J45

*Keywords*

- nonlinear elliptic equations, discontinuous operator, compensated compactness, Young measures, set-valued analysis

*Abstract*

In this paper we prove an existence result for Leray-Lions quasilinear elliptic operator with discontinuous coefficients. The idea of the proof is based on compactness results for the sequences of solutions to regularized problems obtained via the Compensated Compactness, Young measures, and Set-Valued Analysis tools.

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# On a decay rate for a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys

*Authors*

- Vodák, Rostislav

*2010 Mathematics Subject Classification*

- 35Q72 73B30 35B40

*Keywords*

- nonlinear thermo-viscoelasticity, shape memory alloys, phase transitions, asymptotic behaviour, Landau-Ginzburg theory

*Abstract*

In this paper, we investigate the decay rate of stabilization of the solution of the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as a function of the order parametr. We prove that for appropriate constants, which appear in the above-mentioned model, we can decide upon the exponencial decrease of the solution to its attractor for time tending to infinity.

*Appeared in*

- Adv. Math. Sci. Appl. 14 (2004), no. 1, 295--305.

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# Surface waves in two-component poroelastic media on impermeable boundaries -- numerical analysis in the whole frequency domain

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 74J15 76S05 74S99

*Keywords*

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

*Abstract*

In this work the dispersion relation for surface waves on an impermeable boundary of a fully saturated poroelastic medium is investigated numerically in the whole range of frequencies. A linear model of a two-component poroelastic medium is used in the form proposed by K. Wilmański. Similarly to the classical Biot's model it is a continuum mechanical model but it is much simpler.

In the whole range of frequencies there exist two modes of surface waves corresponding to the classical Rayleigh and Stoneley waves. The numerical results for velocities and attenuations of these waves are shown for different values of the bulk permeability coefficient in different ranges of frequencies.

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# Convexity of trace functionals and Schrödinger operators

*Authors*

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

*2010 Mathematics Subject Classification*

- 47H05 46T20 47B10 47F05

*2008 Physics and Astronomy Classification Scheme*

- 31.15.Ew 31.15.Md 05.30.-d

*Keywords*

- trace functionals, convexity, monotonicity, double Stieltjes operator integrals, spectral asymptotics, generalized Fermi level, density-functional theory

*DOI*

*Abstract*

Let 𝐻 be a semi-bounded self-adjoint operator in a separable Hilbert space. For a certain class of positive, continuous, decreasing, and convex functions F we show the convexity of trace functionals tr(𝐹(𝐻+𝑈 - ε (𝑈))) - ε (𝑈), where 𝑈 is a bounded self-adjoint operator on 𝐻 and ε (𝑈) is a normalizing real function－the Fermi level－which may be identical zero. If additionally 𝐹 is continuously differentiable, then the corresponding trace functional is Frechet differentiable and there is an expression of its gradient in terms of the derivative of 𝐹. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals. If, in particular, 𝐻 is a Schrödinger-type operator and 𝑈 a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function ƒ = -𝐹'. Thus, the monotonicity of the particle density in its dependence on the potential 𝑈 of Schrödinger's operator－which has been understood since the late 1980s－follows as a special case.

*Appeared in*

- J. Funct. Anal., 234 (2006) pp. 45--69.

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# Aging in two-dimensional Bouchaud's model

*Authors*

- Ben Arous, Gerard
- Cerny, Jiri
- Mountford, Thomas

*2010 Mathematics Subject Classification*

- 82D30 82C41 60F17

*Keywords*

- Aging, trap model, Levy process, random walk, time change

*Abstract*

Let E_x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on Z^2 is a Markov chain X(t) whose transition rates are given by w_xy=nu exp(-beta E_x) if x, y are neighbours in Z^2. We study the behaviour of two correlation functions: P[X(t_w+t)=X(t_w)] and P[X(t')=X(t_w)forall t'in[t_w,t_w+t]]. We prove the (sub)aging behaviour of these functions when beta >1.

*Appeared in*

- Probability Theory and Related Fields, Online First, DOI 10.1007/s00440-004-0408-1 .

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# Approximation of Wiener integrals with respect to the Brownian bridge by simulation of SDEs

*Authors*

- Milstein, Grigori N.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 65C30 28C20 65C05 60H35

*Keywords*

- Conditional Wiener integrals, Feynman path integrals, numerical integration of stochastic differential equations, Monte Carlo simulation

*Abstract*

Numerical integration of stochastic differential equations together with the Monte Carlo technique is used to evaluate conditional Wiener integrals of exponential-type functionals. An explicit Runge-Kutta method of order four and implicit Runge-Kutta methods of order two are constructed. The corresponding convergence theorems are proved. To reduce the Monte Carlo error, a variance reduction technique is considered. Results of numerical experiments are presented.

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# Dynamic nonparametric filtering with application to finance

*Authors*

- Cheng, Ming-Yen
- Fan, Jianqing
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62P05 60G35

*Keywords*

- adaptive filtering, volatility estimation, GARCH, exponential smoothing, autoregression

*Abstract*

Problems of nonparametric filtering arises frequently in engineering and financial economics. Nonparametric filters often involve some filtering parameters to choose. These parameters can be chosen to optimize the performance locally at each time point or globally over a time interval. In this article, the filtering parameters are chosen via minimizing the prediction error for a large class of filters. Under a general martingale setting, with mild conditions on the time series structure and virtually no assumption on filters, we show that the adaptive filter with filtering parameter chosen by historical data performs nearly as well as the one with the ideal filter in the class, in terms of filtering errors. The theoretical result is also verified via intensive simulations. Our approach is also useful for choosing the orders of parametric models such as AR or GARCH processes. It can also be applied to volatility estimation in financial economics. We illustrate the proposed methods by estimating the volatility of the returns of the S&P500 index and the yields of the three-month Treasury bills.

*Appeared in*

- J. Mach. Learn. Res., (2003) pp. 315-333.

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# An asymptotic maximum principle for essentially linear evolution models

*Authors*

- Baake, Ellen
- Baake, Michael
- Bovier, Anton
- Klein, Markus

*2010 Mathematics Subject Classification*

- 15A18 95D15 60J80.

*Keywords*

- asymptotics of leading eigenvalue, reversibility, mutation-selection models, ancestral distribution, lumping

*Abstract*

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional variational principle in the limit $N to infty$ (where $N$ is the number of types). In order to generalize these results, we consider here a large family of reversible $N times N$ matrices and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order $1/N$. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.

*Appeared in*

- J. Math. Biol. (2004). Math. Biol., 50, no. 1, pp. 83-114, 2005, and, DOI 10.1007/s00285-004-0281-7J .

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# Simulation of phase-controlled mode-beating lasers

*Authors*

- Wünsche, Hans-Jürgen
- Radziunas, Mindaugas
- Bauer, Stefan
- Brox, Olaf
- Sartorius, Bernd

*2010 Mathematics Subject Classification*

- 78-04 78A60 35-04 78-05

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Mi 42.60.By 02.60.Cb

*Keywords*

- semiconductor laser, distributed feedback, modelling, self-pulsations, high speed, all optical signal processing, 3R-regeneration

*Abstract*

Self-pulsations in Phase Controlled Mode Beating lasers (PhaseCOMB) are very attractive for all-optical clock recovery at ultra-high bit rates. In this paper we apply the comprehensive simulation tool LDSL that has been developed by us for studying the self-pulsation features of PhaseCOMB lasers considering the effects of spontaneous emission noise, longitudinal spatial hole burning, and gain dispersion. In particular the importance of mode control for adjusting the PhaseCOMB operating conditions is pointed out. The simulation results are confirmed by measurements on fabricated devices.

