# Mathematical Theory of Porous Media. - Lecture Notes. XXV Summer School on Mathematical Physics, Ravello, September 2000

*Authors*

- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 76S05 73S10 76R50 73D20 35L67

*Keywords*

- Flows in porous media, micromechanics of solids, surface waves, shock waves and singularities

*DOI*

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# Sound and shock waves in porous and granular materials

*Authors*

- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 76S05 73S10 73D20

*Keywords*

- Flows in porous media, filtration, micromechanics of solids, surface waves

*DOI*

*Appeared in*

- V.Ciancio, A.Donato, F.Oliveri, and S.Rionero, editors, Proceedings "WASCOM 99". 10th Conference on Waves and Stability in Continuous Media. Vulcano, Italy, 7-12 June 1999, pages 489-503. World Scientific, Singapur [u.a.], 2001

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# A subdifferential criterion for calmness of multifunctions

*Authors*

- Henrion, René
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 90C31 26E25 49J52

*Keywords*

- calmness, multifunctions, coderivative, constraint sets, nonlinear complementarity problems

*DOI*

*Abstract*

A criterion for the calmness of a class of multifunctions between finite-dimensional spaces is derived in terms of subdifferential concepts developed by Mordukhovich. The considered class comprises general constraint set mappings as they occur in optimization or mappings associated with a certain type of variational systems. The criterion for calmness is obtained as an appropriate reduction of Mordukhovich's well-known characterization of the stronger Aubin property (roughly spoken, one may pass to the boundaries of normal cones or subdifferentials when aiming at calmness).

*Appeared in*

- Journal of Mathematical Analysis and Applications 258 (2001), pp. 110-130

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# A large-deviations approach to gelation

*Authors*

- Andreis, Luisa
- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 05C80 60F10 60K35 82B26

*Keywords*

- Coagulation process, multiplicative coalescent, gelation, phase transition, large deviations, Erdős-Rényi random graph

*DOI*

*Abstract*

A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

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# Modeling and simulation of strained quantum wells in semiconductor lasers

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Kaiser, Hans-Christoph
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 78A60 68U20 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 73.20.Dx 85.60.Bt 78.66.Fd

*Keywords*

- quantum well laser, band structure, kp-method, Schrödinger-Poisson systems, exchange-correlation effects, optical gain, semiconductor laser simulation

*DOI*

*Abstract*

A model allowing for efficiently obtaining band structure information on semiconductor Quantum Well structures will be demonstrated which is based on matrix-valued kp-Schrödinger operators. Effects such as confinement, band mixing, spin-orbit interaction and strain can be treated consistently. The impact of prominent Coulomb effects can be calculated by including the Hartree interaction via the Poisson equation and the bandgap renormalization via exchange-correlation potentials, resulting in generalized (matrix-valued) Schrödinger-Poisson systems. Band structure information enters via densities and the optical response function into comprehensive simulations of Multi Quantum Well lasers. These device simulations yield valuable information on device characteristics, including effects of carrier transport, waveguiding and heating and can be used for optimization.

*Appeared in*

- Mathematics - Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jaeger, H.-J. Krebs, eds., Springer-Verlag Berlin heidelberg, 2003, pp. 377-390

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# Numerical solution of the Neumann problem for nonlinear parabolic equations by probability approach

*Authors*

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

*2010 Mathematics Subject Classification*

- 35K55 60H10 60H30 65M99

*Keywords*

- Semilinear parabolic equations, Neumann problem, probabilistic representations for equations of mathematical physics, weak approximation of solutions of stochastic differential equations in bounded domain

*DOI*

*Abstract*

A number of new layer methods solving the Neumann problem for semilinear parabolic equations is constructed by using probabilistic representations of their solutions. The methods exploit the ideas of weak sense numerical integration of stochastic differential equations in bounded domain. In spite of the probabilistic nature these methods are nevertheless deterministic. Some convergence theorems are proved. Numerical tests are presented.

*Appeared in*

- IMA J. of Numerical Analysis, vol. 22 (2002), no. 4, pp. 599-622, under new title: A probabilistic approach to the solution of Neumann problem for nonlinear parabolic equations.

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# Branching systems with long living particles at the critical dimension

*Authors*

- Fleischmann, Klaus
- Vatutin, Vladimir A.
- Wakolbinger, Anton

*2010 Mathematics Subject Classification*

- 60J80 60G70 60J15

*Keywords*

- branching particle system, critical dimension, limit theorem, long living particles, absolute continuity, random density, superprocess, persistence, mixed Poissonian particle system, residual lifetime process, stable subordinator

*DOI*

*Abstract*

A spatial branching process is considered in which particles have a life time law with a tail index smaller than one. Other than in classical branching particle systems, at the critical dimension the system does not suffer local extinction when started from a spatially homogenous initial population. In fact, persistent convergence to a mixed Poissonian system is shown. The random limiting intensity is characterized in law by the random density in a space point of a related age-dependent superprocess at a fixed time. The proof relies on a refined study of the system starting from asymptotically large but finite initial populations.

*Appeared in*

- Teor. Veroyatnost. i Primenen 47 (2002), pp. 417-451

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# Waveform relaxation methods for stochastic differential equations

*Authors*

- Schneider, Klaus R.
- Schurz, Henri

*2010 Mathematics Subject Classification*

- 34F05 60H10 65C30 65L05 65L20

*Keywords*

- Waveform relaxation methods, stochastic differential equations, stochastic-numerical methods, large scale systems

*DOI*

*Abstract*

An operator equation X = Π X + G in a Banach space 𝓔 of 𝓕_{t}-adapted random elements describing an initial- or boundary value problem of a system of stochastic differential equations (SDEs) is considered. Our basic assumption is that the underlying system consists of weakly coupled subsystems. The proof of the convergence of corresponding waveform relaxation methods depends on the property that the spectral radius of an associated matrix is less than one. The entries of this matrix depend on the Lipschitz-constants of a decomposition of Π. In proving an existence result for the operator equation we show how the entries of the matrix depend on the right hand side of the stochastic differential equations. We derive conditions for the convergence under "classical" vector-valued Lipschitz-continuity of an appropriate splitting of the system of stochastic ODEs. A generalization of these key results under one-sided Lipschitz continuous and anticoercive drift coefficients of SDEs is also presented. Finally, we consider a system of SDEs with different time scales (singularly perturbed SDEs) as an illustrative example.

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# Catalytic and mutually catalytic super-Brownian motions

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Mutually catalytic branching, catalytic super-random walk, catalyst, reactant, superprocess, measure-valued branching, absolute continuity, collision measure, collision local time, self-similarity, martingale problem, segregation of types, coexistence of types, self-duality, finite time extinction, ultimate extinction, biodiversity, cyclic reaction

*DOI*

*Abstract*

Catalytic branching processes describe the evolution of two types of material (populations) called catalyst and reactant. The catalyst evolves autonomously, but catalyzes the reactant. The individuals of both populations share the features of motion, growth and death. In mutually catalytic models however there is an additional feedback from the reactant to the catalyst destroying completely the basic independence assumption of branching theory. Recent results for continuum models of this type are surveyed.

*Appeared in*

- Seminar on Stochastic Analysis, Random Fields and Applications III, Centro Stefano Franscini, Ascona, Switzerland, September 19-24, 1999, R. C. Dalang, M. Dozzi,, F. Russo, (eds.), vol. 52 of Progress in Probability, Birkhaeuser-Verlag, Basel, 2002, pp. 89-110

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# Analysis of relative dispersion of two particles by Lagrangian stochastic models and DNS methods

*Authors*

- Kurbanmuradov, Orazgeldi
- Orszag, Steven A.
- Sabelfeld, Karl K.
- Yeung, P. K.

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Turbulent flows, Lagrangian trajectories, separation vector, DNS method

*DOI*

*Abstract*

Comparisons of the Q1D against the known Lagrangian stochastic well-mixed quadratic form models and the moments approximation models are presented. In the case of modestly large Reynolds numbers turbulence (Re _{λ} ⋍ 240) the comparison of the Q1D model with the DNS data is made. Being in a qualitatively agreemnet with the DNS data, the Q1D model predicts higher rate of separation. Realizability of Q1D model extracted from the transport equation with a quadratic form of the conditional acceleration is shown.

*Appeared in*

- Monte Carlo Methods Appl., 7 (2001) pp. 245--263.

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# Unidirectional transport in stochastic ratchets

*Authors*

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

*2010 Mathematics Subject Classification*

- 60H10 93E30

*Keywords*

- Noise-driven systems, Brownian ratchets, boundary value problems of parabolic type

*DOI*

*Abstract*

Constructive conditions for existence of the unidirectional transport are given for systems with state-dependent noise and for forced thermal ratchets. Using them, domains of parameters corresponding to the unidirectional transport are indicated. Some results of numerical experiments are presented.

*Appeared in*

- Stochastics and Dynamics, vol.1 (2001), no.3, pp. 361-375, under new title: Noice-induced unidirectional transport.

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# A proof of a Shilnikov theorem for C^1-smooth dynamical systems

*Authors*

- Shashkov, Mikhail
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37G20 37G15 37C27 37C29 37C75 37C10 34C20 34C23 34C25 34C37

*Keywords*

- separatrix loop, periodic orbit, homoclinic bifurcations

*DOI*

*Abstract*

Dynamical systems with a homoclinic loop to a saddle equilibrium state are considered. Andronov and Leontovich have shown (see [1939], [1959]) that a generic bifurcation of a two-dimensional C^{1}-smooth dynamical system with a homoclinic loop leads to appearance of a unique periodic orbit. This result holds true in the multi-dimensional setting if some additional conditions are satisfied, which was proved by Shilnikov [1962, 1963, 1968] for the case of dynamical systems of sufficiently high smoothness. In the present paper we reprove the Shilnikov theorem for dynamical systems in C^{1}.

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# Numerical simulation for lossy microwave transmission lines including PML

*Authors*

- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer

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

*2010 Mathematics Subject Classification*

- 35Q60 65F15 65N22

*Keywords*

- Microwave device simulation, Maxwell's equations, PML boundary condition, Finite integration technique, Eigenvalue problem

*DOI*

*Abstract*

Finite-difference analysis of transmission lines including lossy materials and radiation effects leads to a complex eigenvalue problem. A method is presented which preserves sparseness and delivers only the small number of interesting modes out of the complete spectrum. The propagation constants are found solving a sequence of eigenvalue problems of modified matrices with the aid of the shift-and-invert mode of the Arnoldi method. In an additional step non physical Perfectly Matched Layer modes are eliminated.

*Appeared in*

- Lecture Notes in Computaional Science and Engineering, Vol. 18 (2001), Eds. Ursula van Rienen, Michael Günther, Dirk Hecht: Scientific Computing in Electrical Engineering, pp. 267-275

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# Heterogeneous dynamic process flowsheet simulation of chemical plants

*Authors*

- Grund, Friedrich
- Ehrhardt, Klaus
- Borchardt, Jürgen
- Horn, Dietmar

*2010 Mathematics Subject Classification*

- 65L05 80A30 65Y05 65H10

*Keywords*

- Chemical process simulation, Systems of differential-algebraic equations, Large-scale dynamic simulation, Coupled processes, Distributed simulation, Waveform iteration, Broyden update

*DOI*

*Abstract*

For large-scale dynamic simulation problems in chemical process engineering, a heterogeneous simulation concept is described which allows to distribute the solution of the models of coupled dynamic subprocesses to a computer network. The main principle of such a technique is to solve the submodels of an overall model independently of each other on subsequent time intervals. This is done by estimating the vector of input variables of the submodels, calculating the corresponding time behaviour of the output variables concurrently, and matching the time profiles of the interconnecting variables of the process flowsheet iteratively. Therefore, accelerated waveform iteration methods are considered, using Broyden- and block-Broyden-type updates. The simulation concept is investigated especially in the case that the submodels do not provide input-output sensitivities.