*Appeared in*

- IEEE J. Select. Topics Quantum Electron., 9 (2003), pp. 857-864

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# Stochastic particle methods for Smoluchowski coagulation equation: Variance reduction and error estimations

*Authors*

- Kolodko, Anastasia
- Sabelfeld, Karl K.

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Stochastic particle methods, Smoluchowski equation, variance reduction, coagulation-fragmentation process

*Abstract*

Stochastic particle methods for the coagulation-fragmentation Smoluchowski equation are developed and a general variance reduction technique is suggested. This method generalizes the mass-flow approach due to H. Babovsky, and has in focus the desired band of the size spectrum. Estimations of the variance and bias of the method are derived. A comparative cost and variance analysis is made for the known stochastic methods. An applied problem of coagulation-evaporation dynamics in free molecule regime is solved.

*Appeared in*

- Monte Carlo Methods Appl., 9 (2003) pp. 315-339.

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# Towards thermodynamic modeling of nucleation and growth of droplets in crystals

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank

*2010 Mathematics Subject Classification*

- 35R35 74B20 74F20 74F25 76R50 80A22

*Keywords*

- phase transition, Gallium Arsenide, mechanical stresses in mixures, low of mass action

*Abstract*

Stress assisted diffusion in single crystal Gallium Arsenide (GaAs) leads to the formation and growth of unwanted liquid arsenic droplets in a solid matrix. This process happens during the heat treatment of single crystal GaAs, which is needed for its application in opto-electronic devices, and it is of crucial importance to pose and answer the question if the appearance of droplets can be avoided. To this end we start a thermodynamic simulation of this process. Special emphasis is given to the influence of mechanical effects on chemistry, diffusion and interface motion in GaAs.

*Appeared in*

- Free Boundary Problems. Theory and Applications. Internat. Ser. Numer. Math. 147, Birkhäuser, Basel u.s. 2004, pp. 113--130

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# Finite element methods for surface diffusion

*Authors*

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

*2010 Mathematics Subject Classification*

- 35K55 65M12 65M15 65M60 65Z05

*Keywords*

- Surface diffusion, fourth-order parabolic problem, finite elements, a priori error estimates, Schur complement, smooth ing effect, pinch-off

*Abstract*

Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for the parametric case, develop a finite element method, and propose a Schur complement approach to solve the resulting linear systems. We also introduce a new graph formulation and state an optimal a priori error estimate. We conclude with several significant simulations, some with pinch-off in finite time.

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# Numerically stable computation of CreditRisk+

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 91B30 60-08 60E10

*Keywords*

- Credit risk, Probability generating function, Computation of functions of power series, Numerical stability

*Abstract*

The CreditRisk+ model launched by CSFB in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk^{+}, equivalent to an algorithm recently proposed by Giese, based on well-known expansions of the logarithm and the exponential of a power series. We show that it is advantageous to the Panjer recursion advocated in the original CreditRisk^{+} document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the computational complexity of the resulting algorithm is stated.

*Appeared in*

- Journal of Risk (2004) Vol. 6, Nr. 4, pp. 1-10

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# Varying coefficient regression modeling by adaptive weights smoothing

*Authors*

- Polzehl, Jörg

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

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- adaptive weights, local structure, local polynomial regression

*DOI*

*Abstract*

The adaptive weights smoothing (AWS) procedure was introduced in Polzehl and Spokoiny (2000) in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is also fully adaptive and dimension free. Simulations with artificial images show that AWS is superior to classical smoothing techniques especially when the underlying image function is discontinuous and can be well approximated by a piecewise constant function. However, the latter assumption can be rather restrictive for a number of potential applications. Here the AWS method is generalized to the case of an arbitrary local linear parametric structure. We also establish some important results about properties of the AWS procedure including the so called "propagation condition" and spatial adaptivity. The performance of the procedure is illustrated by examples for local polynomial regression in univariate and bivariate situations.

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# Stabilization of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

*Authors*

- Vodák, Rostislav

*2010 Mathematics Subject Classification*

- 35Q30 35B35 76N10

*Keywords*

- stabilization, asymptotic behaviour, Navier-Stokes equations, isothermal fluids

*Abstract*

The aim of this paper is to study the stabilization of solutions to the Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor. We study stabilization from the point of view of the method used in [17], where authors studied the asymptotic behaviour of solutions to barotropic compressible Navier-Stokes equations.

*Appeared in*

- J. Evol. Equ. 4 (2004), no. 2, 213--247, under new title: Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor.

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# On thermodynamic modeling and the role of the second law of thermodynamics in geophysics

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 80-01 74A99 74F10

*Keywords*

- thermodynamics, geophysics, poroelastic materials

*Abstract*

The article contains a brief review of elements of thermodynamic modeling in theoretical geophysics. We motivate the existence of the second law of thermodynamics in macroscopic theoretical physics and demonstrate its evaluation. In particular we show its consequences in the construction of constitutive laws for a two-component poroelastic medium. This construction is also related to microstructural properties verified by means of the second law.

*Appeared in*

- Advanced Mathematical and Computational Geomechanics, D. Kolymbas (ed.), Springer, 3--33, 2003

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# High order central schemes applied to relativistic multicomponent flows

*Authors*

- Qamar, Shamsul

*2010 Mathematics Subject Classification*

- 65M99 65Y20

*Keywords*

- multicomponent flows, relativistic Euler equations, central schemes, higher order accuracy

*Abstract*

The dynamics of inviscid multicomponent relativistic fluids may be modelled by the relativistic Euler equations, augmented by one (or more) additional species equation (s). We use high-resolution central schemes to solve these equations. The equilibrium states for each component are coupled in space and time to have a common temperature and velocity. The current schemes can handle strong shocks and the oscillations near the interfaces are negligible, which usually happens in the multicomponent flows. The schemes also guarantee the exact mass conservation for each component and the exact conservation of total momentum and energy in the whole particle system. The central schemes are robust, reliable, compact and easy to implement. Several one- and two-dimensional numerical test cases are included in this paper, which validate the application of these schemes to relativistic multicomponent flows.

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# Dynamo action in cellular convection

*Authors*

- Seehafer, Norbert
- Demircan, Ayhan

*2010 Mathematics Subject Classification*

- 76U05 76W05

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 42.81.Dp

*Keywords*

- Magnetohydrodynamics, Convection, Dynamo

*Abstract*

The dynamo properties of square patterns in Boussinesq Rayleigh-Benard convection in a plane horizontal layer are studied numerically. Cases without rotation and with weak rotation about a vertical axis are considered, particular attention being paid to the relation between dynamo action and the kinetic helicity of the flow. While the fluid layer is symmetric with respect to up-down reflections, the square-pattern solutions may or may not possess this vertical symmetry. Vertically symmetric solutions, appearing in the form of checkerboard patterns, do not possess a net kinetic helicity and we find them to be incapable of dynamo action at least up to magnetic Reynolds numbers of ≈ 12000. Vertically asymmetric squares, a secondary convection pattern appearing via the skewed varicose instability of rolls and being characterized by rising (descending) motion in the centers and descending (rising) motion near the boundaries, can in turn be devided into such that possess full horizontal square symmetry and others lacking also this symmetry. The flows lacking both the vertical and horizontal symmetries are particularly interesting in that they possess kinetic helicity and show kinematic dynamo action even without rotation. The generated magnetic fields are concentrated in vertically oriented filamentary structures near cell boundaries. The dynamos found in the nonrotating case are, however, always only kinematic, never nonlinear dynamos. Nonlinearly the back-reaction of the magnetic field then forces the flow into the basin of attraction of a roll-pattern solution incapable of dynamo action. But with rotation added parameter regions are found where a subtle balance between the Coriolis and Lorentz forces enables nonlinear dynamo action of stationary asymmetric squares. In some parameter regions this balance leads to nonlinear dynamos with flows in the form of oscillating squares or stationary modulated rolls.

*Appeared in*

- Magnetohydrodynamics 39 (3), 335-342, 2003.

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# Random walk on spheres algorithm for biharmonic equation: Optimization and error estimation

*Authors*

- Sabelfeld, Karl K.
- Shkarupa, Elena

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Random Walk on Spheres algorithm, global estimators, biharmonic equation, optimization and error estimation, multilinear interpolation

*Abstract*

The global algorithm of Random Walk on Spheres suggested by K.K. Sabelfeld is analysed and a kind of optimisation strategy is suggested. The algorithm is applied here to construct a functional version of this method which uses a multilinear interpolation. As an example we have chosen the biharmonic equation governing the bending of a thin elastic plate with the simply supported boundary, however generalizations to other equations can be carried out.

*Appeared in*

- Monte Carlo Methods Appl., 9 (2003) pp. 51-65.

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# Kinetic schemes of selected initial and boundary value problems

*Authors*

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

*2010 Mathematics Subject Classification*

- 76Y05 82C40 82C70

*Keywords*

- Maximum Entropy Principle, Extended Thermodynamics, kinetic schemes, relativistic Euler equations, conservation laws, hyperbolic systems, shock waves, kinetic theory of phonons

*Abstract*

The hyperbolic system that describes heat conduction at low temperatures and the relativistic Euler equations belong to a class of hyperbolic conservation laws that result from an underlying kinetic equation. The focus of this study is the establishment of an kinetic approach in order to solve initial and boundary value problems for the two examples. The ingredients of the kinetic approach are: (i) Representation of macroscopic fields by moment integrals of the kinetic phase density. (ii) Decomposition of the evolution into periods of free flight, which are interrupted by update times. (iii) At the update times the data are refreshed by the Maximum Entropy Principle.

*Appeared in*

- Analysis and Numerics for Conversation Laws, G. Warnecke, ed., Springer, Berlin [u.a.], 2005, pp. 293-306.

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# Dynamical approach to complex regional economic growth based on Keynesian model for China

*Authors*

- Yanchuk, Serhiy
- Kristensen, Gustav
- Sushko, Irina

*2010 Mathematics Subject Classification*

- 91B62 91B28

*Keywords*

- regional economic growth, China, synchronization, Keynesian model

*Abstract*

The paper addresses the problem of complex regional economic growth by using nonlinear Keynesian model with focusing on direct foreign investments effects. We investigate the dynamics of the model for the broad range of parameters which, in particular, contains the parameter values obtained recently by econometric analysis of the data for economic growth in China. For the single-region model we give conditions for which the dynamics of the model will be chaotic or regular. The parameters which prevent the economic stagnation are indicated. Further, we consider the model for two regions with a common trade as a coupling factor. The conditions are given for the two trading systems to exhibit chaotic synchronization, in-phase and out-of-phase behavior.

*Appeared in*

- Chaos Soliton Fractals, 18 (2003), pp. 937-952

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# Electro-reaction-diffusion systems with nonlocal constraints

*Authors*

- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35B40 35B45 35D05 35K35 35K57 35R05 78A35 80A30

*Keywords*

- Drift-diffusion systems, reaction-diffusion systems, heterostructures, energy estimates, global estimates, asymptotic behaviour, global existence, fixed point theorems

*Abstract*

The paper deals with equations modelling the redistribution of charged particles by reactions, drift and diffusion processes. The corresponding model equations contain parabolic PDEs for the densities of mobile species, ODEs for the densities of immobile species, a possibly nonlinear, nonlocal Poisson equation and some nonlocal constraints. Based on applications to semiconductor technology these equations have to be investigated for non-smooth data and kinetic coefficients which depend on the state variables. In two space dimensions we discuss the steady states of the system, we prove energy estimates, global a priori estimates and give a global existence result.

*Appeared in*

- Mathematischen Nachrichten 277(2004), pp.14-46

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# On Biot-like models and micro-macro transitions for poroelastic materials

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 80A17 74A20 74F10

*Keywords*

- Thermodynamics, Biot's model, poroelastic materials, Gassmann relations

*Abstract*

The paper is devoted to the thermodynamic derivation of a two-component poroelastic model with balance equation of porosity as a prototype of Biot's model. It is shown that a constitutive dependence on the porosity gradient yields the possibility of the construction of the linear Biot's model of poroelastic materials provided one negelcts the relaxation of porosity. A procedure of micro-macrotransition for a homogeneous microstructure yields Gassmann relations as an approximation of the thermodynamic model. Simultaneously it is shown that the model with the porosity balance equation can be applied to porous materials with a rather rigid skeleton. In the case of soft granular materials there exists a correction to porosity changes which follow from an appropriate modification of the porosity source.

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# Pattern formation in intracortical neuronal fields

*Authors*

- Hutt, Axel
- Bestehorn, Michael
- Wennekers, Thomas

*2010 Mathematics Subject Classification*

- 92C20 45G10

*2008 Physics and Astronomy Classification Scheme*

- 87.10.+e 05.65.+b 05.70.Jk

*Keywords*

- Neural fields, nonlinear integral equations, propagation delay, bifurcat ion analysis, pattern formation

*Abstract*

The present article introduces a neuronal field model for both excitatory and inhibitory connections. A single integro-differential equation with delay is derived and studied at a critical point by stability analysis, which yields conditions for static periodic patterns and wave instabilities. It turns out that waves only occur below a certain threshold of the activity propagation velocity. An additional brief study exhibits increasing phase velocities of waves with decreasing slope subject to increasing activity propagation velocities, which are in accordance to experimental results. Numerical studies near and far from instability onset supplement the work.