*Appeared in*

- W. Jaeger and H.-J. Krebs, editors: Mathematics - Key Technology for the Future . Joint Projects Between Universities and Industry, pages 184-193. Springer-Verlag Berlin Heidelberg, 2003

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# Parallel numerical methods for large-scale DAE systems

*Authors*

- Borchardt, Jürgen

*2010 Mathematics Subject Classification*

- 65Y05 80A30 65L05 65H10 65F50

*Keywords*

- Systems of differential-algebraic equations, Block partitioned systems, Newton-type methods, Parallelization, Chemical process simulation, Dynamic simulation of distillation plants

*DOI*

*Abstract*

For plantwide dynamic simulation in chemical process industry, parallel numerical methods using a divide and conquer strategy are considered. An approach for the numerical solution of initial value problems for large systems of differential algebraic equations (DAEs) arising from industrial applications and its realization on parallel computers with shared memory is discussed. The system is partitioned into blocks and then it is extended appropriately, such that block-structured Newton-type methods can be applied which enable the application of relaxation techniques. This approach has gained considerable speedup factors for the dynamic simulation of various large-scale distillation plants, covering systems with up to 60 000 equations.

*Appeared in*

- Computers and Chemical Engineering, vol. 25 (2001), pp. 951-961

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# The long-time behaviour of the thermoconvective flow in a porous medium

*Authors*

- Efendiev, Messoud A.
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Zelik, Sergei V.

*2010 Mathematics Subject Classification*

- 37L25 37L30 35B40 35B45 65M99

*Keywords*

- equations of coupled heat and fluid flow in a porous medium, Boussinessq approximation, global attractor, upper and lower bounds, Rayleigh number, Hausdorff and fractal dimension, finite volumes, numerical solution

*DOI*

*Abstract*

For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations posesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system.

*Appeared in*

- Mathematical Methods in the Applied Sciences, 27(4), pp. 907-930, 2004

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# Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models

*Authors*

- Kurbanmuradov, Orazgeldi
- Rannik, Üllar
- Sabelfeld, Karl K.
- Vesala, Timo

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Turbulent flows, Lagrangian trajectories, forward and backward random estimators, concentration and fluxes

*DOI*

*Abstract*

Forward and backward stochastic Lagrangian trajectory simulation methods for calculation of the mean concentration of scalars and their fluxes for sources arbitrarily distributed in space and time are constructed and justified. Generally, absorption of scalars by medium is taken into account. A special case of the source structure, when the scalar is generated by a plane source, say, located close to the ground, is treated. This practically interesting particular case is known in the literature as the footprint problem.

*Appeared in*

- Math. Comput. Simulation, 54 (2001) pp. 459--476.

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# A Riemann problem for poroelastic materials with the balance equation for porosity. Part II

*Authors*

- Radkevich, Evgeniy V.
- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 41A60 35Q51 35L45 76S05

*Keywords*

- Porous and granular media, solitons, Riemann Problem, asymptotic expansions

*DOI*

*Abstract*

In the first part of this work we considered the properties of the first order approximation with respect to a small parameter β. In this part we present the second order approximation which describes the evolution of amplitudes and, consequently, establishes conditions of stability.

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# The Brockett problem in the theory of nonstationary stabilization of linear differential equations

*Authors*

- Leonov, Gennadi A.

*2010 Mathematics Subject Classification*

- 93D15 34H05

*Keywords*

- stabilization, transfer function, pendulum, invariant manifold

*DOI*

*Abstract*

In the present work a review of algorithms for nonstationary linear stabilization is given. In many cases these algorithms, together with the criterion of nonstabilizing, allow us to obtain a solution of the Brockett problem.

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# Metastability and small eigenvalues in Markov chains

*Authors*

- Bovier, Anton
- Eckhoff, Michael
- Gayrard, Véronique
- Klein, Markus

*2010 Mathematics Subject Classification*

- 60J10 60K35

*Keywords*

- Markov chains, metastability, eigenvalue problems, exponential distribution

*DOI*

*Abstract*

In this letter we announce rigorous results that elucidate the relation between metastable and low-lying eigenvalues in Markov chains in a much more general setting and with considerable greater precision as was so far available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable state, and sharp estimates on the convergence of probability distribution of the metastable transition times to the exponential distribution.

*Appeared in*

- J. Phys A 33, (2000), L447-L451

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# Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals

*Authors*

- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 35J55 46E35 35B65

*Keywords*

- Bounded measurable coefficients, Sets with Lipschitz boundary, Regular sets, Non-homogeneous mixed boundary conditions, Regularity up to the boundary of weak solutions, Arbitrary space dimension

*DOI*

*Abstract*

In this paper linear elliptic boundary value problems of second order with non-smooth data (bounded measurable coefficients, sets with Lipschitz boundary, regular sets, non-homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev-Campanato spaces of functions and functionals, respectively.

*Appeared in*

- Mathematische Nachrichten, 2002, Vol. 243, pp.19-42

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# On polynomial collocation for Cauchy singular integral equations with fixed singularities

*Authors*

- Junghanns, Peter
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 45L10 65R20

*Keywords*

- Cauchy singular integral equation, Mellin kernel, polynomial collocation, stability, convergence

*DOI*

*Abstract*

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weighted L^{2} spaces, we derive necessary and sufficient conditions.

*Appeared in*

- Integral Equations and Operator Theory 43, 2002, pp. 155-176

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# Numerical study of a stochastic particle method for homogeneous gas phase reactions

*Authors*

- Kraft, Markus

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

*2010 Mathematics Subject Classification*

- 65C35 60K40

*Keywords*

- Stochastic particle method, combustion, convergence, efficiency

*DOI*

*Abstract*

In this paper we study a stochastic particle system that describes homogeneous gas phase reactions of a number of chemical species. First we introduce the system as a Markov jump process and discuss how relevant physical quantities are represented in terms of appropriate random variables. Then, we show how various deterministic equations, used in the literature, are derived from the stochastic system in the limit when the number of particles goes to infinity. Finally, we apply the corresponding stochastic algorithm to a toy problem, a simple formal reaction mechanism, and to a real combustion problem. This problem is given by the isothermal combustion of a homogeneous mixture of hepthane and air modelled by a detailed reaction mechanism with 107 chemical species and 808 reversible reactions. Heptane as described in this chemical mechanism serves as model-fuel for different types of internal combustion engines. In particular, we study the order of convergence with respect to the number of simulation particles, and illustrate the limitations of the method.

*Appeared in*

- Comput. Math. Appl. 45 (2003), pp. 329-349

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# A new simple bifurcation of a periodic orbit of "blue sky catastrophe'' type

*Authors*

- Shilnikov, Leonid
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37G15 37C29 37G05 34C20 37C75 37G35 37C27 37C10

*Keywords*

- boundaries of stability, saddle-node, homoclinic orbits, nonlocal bifurcations, embedding into the flow

*DOI*

*Abstract*

In this paper, we study a global bifurcation of codimension one connected with the disappearance (for positive values of a parameter μ) of a saddle-node periodic orbit L_{0} under the condition that all orbits from the locally unstable manifold W^{u} of L_{0} tend to L_{0} as t → +∞. Conditions are presented which guarantee the blue sky catastrophe: the appearance of a stable periodic orbit L_{μ} which exists for any small positive values of μ but its length and period unboundedly increase as μ → + 0.

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# Thermodynamic design of energy models of semiconductor devices

*Authors*

- Albinus, Günter
- Gajewski, Herbert
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 82D37 35K55 47H05 47H50 80A20 80A30

*Keywords*

- Energy model, semiconductor devices, carrier temperatures, convex thermodynamic potentials, systems of conservation laws, systems of nonlinear parabolic equations, Lyapunov functions, convex analysis, monotone operators

*DOI*

*Abstract*

In this preprint a system of evolution equations for energy models of a semiconductor device is derived on an deductive way from a generally accepted expression for the free energy. Only first principles like the entropy maximum principle and the principle of partial local equilibrium are applied. Particular attention is paid to include the electrostatic potential self-consistently. Dynamically ionized trap levels and models with carrier temperatures are regarded. The system of evolution equations is compatible with the corresponding entropy balance equation that contains a positively definite entropy production rate.

*Appeared in*

- Nonlinearity, 15 (2002) pp. 367--383.

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# Interpolation for function spaces related to mixed boundary value problems

*Authors*

- Griepentrog, Jens André
- Gröger, Konrad
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 46B70 46E35 35J25

*Keywords*

- Interpolation theory, function spaces

*DOI*

*Abstract*

Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.

*Appeared in*

- Mathematische Nachrichten, 2002, Vol.241, pp. 110-120

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# Analysis and optimization of nonsmooth mechanical structures

*Authors*

- Ignat, Anca
- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49Q10 35J35 34A55

*Keywords*

- Lipschitz arches, discontinuous plate thickness, shape optimization

*DOI*

*Abstract*

It is our aim to give a new treatment for some classical models of arches and plates and for their optimization. In particular, our approach allows to study nonsmooth arches, while the standard assumptions from the literature require W^{3,∞}-regularity for the parametric representation. Moreover, by a duality-type argument, the deformation of the arches may be explicitly expressed by integral formulas.

As examples for the shape optimization problems under study, we mention the design of the middle curve of a clamped arch or of the thickness of a clamped plate such that, under a prescribed load, the obtained deflection satisfies certain desired properties. In all cases, no smoothness is required for the design parameters.

*Appeared in*

- SIAM J. Control Optim., 40 (2001), pp. 1107--1133

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# Multigrid optimization in applications

*Authors*

- Dreyer, Thomas
- Maar, Bernd
- Schulz, Volker

ORCID: 0000-0001-7665-130X

*2010 Mathematics Subject Classification*

- 65K10 65M55 90C06

*Keywords*

- Partially reduced SQP methods, multigrid methods, turbine blade optimization, topology optimization

*DOI*

*Abstract*

Iterative techniques are a key methodology for the numerical solution of optimization problems in differential equations. In two practical application problems with different characteristics, this paper shows, how multigrid methods can be applied efficiently to this problem class. Problem formulations, solution approaches as well as numerical results are presented.

*Appeared in*

- J. Comput. Appl. Math., 120(2000), No. 1-2 (Special Issue), pp. 67-84

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# Efficient computation of option price sensitivities using homogeneity and other tricks

*Authors*

- Reiß, Oliver
- Wystup, Uwe

*2010 Mathematics Subject Classification*

- 91-08 91B28

*Keywords*

- Calculation of Greeks, Derivatives of option prices, Homogeneity properties of financial markets

*DOI*

*Abstract*

No front-office software can survive without providing derivatives of options prices with respect to underlying market or model parameters, the so called Greeks. We present a list of common Greeks and exploit homogeneity properties of financial markets to derive relationships between Greeks out of which many are model-independent. We apply the results to European style options, rainbow options, as well as options priced in Heston's stochastic volatility model and avoid exorbitant and time-consuming computations of derivatives which even strong symbolic calculators fail to produce.

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# Modelling and simulation of power devices for high-voltage integrated circuits

*Authors*

- Hünlich, Rolf
- Albinus, Günter
- Gajewski, Herbert
- Glitzky, Annegret
- Röpke, Wilfried
- Knopke, Jürgen

*2010 Mathematics Subject Classification*

- 35B40 35B45 35K55 35K57 80A20 80A30 82D37

*Keywords*

- Power devices, integrated circuits, process simulation, device simulation, drift-diffusion systems, reaction-diffusionsystems, heat flow equation, thermodynamic potentials, conservation laws, Lyapunov functions

*DOI*

*Abstract*

Process and device simulators turned out to be important tools in the design of high-voltage integrated circuits and in the development of their technology. The main goal of this project was the improvement of the device simulator WIAS-TeSCA in order to simulate different power devices in high-voltage integrated circuits developed by the industrial partner. Some simulation results are presented. Furthermore, we discuss some aspects of the mathematics of relevant model equations which device and process simulations are based on.

*Appeared in*

- Mathematics - Key Technolofy for the Future. Joint Projects Between Universities and Industry, W. Jaeger, H.-J. Krebs, eds., Springer-Verlag Berlin Heidelberg, 2003, pp. 401-411

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# Surface waves at an interface separating two saturated porous media

*Authors*

- Edelman, Inna
- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 35C20 35L50 73D20

*Keywords*

- partial differential equations, waves in porous media

*DOI*

*Abstract*

Surface waves at an interface between two saturated porous media of different structure are investigated. Existence and peculiarities of surface wave propagation are revealed. Four types of surface waves are proven to be possible: true Stoneley surface wave propagating almost without dispersion, leaky slow pseudo-Stoneley wave, leaky generalized Rayleigh wave, and one more new leaky mode. True Stoneley, leaky pseudo-Stoneley, and generalized Rayleigh waves are similar to those waves, which exist at an interface between a saturated porous medium and a liquid. Existence of generalized Rayleigh wave or new mode depends crucially on the parameters of the skeletons.