*Appeared in*

- Network: Comput. Neural. Syst. 14, pp. 351-368, (2003)

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# Detection of mutual phase synchronization in multivariate signals and application to phase ensembles and chaotic data

*Authors*

- Hutt, Axel
- Daffertshofer, Andreas
- Steinmetz, Ulrich

*2010 Mathematics Subject Classification*

- 34C15 34C28 34D30

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Sk 05.45.Xt 05.10.-a

*Keywords*

- Multivariate analysis, transient behavior, phase synchronization, chaotic systems

*Abstract*

The work presents a novel method for the detection of mutual phase synchronization in non-stationary time series. We show how the application of a cluster algorithm that considers spatio-temporal structures of data follows from the general condition of phase-synchronized data. In view of the topology of phasic data, we re-formulate the 𝖪-Means cluster algorithm on a flat torus and apply a segmentation index derived in an earlier work (Physica D 177,203-232(2003)). This index is extended by means of averaging in order to reflect phase synchronization in ensembles of multivariate time series. The method is illustrated using simulated multivariate phase dynamics and arrays of chaotic systems, in which temporal segments of phase-synchronized states are registered. A comparison with results from an existing bivariate synchronization index reveals major advantages of our method.

*Appeared in*

- Physical Review E 68, 036219 (2003)

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# Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme*

*Authors*

- Klein, Olaf

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

*2010 Mathematics Subject Classification*

- 45K05 65M99 35K05 35K55 65N22 47H10 80A20

*Keywords*

- Integro-partial differential equations, Finite volume method, Nonlinear parabolic PDEs, Integral operators, Nonlocal interface conditions, Diffuse-gray radiation, Maximum principle

*Abstract*

This article presents a finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions modeling diffuse-gray radiation between the surfaces of (both open and closed) cavities. The model is considered in three space dimensions, modifications for the axisymmetric case are indicated. Proving a maximum principle as well as existence and uniqueness for roots to a class of discrete nonlinear operators that can be decomposed into a scalar-dependent sufficiently increasing part and a benign rest, we establish a discrete maximum principle for the finite volume scheme, yielding discrete L^{∞}-L^{∞} a priori bounds as well as a unique discrete solution to the finite volume scheme.

*Appeared in*

- Mathematical Models and Methods in Applied Sciences, 15 (2005), 227-258

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# Component identification and estimation in nonlinear high-dimensional regression models by structural adaption

*Authors*

- Samarov, Alexander
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Vial, Celine

*2010 Mathematics Subject Classification*

- 62H30 62J02

*Keywords*

- structural adaptation, partially linear model, component analysis

*Abstract*

This article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown. The target of analysis is identification of the set of regressors which enter in a nonlinear way in the model function, and the complete estimation of the model including slope coefficients of the linear component and the link function of the nonlinear component. The procedure also allows for selecting the significant regression variables. As a by-product, we develop a test of linear hypothesis against a partially linear alternative, or, more generally, a test that the nonlinear component is M-dimensional for M = 0,1,2,....

The approach proposed in this article is fully adaptive to the unknown model structure and applies under mild conditions on the model. The only important assumption is that the dimensionality of nonlinear component is relatively small. The theoretical results indicate that the procedure provides a prescribed level of the identification error and estimates the linear component with the accuracy of order 𝑛^{-1/2}. A numerical study demonstrates a very good performance of the method even for small or moderate sample sizes.

*Appeared in*

- J. Amer. Statist. Assoc., 100 (2005) pp. 429--445.

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# On jump conditions at phase boundaries for ordered and disordered phases

*Authors*

- Dreyer, Wolfgang

*2010 Mathematics Subject Classification*

- 74N20 74N25 80A22

*Keywords*

- Free boundaries, phase transitions, ordered and disordered systems

*Abstract*

This is a study on jump conditions across the interface between two adjacent phases. The interface behaves as a free boundary, and in sharp interface models jump conditions are used to determine the values of thermodynamic fields at the free boundaries. In this study the jump conditions are derived from balance equations for singular surfaces that do not have singular lines, i.e. triple junctions are not considered here. At first we present the most general form of jump conditions to give a general framework, from where we consider various special cases with a focus on the influence of mechanical fields on the interfacial processes. The special cases include the Hoffmann/Cahn capillarity vector theory and jump conditions for interfaces where order/disorder transitions are involved. Furthermore we discuss interfacial chemical reaction laws, and in particular the creation and annihilation of vacancies at a liquid/solid interface.

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# Macroscopic modeling of porous and granular materials --- microstructure, thermodynamics and some boundary-initial value problems

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 35L50 74J15 80A17 74A20 74F10

*Keywords*

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

*Abstract*

This work contains the material presented in the key lecture during the Congress Cancam 2003 (Calgary, Canada). It contains a review of the recent development of thermodynamic modeling of porous and granular materials. We present briefly main features of the thermodynamic construction of a nonlinear poroelastic model but the emphasis is put on the analysis of a linear two-component model. In particular we indicate similarities and differences of the thermodynamic model with the classical Biot's model of porous materials. We analyze jacketed and ujacketed Gedankenexperiments which provide a micro-macrotransition procedure for compressibilities. This gives rise to Gassmann-like relations which are incorporated in wave analysis. An acoustic waves analysis is presented in some details. In particular we show the construction of bulk monochromatic waves as well as some surface waves and indicate their practical applications in testing of soils.

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# Stochastic Eulerian model for the flow simulation in porous media

*Authors*

- Kolyukhin, Dmitry R.
- Sabelfeld, Karl K.

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

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

*Abstract*

This work deals with the stochastic flow simulation in statistically isotropic and anisotropic saturated porous media in 3D case. The hydraulic conductivity is assumed to be a random field with lognormal distribution. Under the assumption of smallness of fluctuations in the hydraulic conductivity we construct a stochastic Eulerian model for the incompressible flow as a divergenceless Gaussian random field with a spectral tensor of a special structure derived from Darcy's law. A randomized spectral representation is then used to simulate this random field. Numerical results are compared with the analytical results obtained by the small pertrubation expansion. A series of test calculations confirmed the high accuracy and computational efficiency of the method. Comparisons with asymptotically exact results show a good agreement.

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# Minimax nonparametric hypothesis testing for small type I errors

*Authors*

- Ingster, Yuri I.
- Suslina, Irina A.

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Minimax hypothesis testing, nonparametric signal detection, adaptive hypothesis testing, intermediate efficiency

*Abstract*

Under the white Gaussian noise model with the noise level ε → 0, we study minimax nonparametric hypothesis testing problem 𝐻_{0} : ƒ = 0 on unknown function ƒ ∈ L_{2}(0,1). We consider alternative sets that are determined a regularity constraint in the Sobolev norm and we suppose that signals are bounded away from the null either in L_{2}-norm or in L_{∞}-norm. Analogous problems are considered in the sequence space. If type I error probability α ∈ (0,1) is fixed, then these problems were studied in book [13]. In this paper we consider the case α → 0. We obtain either sharp distinguishability conditions or sharp asymptotics of the minimax type II error probability in the problem. We show that if α is "not too small", then there exists natural extension of results [13], whenever if α is "very small", then we obtain classical asymptotics and distinguishability conditions for small α. Adaptive problems are studied as well.