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# On the approximation of kinetic equations by moment systems

*Authors*

- Dreyer, Wolfgang
- Junk, Michael
- Kunik, Matthias

*2010 Mathematics Subject Classification*

- 82C70 35L30 82B40

*Keywords*

- maximum entropy, moment methods, Fokker-Planck equation, exact solution, Grad expansion, moment realizability

*DOI*

*Abstract*

The aim of this article is to show that moment approximations of kinetic equations based on a Maximum Entropy approach can suffer from severe drawbacks if the kinetic velocity space is unbounded. As example, we study the Fokker Planck equation where explicit expressions for the moments of solutions to Riemann problems can be derived. The quality of the closure relation obtained from the Maximum Entropy approach as well as the Hermite/Grad approach is studied in the case of five moments. It turns out that the Maximum Entropy closure is even singular in equilibrium states while the Hermite/Grad closure behaves reasonably. In particular, the admissible moments may lead to arbitrary large speeds of propagation, even for initial data arbitrary close to global eqilibrium.

*Appeared in*

- Nonlinearity, 14(4):881-906, 2001

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# On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies

*Authors*

- Gonchenko, Sergey V.
- Gonchenko, Vladimir S.

*2010 Mathematics Subject Classification*

- 58F12 58F13

*Keywords*

- homoclinic tangency, invariant curve, Andronov-Hopf bifurcation, strange attractors, Newhouse regions

*DOI*

*Abstract*

The bifurcation of the birth of a closed invariant curve in the two-parameter unfolding of a two-dimensional diffeomorphism with a homoclinic tangency of invariant manifolds of a hyperbolic fixed point of neutral type (i.e. such that the Jacobian at the fixed point equals to 1) is studied. The existence of periodic orbits with multipliers e^{±iψ} (0 < ψ < π) is proved and the first Lyapunov value is computed. It is shown that, generically, the first Lyapunov value is non-zero and its sign coincides with the sign of some separatrix value (i.e. a function of coefficients of the return map near the global piece of the homoclinic orbit).

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# Iterative solution of systems of linear equations in microwave circuits using a block quasi-minimal residual algorithm

*Authors*

- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer

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

*2010 Mathematics Subject Classification*

- 35Q60 65F10 65F15 65N22

*Keywords*

- Microwave device simulation, Scattering matrix, Maxwell's equations, Boundary value problem, Finite integration technique, Eigenvalue problem, System of linear algebraicequations, Multiple right-hand sides

*DOI*

*Abstract*

The electric properties of monolithic microwave integrated circuits that are connected to transmission lines are described in terms of their scattering matrix using Maxwell's equations. Using a finite-volume method the corresponding three-dimensional boundary value problem of Maxwell's equations in the frequency domain can be solved by means of a two-step procedure. An eigenvalue problem for non-symmetric matrices yields the wave modes. The eigenfunctions determine the boundary values at the ports of the transmission lines for the calculation of the fields in the three-dimensional structure. The electromagnetic fields and the scattering matrix elements are achieved by the solution of large-scale systems of linear equations with indefinite complex symmetric coefficient matrices. In many situations, these matrix problems need to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. The block quasi-minimal residual algorithm is a block Krylov subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences.

*Appeared in*

- Lecture Notes in Computational Science and Engineering, Vol. 18 (2001), Eds. Ursula van Rienen, Michael Günther, Dirk Hecht: Scientific Computing in Electrical Engineering, pp. 325-333

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# Direct and inverse problems for diffractive structures - optimization of binary gratings

*Authors*

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

*2010 Mathematics Subject Classification*

- 78-99 78A45 35J20 65N30 49J20

*Keywords*

- Diffraction by periodic structures, Helmholtz equation, transmission problems, strongly elliptic variational formulation, generalized FEM, optimal design problems, gradient methods

*DOI*

*Abstract*

The goal of the project is to provide flexible analytical and numerical tools for the optimal design of binary and multilevel gratings occurring in many applications in micro-optics. The direct modeling of these diffractive elements has to rely on rigorous grating theory, which is based on Maxwell's equations. We developed efficient and accurate direct solvers using a variational approach together with a generalized finite element method which appears to be well adapted to rather general diffractive structures as well as complex materials. The optimal design problem is solved by minimization algorithms based on gradient descent and the exact calculation of gradients with respect to the geometry parameters of the grating.

*Appeared in*

- Mathematics - Key Technology for the Future II, Springer, Berlin, 2003

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# Homoclinic orbits: Since Poincaré till today

*Authors*

- Shilnikov, Leonid

*2010 Mathematics Subject Classification*

- 37C29 01A65 37G25 37C15 37D45

*Keywords*

- strange attractors, spiral chaos, Smale horseshoe, geodesic flows, Hamiltonian systems, homoclinic tangency, Newhouse phenomenon, bifurcation

*DOI*

*Abstract*

The history and the contemporary results in homoclinic orbits are reported.

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# Forward and backward Lagrangian stochastic models of turbulent transport

*Authors*

- Sabelfeld, Karl
- Shalimova, Irina

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Turbulent flows, Lagrangian trajectories, forward and backward algorithms, well-mixed condition

*DOI*

*Abstract*

The key question analysed in the paper is: under what condition are the Lagrangian stochastic models stochastically reversible? We show that this property is deeply related to Thomson's well-mixed condition. Direct and backward in time Monte Carlo algorithms are suggested. A consistency principle in the turbulent transport problems and in the financial mathematics is analysed to find out an analogy in constructing Lagrangian stochastic models in these two different fields.

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# Dynamical phenomena near a homo- or heteroclinic connection involving saddle-foci in a Hamiltonian system

*Authors*

- Lerman, Lev M.

*2010 Mathematics Subject Classification*

- 34C37 37J45

*Keywords*

- Hamiltonian, saddle-focus, homoclinic, heteroclinic, bifurcation, generating function

*DOI*

*Abstract*

The main features of the orbit behavior for a Hamiltonian system in a neighborhood of a homoclinic orbit to a saddle-focus equilibrium or of a contour made up of two saddle-foci and two heteroclinic orbits to them are presented. These features includes description of hyperbolic subsets and main bifurcations when varying a value of the Hamiltonian and parameters. The proofs of results about bifurcations are given.

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# Two notes on continuous modelling of porous media

*Authors*

- Wilmański, Krzysztof
- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 76S05 73B05

*Keywords*

- porous media, objectivity, consolidation

*DOI*

*Abstract*

The note is devoted to the analysis of contributions of relative accelerations to partial momentum balance equations of multicomponent models of porous materials. We show that such contributions violate the principle of material ob jectivity. Even if we ignore this principle contributions of relative accelerations are either undistinguishable from other contributions or yield unacceptable modes of propagation of sound waves or both. Consequently we conclude that such contributions should be ignored completely in the construction of macroscopical models of porous materials.

*Appeared in*

- Rational Continua, Classical and New, P. PODIO-GUIDUGLI, M. BROCATO (eds.), 183-195, Springer-Verlag, Italia Srl, Milano, 2003

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# Dependence of adsorption/diffusion processes in porous media on bulk and surface permeabilities

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 76S05 73Q05 76R50

*Keywords*

- Adsorption, diffusion, flows in porous media, parameter analysis

*DOI*

*Abstract*

The paper contains a brief summary of a macroscopic continuum model for adsorption in porous materials which is an extension of the model for porous bodies by K. Wilmanski on mass exchange processes. We consider the flow of a fluid/adsorbate mixture through channels of a solid component. The fluid serves as carrier for an adsorbate whose mass balance equation contains a source term. Due to low adsorbate concentration we deal with a physical adsorption process which means that particles of the adsorbate stick to the skeleton due to weak van der Waals forces. The model contains two different permeability parameters whose nature is completely different: The first one, the usual bulk permeability coefficient, describes the resistance of the skeleton to the flow of the fluid/adsorbate mixture. The second one describes the surface resistance to the outflow of the mixture from the solid. This work shows within a simple example the range of these parameters and the dependence of adsorption/diffusion on them.

*Appeared in*

- Dependence of Adsorption/Diffusion Processes in Porous Media on Bulk and Surface Permeabilities, Arch. Mech., 53, 3, 303-320, 2001

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# Global properties of pair diffusion models

*Authors*

- Glitzky, Annegret
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 35B40 35B45 35K57 35R05 78A35

*Keywords*

- Drift-diffusion systems, reaction-diffusion systems, heterostructures, energy estimates, global estimates, asymptotic behaviour

*DOI*

*Abstract*

The paper deals with global properties of pair diffusion models with non-smooth data arising in semiconductor technology. The corresponding model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs containing drift, diffusion and reaction terms. The corresponding equations for the immobile species are ODEs involving reaction terms only. Starting with energy estimates obtained by methods of convex analysis we establish global upper and lower bounds for solutions of the initial boundary value problem. We use Moser iteration for the diffusing species, the non-diffusing species are treated separately. Finally, we study the asymptotic behaviour of solutions.

*Appeared in*

- Adv. Sci. Appl. 11 (2001), pp. 293-321

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# General ether theory

*Authors*

- Schmelzer, Ilja

*2010 Mathematics Subject Classification*

- 83D05 83C45

*Keywords*

- quantum gravity, Bell's inequality, alternative theories of gravity

*DOI*

*Abstract*

The paper is an introduction into General Ether Theory (GET). We start with few assumptions about an universal "ether" in a Newtonian space-time which fulfils

∂_{t}ρ + ∂_{i} (ρv^{i}) = 0

∂_{t}(ρv^{j})+∂_{i}(ρv^{i}v^{j}+p^{ij} = 0 For an "effective metric" g_{µν} we derive a Lagrangian where the Einstein equivalence principle is fulfilled:

L = L_{GR} - (8πG)^{-1}(Υg^{00} - Ξ(g^{11} + g^{22} + g^{33}))√-g

We consider predictions (stable frozen stars instead of black holes, big bounce instead of big bang singularity, a dark matter term), quantization (regularization by an atomic ether, superposition of gravitational fields), related methodological questions (covariance, EPR criterion, Bohmian mechanics).

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# Maximum likelihood estimate for nonparametric signal in white noise by optimal control

*Authors*

- Milstein, Grigori N.
- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62G05 60H10 49K15

*Keywords*

- nonparametric diffusion model, maximum likelihood method, optimal estimation.

*DOI*

*Abstract*

The paper is devoted to questions of constructing the maximum likelihood estimate for a nonparametric signal in white noise by considering corresponding problems of optimal control. For signals with bounded derivatives, sensitivity theorems are proved. The theorems state a stability of the maximum likelihood estimate with respect to changing output data. They make possible to reduce the original problem to a standard problem of optimal control which is solved by iterative procedure. For signals of Sobolev type the maximum likelihood estimate is obtained to within a parameter which can be found from a transcendental equation.

*Appeared in*

- Statistic and Probability Letters, vol. 55/2 (2001), pp. 193-203

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# A Riemann problem for poroelastic materials with the balance equation for porosity. Part I

*Authors*

- Radkevich, Evgeniy V.
- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 41A60 35Q51 35L45 76S05

*Keywords*

- Porous and granular media, solitons, Riemann Problem, asymptotic expansions

*DOI*

*Abstract*

The paper is devoted to the asymptotic analysis of the multicomponent model of poroelastic materials in which the porosity is described by its own field equation. The model is weakly nonlinear due to the kinematic contributions and a nonlinear dependence of material parameters on an equilibrium porosity. It is shown that the model contains two small parameters. The first one describes the coupling of the skeleton (solid component) with the fluid and its contribution through partial stresses is similar to a dynamical pressure of extended thermodynamics. Mathematically it leads to a dispertion effect similar to this appearing in the Korteweg - de Vries equation. The second small parameter describes a relaxation of porosity. We consider two cases. In the first one the order of magnitude of both small parameters is the same. This seems to correspond to usual porous materials. In the second case the dimensionless relaxation time is propotional to the square of the other parameter. We call such materials granular-like porous. They seem to correspond to compact granular materials with hard and smooth particles. We prove the existence of soliton-like solutions for the porosity and kink-like solutions for the partial velocities under a natural entropy-like selection condition which is also presented in the paper. The proof is based on the asymptotic analysis in which two steps of approximations were investigated. We show that the diffusive interaction force of components - a kind of an internal friction - yields decaying amplitudes of discontinuities. We show as well that in one class of Riemann problems a Saffman-Taylor instability appears. The paper is divided into two parts solely for technical reasons. Therefore the references appear after the second part. In the Appendix we show a few examples of a numerical simulation of a two-dimensional Riemann problem. These were obtained by dr. O. A. Vassilieva (Moscow University). The full numerical analysis shall be presented separately.