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# Stress analysis and bending tests for GaAs wafer

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank
- Eichler, Stefan
- Jurisch, Manfred

*2010 Mathematics Subject Classification*

- 74B2 74K20 74M20 74M15

*Keywords*

- non-linear elasticity, anisotropy, plate theory, von Kármán, Hertzian contact

*Abstract*

Wafer made from single crystal Gallium Arsenide (GaAs) are used as substrate materials in micro- and opto- electronic devices. During the various processes of manufacturing, the wafer are subjected to mechanical loads which may lead to fracture. The characterization of the fracture toughness of the wafer needs bending tests and a theoretical calculation of various stress distributions within the wafer.

In this study we show that the nonlinear von Kármán theory may serve as an appropriate tool to calculate the stress distributions as functions of the external load, while the Kirchhoff theory has turned out to be completely inappropriate. Our main focus is devoted to (i) calculation of the contact area between the load sphere and the wafer, (ii) study of the influence of the anisotropic character of the material, (iii) study of the important geometric nonlinearity. Finally we compare the calculated and theoretical load-flexure relations in order to demonstrate the high accuracy of the von Kármán theory and its Finite Element implementation.

*Appeared in*

- Microelectronics Reliability, 46 (2006) pp. 822--835.

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# Analytic-numerical investigation of delayed exchange of stabilities in singularly perturbed parabolic problems

*Authors*

- Nefedov, Nikolai N.
- Radziunas, Mindaugas
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 35B25 35K20

*Keywords*

- singularly perturbed parabolic equation, Neumann boundary value problem, delayed exchange of stabilities, interior boundary layer

*Abstract*

We consider a class of singularly perturbed parabolic problem in case of exchange of stabilities, that is, the corresponding degenerate equation has two intersecting roots. We present an analytic result about the phenomenon of delayed exchange of stabilities and compare it with numerical investigations of some examples.

*Appeared in*

- Comp. Math. and Math. Physics 44(7), pp. 1213-1220, (2004)

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# Global behavior and asymptotic reduction of a chemical kinetics system with continua of equilibria

*Authors*

- Schneider, Klaus R.
- Kalachev, Leonid V.

*2010 Mathematics Subject Classification*

- 34D05 34E15 34C60 92C45

*Keywords*

- chemical dynamics system, model reduction, invariant manifold, singular perturbations, boundary function method

*Abstract*

We consider a model chemical kinetics system describing the dynamics of species concentrations taking part in a consecutive-competitive reaction in a continuously stirred tank reactor. Corresponding dynamical system has a continua of equilibria. The solution of the system tends to a particular equilibrium depending on the initial conditions. Global behavior of the system and its reductions via invariant manifold theory and the boundary function methods are studied.

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# A quantum transmitting Schrödinger-Poisson system

*Authors*

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

*2010 Mathematics Subject Classification*

- 34B24 34L40 47B44 81U20 82D37

*2008 Physics and Astronomy Classification Scheme*

- 85.35.-p

*Keywords*

- Quantum phenomena, current carrying state, inflow boundary condition, dissipative operators, open quantum systems, carrier and current densities, density matrices, quantum transmitting boundary method

*Abstract*

We consider a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schroedinger operator is defined on the bounded domain with transparent boundary conditions. This allows to model a non-zero current flow trough the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

*Appeared in*

- Reviews in Mathematical Physics, 2004, Vol. 16, No. 3, pp. 281-330

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# On the Gibbs phase rule in the Pirogov-Sinai regime

*Authors*

- Bovier, Anton
- Merola, Immacolata
- Presutti, Ericco
- Zahradník, Miloš

*2010 Mathematics Subject Classification*

- 82A25

*Keywords*

- Gibbs phase rule, Pirogov-Sinai theory, contour models, Kac-interactions

*Abstract*

We consider extended Pirogov-Sinai models including lattice and continuum particle systems with Kac potentials. Calling λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term -λα, we prove that if a Pirogov-Sinai phase transition with order parameter λ occurs at λ = 0, then this is the only point in an interval of values of λ centered at 0, where phase transitions occur.

*Appeared in*

- J. Statist. Phys, 114 (2004), pp. 1235--1267

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# Qualitative theory and identification of a class of mechanical systems

*Authors*

- Volkova, Viktorija E.
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34A55 70E99

*Keywords*

- identification, mechanical system, second order differential equations

*Abstract*

We consider mechanical systems which can be described by the nonlinear differential equation my+h(y,y)+r(y)=0 and which are characterized by the property that there exists no information or only partial information on the damping force h or the restoring force r. We characterize several classes of such systems where by applying a periodic force and by measuring acceleration, velocity and displacement, the functions h or r can be easily identified.

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# Perturbation analysis of chance-constrained programs under variation of all constraint data

*Authors*

- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15 90C31

*Keywords*

- probabilistic constraints, chance constraints, stability, stochastic optimization

*Abstract*

We consider stability of solutions to optimization problems with probabilistic constraints under perturbations of all constraint data (probability level, probability measure, deterministic constraints, random set mapping). Constraint qualifications ensuring stability are derived for each of the single parameters. Examples illustrating the necessity of the stated conditions as well as the limitations of the given results are provided.

*Appeared in*

- R. Henrion, Dynamic Stochastic Optimization, K. Marti, ed., vol. 532 of Lecture Notes in Economics and Mathematical Systems, Springer, Heidelberg, 2004, pp. 257--274

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# The numerical solution of an inverse periodic transmission problem

*Authors*

- Bruckner, Gottfried
- Elschner, Johannes

*2010 Mathematics Subject Classification*

- 78A46 78M50 35R30 35J05

*Keywords*

- Diffraction grating, TE transmission problem, profile reconstruction, Tikhonov regularization, optimization method

*Abstract*

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We discuss convergence and implementation of an optimization method for solving the inverse TE transmission problem, following an approach first developed by Kirsch and Kress for acoustic obstacle scattering. The convergence analysis includes the case of Lipschitz grating profiles and relies on variational methods and solvability properties of periodic boundary integral equations. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm.

*Appeared in*

- Math. Methods Appl. Sci., 28 (2005) pp. 757--778.

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# Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension

*Authors*

- Dreyer, Wolfgang
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74D10 74F05 74F10 77N25

*Keywords*

- Matched asymptotics, boundary integral method, numerics

*Abstract*

We consider the problem of phase separation in eutectic alloy such e.g. SnPb. For this we derive a phase field model from a atomistic point of view. We find the surface energy to be anisotropic, having in general a nonlinear dependence on concentration. We use matched asymptotic analysis to obtain a corresponding sharp interface model. The resulting expression for the surface tension agrees with that found on the basis of classical thermodynamics for jump conditions at singular interfaces. A boundary integral formulation of the sharp interface model enables us to numerically describe the motion and deformation of the binary alloy.

*Appeared in*

- Interfaces Free Boundaries 7 (2005), pp. 199--227.

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# On the noise-induced passage through an unstable periodic orbit I: Two-level model

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H20 60H10 34F05

*Keywords*

- Stochastic exit problem, diffusion exit, first-exit time, large deviations, metastability, level-crossing problem, limit cycle, synchronization, phase slip, cycling, stochastic resonance

*Abstract*

We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips between synchronized oscillators, or when studying stochastic resonance far from the adiabatic limit. We introduce a simple, piecewise linear model equation, for which the distribution of first-passage times can be precisely computed. In particular, we obtain a quantitative description of the phenomenon of cycling: The distribution of first-passage times rotates around the unstable orbit, periodically in the logarithm of the noise intensity, and thus does not converge in the zero-noise limit. We compute explicitly the cycling profile, which is universal in the sense that in depends only on the product of the period of the unstable orbit with its Lyapunov exponent.

*Appeared in*

- J. Statist. Phys., 114 (2004), pp. 1577-1618

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# Maximal temperature of safe combustion in case of an autocatalytic reaction

*Authors*

- Schneider, Klaus R.
- Shchepakina, Elena A.

*2010 Mathematics Subject Classification*

- 34E15 80A25

*Keywords*

- singular perturbations, canards, combustion

*Abstract*

We consider the problem of thermal explosion of a gas mixture in the case of an autocatalytic combustion reaction in a homogeneous medium. We determine the maximal temperature on the trajectories located in the transition region between the slow combustion regime and the explosive one.

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# Heat equation with strongly inhomogeneous noise

*Authors*

- Zähle, Henryk

*2010 Mathematics Subject Classification*

- 60H15 35R60 60H20 60J80

*Keywords*

- stochastic partial differential equation, martingale problem, singularmeasure, catalytic super-Brownian motion

*Abstract*

We consider the stochastic heat equation in dimension one with singular drift and driven by an inhomogeneous space-time white noise whose quadratic variation measure is not absolutely continuous w.r.t. Lebesgue measure, neither in space nor in time. Under various assumptions we give statements on strong and weak existence as well as strong and weak uniqueness of continuous solutions.

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# Dynamics of two mutually coupled semi conductor lasers: Instantaneous coupling limit

*Authors*

- Yanchuk, Serhiy
- Schneider, Klaus R.
- Recke, Lutz

*2010 Mathematics Subject Classification*

- 34C15 34K60 34C11 34C14 34K20 34C60

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 42.55.Px

*Keywords*

- two coupled lasers, complete synchronization, antisynchronization, detuning

*Abstract*

We consider two semiconductor lasers coupled face to face under the assumption that the delay time of the injection is small. The model under consideration consists of two coupled rate equations, which approximate the coupled Lang-Kobayashi system as the delay becomes small. We perform a detailed study of the synchronized and antisynchronized solutions for the case of identical systems and compare results from two models: with the delay and with instantaneous coupling. The bifurcation analysis of systems with detuning reveals that self-pulsations appear via bifurcations of stationary (i.e. continuous wave) solutions. We discover the connection between stationary solutions in systems with detuning and synchronous (also antisynchronous) solutions of coupled identical systems. We also identify a codimension two bifurcation point as an organizing center for the emergence of chaotic behavior.

*Appeared in*

- Phys. Rev. E., 69 (2004), pp. 056221-1 - 056221-12

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# An example of a resonant homoclinic loop of infinite cyclicity

*Authors*

- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37G20 34C07 34C37

*Keywords*

- codimension-3 homoclinic bifurcation, invariant manifold, swallow tail, limit cycle

*Abstract*

We describe a codimension-3 bifurcational surface in the space of 𝐶^{𝑟}-smooth (𝑟 ≥ 3) dynamical systems (with the dimension of the phase space equal to 4 or higher) which consists of systems which have an attractive two-dimensional invariant manifold with an infinite sequence of periodic orbits of alternating stability which converge to a homoclinic loop.

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# High-frequency pulsations in DFB-lasers with amplified feedback

*Authors*

- Brox, Olaf
- Bauer, Stefan
- Radziunas, Mindaugas
- Wolfrum, Matthias
- Sieber, Jan
- Kreissl, Jochen
- Sartorius, Bernd
- Wünsche, Hans-Jürgen

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf

*Keywords*

- semiconductor laser, optical feedback, pulsations

*Abstract*

We describe the basic ideas behind the concept of DFB-lasers with short optical feedback for the generation of high-frequency self-pulsations (SPs) and show the theoretical background describing realized devices. It is predicted by theory that the SP frequency increases with increasing feedback strength. To provide evidence for this we propose a novel device design which employs an amplifier section in the integrated feedback cavity of a DFB-laser. We present results from numerical simulations and experiments. It has been shown experimentally that a continuous tuning of the SP frequency from 12 to 45GHz can be adjusted via the control of the feedback strength. The numerical simulations which are in good accordance with experimental investigations give an explanation for a self stabilizing effect of the SPs due to the additional carrier dynamic in the integrated feedback cavity.

*Appeared in*

- IEEE J. Select Topics Quantum Electron., 39 (2003), pp. 1381-1387

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# Finite element method for epitaxial growth with thermodynamic boundary conditions

*Authors*

- Bänsch, Eberhard
- Haußer, Frank
- Voigt, Axel

*2010 Mathematics Subject Classification*

- 35Q99 35R35 65N30 65Z05 74S05

*Keywords*

- epitaxial growth, island dynamics, free or moving boundary problem, adatom diffusion, surface diffusion, mean curvature flow, Gibbs-Thomson, finite elements, adaptivity, front tracking

*Abstract*

We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height, thermodynamic boundary conditions on terrace boundaries including anisotropic line tension, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional (possibly anisotropic) "surface" diffusion of edge-adatoms along the step-edges. The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the boundary evolution. A penalty method is used in order to incorporate the boundary conditions. The evolution of the terrace boundaries includes both the weighted/anisotropic mean curvature flow and the weighted/anisotropic surface diffusion. Its governing equation is solved by a semi-implicit front-tracking method using parametric finite elements.

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# PID-control of laser surface hardening of steel

*Authors*

- Hömberg, Dietmar
- Weiss, Wolf

*2010 Mathematics Subject Classification*

- 35K05 93C20

*Keywords*

- Feedback Control, Heat Treatment, Phase Transitions

*Abstract*

We discuss control strategies for the laser surface hardening of steel. The goal is to acchieve a prescribed hardening depth avoiding surface melting. Our mathematical model consists of a system of ODEs for the phase volume fractions coupled with the heat equation. The system is solved semi-implicitely using the finite element method. To obtain a uniform hardening depth the first attempt is to use PID control to achieve a constant temperature in the hot spot of the laser beam on the surface. However, the numerical results prove that this is not sufficient. We show that the best strategy is to control the temperature close to the lower boundary of the hardening zone. Then one can compute the optimal temperature in the hot spot of the beam and use it as the set-point for the pyrometer control of the real process.

*Appeared in*

- IEEE Trans. Control Syst. Technol., 14 (2006) pp. 896--904.

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# Convergence towards equilibrium of Probabilistic Cellular Automata

*Authors*

- Louis, Pierre-Yves

*2010 Mathematics Subject Classification*

- 60G60 60J10 60K35 82C20 82C26

*Keywords*

- Probabilistic Cellular Automata, Interacting Particle Systems, Coupling, Attractive Dynamics, %Stochastic Ordering, Weak Mixing Condition, Ergodicity, Exponential rate of convergence, Gibbs measure

*Abstract*

We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, and under some assumption ( 𝒜) we establish ergodicity for general attractive probabilistic cellular automata on S^{ℤd}, where S is finite: this means the convergence towards equilibrium of these Markovian parallel dynamics, in the uniform norm, exponentially fast. For a class of reversible PCA dynamics on {-1,+1}^{ℤd}, with a naturally associated Gibbsian potential 𝜑, we prove that a Weak Mixing condition for 𝜑 implies the validity of the assumption (𝒜), thus the 'exponential ergodicity' of the dynamics towards the unique Gibbs measure associated to 𝜑 holds. On some particular examples of this PCA class, we verify that our assumption (𝒜) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some precise PCA, the 'exponential ergodicity' holds as soon as there is no phase transition.