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# Practical shape optimization for turbine and compressor blades

*Authors*

- Bock, Hans Georg
- Egartner, Wolfgang
- Kappis, Wolfgang
- Schulz, Volker

ORCID: 0000-0001-7665-130X

*2010 Mathematics Subject Classification*

- 90C55 65H05 65Y20 76D55

*Keywords*

- Turbomachinery design, shape optimization, partially reduced SQP methods, working range optimization

*DOI*

*Abstract*

The shape optimization of blades is a crucial step within the design cycle of a whole turbomachine. This paper is a report on a joint project between academia and industry leading to an efficient solution software for this problem to be used in the daily work of concerned engineers. The problem description and solution method, characterized as a partially reduced SQP method, as well as numerical results are presented.

*Appeared in*

- Optimization and Engineering, 3 (2002), pp. 395-414

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# Pathwise description of dynamic pitchfork bifurcations with additive noise

*Authors*

- Berglund, Nils
- Gentz, Barbara

*2010 Mathematics Subject Classification*

- 37H20 60H10 34E15 93E03

*Keywords*

- Dynamic bifurcation, pitchfork bifurcation, additive noise, bifurcation delay, singular perturbations, stochastic differential equations, random dynamical systems, pathwise description, concentration of measure

*DOI*

*Abstract*

The slow drift (with speed *e*) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity *s*, by giving precise estimates on the behaviour of the individual paths. We show that until time *e*^{1/2} after the bifurcation, the paths are concentrated in a region of size *s*/*e*^{1/4} around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [*e*^{1/2}, *c* (*e* log *s* )^{1/2}], after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents.

*Appeared in*

- Probab. Theory Related Fields 122 (2002) 3, 341-388

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# Hybrid method and vibrational stability for nonlinear singularly perturbed systems under parametric excitations

*Authors*

- Strygin, Vadim V.

*2010 Mathematics Subject Classification*

- 34E15

*Keywords*

- Hybrid asymptotical method, averaging, boundary functions, vibrational stability

*DOI*

*Abstract*

The well-known classic feedback and feedforward techniques are the main tools for investigations of the control problems. Unlike these strategies, the vibrational control technique, introduced by S.M. Meerkov [1], has proven to be a viable alternative to conventional feedback and feedforward strategies in stabilization problems when the outputs, states and disturbances are difficult to access. Mathematical modelling of such systems is closely connected with nonlinear singularly perturbed systems under parametric excitations. In this paper a new asymptotical method based on the periodic solution theory, averaging method and boundary functions method, is presented. Due to it, a vibrational control problem can be investigated. The given example shows the "parasitic" parameters loss in such systems to be extremly dangerous.

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# Radiation- and convection-driven transient heat transfer during sublimation growth of silicon carbide single crystals

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter
- Sprekels, Jürgen
- Wilmański, Krzysztof

*2010 Mathematics Subject Classification*

- 80A20 65M99 35K55 65C20 80A15

*Keywords*

- Modeling, sublimation growth, SiC single crystal, heat transfer, radiation, convection, numerical simulation, partial differential equations

*DOI*

*Abstract*

This article presents transient numerical simulations of heat transfer occurring during sublimation growth of SiC single crystals via the Modified Lely Method, investigating the respective influence of radiative and convective contributions. We give a concise treatment of the radiation model including semi-transparency via the energy-band approach and we briefly describe the corresponding numerical methods. A complete documentation of the used material data is included.

*Appeared in*

- Journal of Crystal Growth, 222 (2001), pp. 832-851

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# A new kinetic equation for dense gases

*Authors*

- Garcia, Alejandro L.
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C40

*Keywords*

- Kinetic theory, Enskog equation, direct simulation Monte Carlo, Boltzmann equation, consistent Boltzmann algorithm, dense gases

*DOI*

*Abstract*

This paper establishes a theoretical foundation for the Consistent Boltzmann Algorithm by deriving the limiting kinetic equation. Besides its relation to the algorithm, this new equation serves as a useful alternative to the Enskog equation in the kinetic theory of dense gases. For a simplified model, the limiting equation is solved numerically, and very good agreement with the predictions of the theory is found.

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# Existence and approximation of solutions to an anisotropic phase field system of Penrose-Fife type

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603

*2010 Mathematics Subject Classification*

- 65M12 35K50 80A22 35K60 35R35

*Keywords*

- Phase-field model, Penrose-Fife model, anisotropie, semidiscretization, convergence

*DOI*

*Abstract*

This paper is concerned with a phase field system of Penrose-Fife type for a non-conserved order parameter χ with a kinetic relaxation coefficient depending on the gradient of χ. This system can be used to model the dendritic solidification of liquids. A time discrete scheme for an initial-boundary value problem tothis system is presented. By proving the convergence of this scheme, the existence of a solution to the problem is shown.

*Appeared in*

- Interfaces and Free Boundaries, 4(2002)1, pp. 47-70, with new title:Existence and approximation of solutions to ananisotropic phase field system for the kinetics of phase transition.

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# An adaptive, rate-optimal test of linearity for median regression models

*Authors*

- Horowitz, Joel L.
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Hypothesis testing, local alternative, uniform consistency

*DOI*

*Abstract*

This paper is concerned with testing the hypothesis that a conditional median function is linear against a nonparametric alternative with unknown smoothness. We develop a test that is uniformly consistent against alternatives whose distance from the linear model converges to zero at the fastest possible rate. The test accommodates conditional heteroskedasticity of unknown form. The numerical performance and usefulness of the test are illustrated by the results of Monte Carlo experiments and an empirical example.

*Appeared in*

- Econometrica, 69, no. 3 (2201) pp. 599 - 631 under the title: An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative.