*Appeared in*

- Electronic Communications in Probability, vol. 9, pp 119-131, 7.10.2004

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# Pareto approximation of the tail by local exponential modeling

*Authors*

- Grama, Ion G.
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G32 62G05

*Keywords*

- Hill estimator, tail index, adaptive choice, model Hill estimator, extreme values

*Abstract*

We give a new adaptive method for selecting the number of upper order statistics used in the estimation of the tail of a distribution function. Our approach is based on approximation by an exponential model. The selection procedure consists in consecutive testing for the hypothesis of homogeneity of the estimated parameter against the change-point alternative. The selected number of upper order statistics corresponds to the first detected change-point. Our main results are non-asymptotic and state optimality of the proposed method in the "oracle" sense.

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# A condition for weak disorder for directed polymers in random environment

*Authors*

- Birkner, Matthias

*2010 Mathematics Subject Classification*

- 60K37 82B44

*Keywords*

- Directed polymer in random environment, weak disorder, size-biasing

*Abstract*

We give a sufficient criterion for the weak disorder regime of directed polymers in random environment, which extends a well-known second moment criterion. We use a stochastic representation of the size-biased law of the partition function.

*Appeared in*

- Electron. Comm. Probab., 9 (2004) pp. 22-25

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# Monte Carlo methods for pricing and hedging American options

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H30 65C30 91B28

*Keywords*

- Pricing and hedging of American options, Monte Carlo simulation, Determination of the exercise boundary

*Abstract*

We introduce a new Monte Carlo method for constructing the exercise boundary of an American option in a generalized Black-Scholes framework. Based on a known exercise boundary, it is shown how to price and hedge the American option by Monte Carlo simulation of suitable probabilistic representations in connection with the respective parabolic boundary value problem. The methods presented are supported by numerical simulation experiments.

*Appeared in*

- International Journal of Theoretical and Applied Finance Vol. 7, No. 5, 591-614(2004) under the title ''A new Monte Carlo method for American options''

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# Stationary solutions of two-dimensional heterogeneous energy models with multiple species

*Authors*

- Glitzky, Annegret
- Hünlich, Rolf

*Keywords*

- energy models, mass, charge and energy transport in heterostructures, strongly coupled elliptic systems, mixed boundary conditions, Implicit Function Theorem, existence, uniqueness, regularity

*Abstract*

We investigate stationary energy models in heterostructures consisting of continuity equations for all involved species, of a Poisson equation for the electrostatic potential and of an energy balance equation. The resulting strongly coupled system of elliptic differential equations has to be supplemented by mixed boundary conditions. If the boundary data are compatible with thermodynamic equilibrium then there exists a unique steady state. We prove that in a suitable neighbourhood of such a thermodynamic equilibrium there exists an unique steady state, too. Our proof is based on the Implicit Function Theorem and on regularity results for systems of strongly coupled elliptic differential equations with mixed boundary conditions and non-smooth data.

*Appeared in*

- Banach Center Publ. bf 66 (2004), pp. 135--151.

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# A new model for passive mode-looking in a semiconductor laser

*Authors*

- Vladimirov, Andrei
- Turaev, Dmitry
- Kozyreff, Gregory

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

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

*Abstract*

We propose a new model for passive mode-locking that is a set of ordinary delay differential equations. We assume the ring cavity geometry and a Lorentzian spectral filtering of the pulses, but do not use small gain and loss and weak saturation approximations. By means of a continuation method we study mode-locking solutions and their stability. We found that stable mode-locking can exist even when the non-lasing state between pulses becomes unstable.

*Appeared in*

- Optics Letters, 29, #11, 1221-1223 (2004)

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# On unique solvability of nonlocal drift-diffusion type problems

*Authors*

- Gajewski, Herbert
- Skrypnik, Igor V.

*2010 Mathematics Subject Classification*

- 35B45 35K15 35K20 35K65

*Keywords*

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

*Abstract*

We prove global existence and uniqueness of bounded weak solutions to Cauchy--Neumann problems for degenerate parabolic equations with drift terms determined by integral equations instead of by elliptic boundary problems as in the corresponding local case. Such problems arise as mathematical models of various transport processes driven by gradients of local particle concentrations and nonlocal interaction potentials. Examples are transport of charge carriers in semiconductors and phase separation processes in alloys.

*Appeared in*

- Nonlinear Anal., 56 (2004) pp. 803--830.

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# Optimal design of mechanical structures

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49J20 49J40 49K20

*Keywords*

- linear elastic curved systems, shape optimization, control-into-coefficients

*Abstract*

We prove new properties for the linear isotropic elasticity system and for thickness minimization problems. We also present very recent results concerning shape optimization problems for three-dimensional curved rods and for shells. The questions discussed in this paper are related to the control variational method and to control into coefficients problems.

*Appeared in*

- Chapter 18 of ``Control Theory of Partial Differential Equations'' (O. Imanuvilov, G. Leugering, R. Triggiani, B.-Y. Zhang, eds.), Pure and Applied Mathematics Series 242, 2005, Chapman & Hall, Boca Raton, pp. 239-271.

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# Solutions for quasilinear nonsmooth evolution systems in L^p

*Authors*

- Elschner, Johannes
- Maz'ya, Vladimir
- Rehberg, Joachim
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 35K55 35D10 35R05 35K45 35K50 35J25

*Keywords*

- Quasilinear parabolic systems, elliptic boundary value problems, polyhedral domains, piecewise constant coefficients, regularity of solutions

*Abstract*

We prove that nonsmooth quasilinear parabolic systems admit a local, strongly differentiable (with respect to time) solution in L^{p} over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems with anisotropic materials. These results are based on sharp pointwise estimates for Green's function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction diffusion systems. The treatment of such equations in an L^{p} context seems to be new and allows (by Gauss' theorem) to define properly the normal component of currents across the boundary.

*Appeared in*

- Arch. Rational Mech. Anal. 171 (2004), pp. 219-262

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# Supercontinuum generation by the modelation instability

*Authors*

- Demircan, Ayhan
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg, 42.81.Dp

*Keywords*

- Nonlinear Schroedinger Equation, Optical Fiber, Four-Wave-Mixing

*Abstract*

We report on a numerical study of supercontinuum generation in a single-mode optical fiber by the modulation instability. An ultrabroadband octave-spanning continuum is generated for femtosecond pulses with subkilowatt peak power. In particular, we investigate the influence of higher-order effects such as third- and fourth-order dispersion, self-steepening and intrapulse Raman scattering on the supercontinuum generation.

*Appeared in*

- Optics Communications, 244 (6), 181-185, 2005.

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# Coarse-graining techniques for (random) Kac models

*Authors*

- Bovier, Anton
- Külske, Christof

*2010 Mathematics Subject Classification*

- 82B44 82B20 82B28

*2008 Physics and Astronomy Classification Scheme*

- 05.10.Cc

*Keywords*

- Kac models, phase transitions, random field model, renormalization group, coarse-graining

*Abstract*

We review our recent results on the low temperature behavior of Kac models. We discuss translation-invariant models and the Kac version of the random field model. For the latter we outline, how various coarse-graining techniques can be used to prove ferromagnetic ordering in dimensions $dgeq 3$, small randomness, and low temperatures, uniformly in the range of the interaction.

*Appeared in*

- Interacting Stochastic Systems, J. Deuschel, A. Greven (eds.), Springer, Berlin--Heidelberg, 2005, pp. 11--28

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# Blue sky catastrophe in singularly perturbed systems

*Authors*

- Shilnikov, Andrey
- Shilnikov, Leonid
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37G15 34E15 37C27 34C26

*Keywords*

- saddle-node, global bifurcations, stability boundaries, slow-fast system, bursting oscillations, spikes, excitability

*Abstract*

We show that the blue sky catastrophe, which creates a stable periodic orbit whose length increases with no bound, is a typical phenomenon for singularly-perturbed (multi-scale) systems with at least two fast variables. Three distinct mechanisms of this bifurcation are described. We argue that it is behind a transition from periodic spiking to periodic bursting oscillations.

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# Gibbs properties of the fuzzy potts model on trees and in mean fields

*Authors*

- Häggström, Olle
- Külske, Christof

*2010 Mathematics Subject Classification*

- 82B20 82B26

*2008 Physics and Astronomy Classification Scheme*

- 05.50.+q 02.50.Cw

*Keywords*

- Potts model, fuzzy Potts model, Gibbs measures, non-Gibbsian measures, trees, mean-field models

*Abstract*

We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general trees that non-Gibbsianness of the fuzzy Potts model happens exactly for those temperatures where the underlying Potts model has multiple Gibbs measures.

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# A stochastic log-Laplace equation

*Authors*

- Xiong, Jie

*2010 Mathematics Subject Classification*

- 60G57 60H15 60J80

*Keywords*

- Superprocess, random environment, Wong-Zakai approximation, particle system representation, stochastic partial differential equation

*Abstract*

We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong (1999). We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler (2001).

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# On immediate-delayed exchange of stabilities and periodic forced canards

*Authors*

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

*2010 Mathematics Subject Classification*

- 34E15 34E05 34C25

*Keywords*

- exchange of stabilities, periodic forced canards, singularly perturbed ordinary differential equations

*Abstract*

We study scalar singularly perturbed non-autonomous ordinary differential equations whose associated equations feature the property of exchange of stabilities, i.e., the set of their equilibria consists of at least two intersecting curves. By means of the method of asymptotic lower and upper solutions we derive conditions guaranteeing that the solution of initial value problems exhibit the phenomenon of immediate exchange of stabilities as well as the phenomenon of delayed exchange of stabilities. We use this result to prove the existence of forced canard solutions.

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# Delayed loss of stability in systems with degenerate linear parts

*Authors*

- Rachinskii, Dimitri
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34D15 37G15

*Keywords*

- singular perturbation, delayed loss of stability, periodic solution, Hopf bifurcation

*Abstract*

We study singularly perturbed scalar and planar differential equations with linear parts independent of time. The associated autonomous equations undergo a bifurcation of equilibria in scalar case and the Hopf bifurcation in case of planar systems at a bifurcation point where the zero equilibrium loses stability. We suggest natural sufficient conditions for the phenomenon of delayed loss of stability for the singularly perturbed equations and estimate the asymptotic delay. Bifurcation points, stability of the zero equilibrium, and the asymptotic delay are determined by superlinear terms in the expansions of the right-hand sides of the associated and singularly perturbed equations.

*Appeared in*

- Z. Anal. Anwend., 22 (2003), pp. 433-453

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# Optimization problems for curved mechanical structures

*Authors*

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

*2010 Mathematics Subject Classification*

- 49Q10 74P10 49Q12

*Keywords*

- Shells and curved rods, minimal regularity, optimal design

*Abstract*

We study the optimization of three dimensional curved rods and of shells under minimal regularity assumptions for the geometry. The results that we establish concern the existence of optimal shapes and the sensitivity analysis. We also compute several numerical examples for the curved rods. The models that we use have been investigated in our previous work [11], [16] and a complete study of the Kirchhoff-Love arches and their optimization has been performed in [10].

*Appeared in*

- SIAM J. Control Optim., Volume 44 (2005), pp. 743-775

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# Synchronization of weakly stable oscillators and semiconductor laser arrays

*Authors*

- Kozyreff, Gregory
- Mandel, Paul
- Vladimirov, Andrei

*2010 Mathematics Subject Classification*

- 78A60 34C15 70K50

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf 42.55.Px 05.45.Xt

*Keywords*

- semiconductor laser, synchronization, coupled oscillators, bifurcations

*Abstract*

We study the synchronization properties of an array of non identical globally coupled limit cycle oscillators. Above a critical coupling strength, some oscillators undergo a selfpulsing instability. We study analytically the synchronization conditions below and above this instability threshold, thus removing the usual restriction of limit cycle stability. Selfpulsing decreases the order parameter and synchronization degradation can be reduced by delaying the coupling among the oscillators. Semiconductor lasers coupled by an external mirror are used as a convenient realization of that model.

*Appeared in*

- Europhys. Lett., 61 (2003) pp. 613--619.

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# Optimizing the temperature profile during sublimation growth of SiC single crystals: Control of heating power, frequency, and coil position

*Authors*

- Meyer, Christian
- Philip, Peter

*2010 Mathematics Subject Classification*

- 80M50 80A20 65Z05 65K10 49-04

*2008 Physics and Astronomy Classification Scheme*

- 81.10.Bk 02.60.Pn 02.60.Cb 44.05.+e 84.32.Hh

*Keywords*

- Numerical simulation. Sublimation growth. Physical vapor transport. Modified Lely method. SiC single crystal. Nelder-Mead method.Optimization. RF heating.

*Abstract*

We use a numerical optimization method to determine the control parameters frequency, power, and coil position for the radio frequency (RF) induction heating of the growth apparatus during sublimation growth of SiC single crystals via physical vapo transport (PVT) (also called the modified Lely method). The control parameters are determined to minimize a functional, tuning the radial temperature gradient on the single crystal surface as well as the vertical temperature gradient between SiC source and seed, both being crucial for high-quality growth. The optimization is subject to constraints with respect to a required temperature difference between source and seed, a required temperature range at the seed, and an upper bound for the temperature in the entire apparatus. The numerical computations use a stationary mathematical model for the heat transport, including heat conduction, radiation, and RF heating to solve the forward problem, and a Nelder-Mead method for optimization. A minimal radial temperature gradient is found to coincide with a minimal temperature at the single crystal surface, and a maximal temperature gradient between source and seed is found to coincide with a low coil position.

*Appeared in*

- Crystal Growth & Design, 5 (2005), 1145 - 1156.

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