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# Uniqueness of determining a periodic structure from discrete far field observations

*Authors*

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

*2010 Mathematics Subject Classification*

- 35R30 35J05 78A46

*Keywords*

- inverse problems, uniqueness, Helmholtz equation, diffractive optics, discrete observations

*DOI*

*Abstract*

This paper is devoted to the uniqueness in the inverse scattering problem of determining a perfectly reflecting periodic surface from far field observations on a discrete set. Our proof is based on the unique continuation of the solution to the Helmholtz equation from a discrete set.

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# Statistical inference for time-inhomogeneous volatility models

*Authors*

- Mercurio, Danilo
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62M10 62P20

*Keywords*

- stochastic volatility model, adaptive estimation, local homogeneity

*DOI*

*Abstract*

This paper offers a new approach for estimation and few-step ahead forecasting of the volatility of financial time series. No assumption is made about the parametric form of the processes, on the contrary we only suppose that the volatility can be approximated by a constant over some interval. In such a framework the main problem consists in filtering this interval of time homogeneity, then the estimate of the volatility can be simply obtained by local averaging. We construct an algorithm which can perform this task and investigate it both from the theoretical point of view and through Monte Carlo simulations. Finally the procedure is applied to some exchange rate data sets and a comparison with a standard GARCH model is also provided. Both models appear to be able of explaining many of the features of the data, nevertheless the new approach based on local constant approximation seems to be slightly superior as far as the out of sample results are taken into consideration.

*Appeared in*

- Ann. Statist., vol. 12 (2004), no. 2, pp. 577-602

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# Stochastic particle approximations for Smoluchowski's coagulation equation

*Authors*

- Eibeck, Andreas
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K40 65C35

*Keywords*

- Stochastic particle method, coagulation equation, variance reduction, gelation phenomena

*DOI*

*Abstract*

This paper studies stochastic particle approximations for Smoluchowski's coagulation equation. A new stochastic algorithm with reduced variance is proposed. Its convergence behaviour is investigated, when the number of simulation particles tends to infinity. Under appropriate assumptions on the coagulation kernel, the limit is the unique solution of the coagulation equation. Then detailed numerical experiments are performed, testing the applicability and efficiency of the algorithm. In particular, the gelation phenomenon (loss of mass in the coagulation equation) is studied numerically for several kernels. A striking feature of the new algorithm is a better convergence after the gelation point, providing a tool for detecting the mass of the gel.

*Appeared in*

- Ann. Appl. Probab. 11 (2001), pp. 1137-1165

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# Conical diffraction by periodic structures: Variation of interfaces and gradient formulas

*Authors*

- Elschner, Johannes
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 35J20 49J20

*Keywords*

- Conical diffraction, shape optimization, Helmholtz equation

*DOI*

*Abstract*

This paper studies the dependence of solutions to conical diffraction problems upon geometric parameters of non-smooth profiles and interfaces between different materials of diffractive gratings. This problem arises in the design of those optical devices to diffract time-harmonic oblique incident plane waves to a specified far-field pattern. We prove the stability of solutions and give analytic formulas for the derivatives of reflection and transmission coefficients with respect to Lipschitz perturbations of interfaces. These derivatives are expressible as contour integrals involving the direct and adjoint solutions of conical diffraction problems.

*Appeared in*

- Math. Nachr. 252 (2003), pp. 24-42

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# Duality principle for discrete linear inclusions

*Authors*

- Vladimirov, Alexander A.

*2010 Mathematics Subject Classification*

- 58E35 47H30

*Keywords*

- Matrix product, stability, oblique projection, Skorokhod problem

*DOI*

*Abstract*

Two properties of finite sets {A_{j}} of n x n-matrices are introduced: P-stability and BV-stability. These properties can be interpreted as two kinds of robustness of orbits of the form x_{i}+1 = A_{ji}x_{i} + u_{i} with respect to disturbances {u_{i}}. Duality between these properties is established, which proves that they are equivalent, respectively, to the right convergent product (RCP) property and the left convergent product (LCP) property of finite sets of matrices. The results can be applied, in particular, in the theory of polyhedral Skorokhod problems and sweeping processes with oblique reflection.

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# A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE

*Authors*

- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 35K57 37L30 35B41 34C37

*Keywords*

- calar semilinear parabolic PDE, order structures; attractors, heteroclinic connections, meandric permutations, nodal properties

*DOI*

*Abstract*

We address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations

u_{t} = u_{xx} + ƒ (x, u, u_{x}), 0 < x < 1

on a bounded interval with Neumann conditions. Introducing a sequence of order relations, we prove a new and simple criterion for the existence of heteroclinic connections, using only information about nodal properties of solutions to the stationary ODE problem. This result allows also for a complete classiffication of possible attractors in terms of the permutation of the equilibria, given by their order at the two boundaries of the interval.

*Appeared in*

- J. Differential Equations, 183, (2002) pp. 56-78

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# Exponentially sensitive internal layer solutions of one-side and their asymptotic expansions

*Authors*

- Bohé, Adriana

*2010 Mathematics Subject Classification*

- 34B15 34E15

*Keywords*

- Internal layers, exponentially sensitive boundary value problems, Gevrey expansions

*DOI*

*Abstract*

We consider a singularly perturbed boundary value problem with Dirichlet conditions and study the sensitivity of the internal layers solutions with respect to small changes in the boundary data. Our approach exploits the existence of smooth invariant manifolds and their asymptotic expansions in the small parameter of perturbation. We show that the phenomenon is extremely sensitive since the shock layers are only obtained by exponentially small perturbations of the boundary data.

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# New exact ground states for one-dimensional quantum many-body systems

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Wagner, Heinz-Jürgen

*2010 Mathematics Subject Classification*

- 81Q05 81V70

*Keywords*

- ground state, wave functions of product type, Calogero-Sutherland systems

*DOI*

*Abstract*

We consider one-dimensional quantum many-body systems with pair interactions in external fields and (re)investigate the conditions under which exact ground state wave functions of product type can be found. Contrary to a claim in the literature that an exhaustive list of such systems is already known, we show that this list can still be enlarged considerably. In particular, we are able to calculate exact ground state wave functions for a class of quantum many-body systems with Ax^{-2} + Bx^{2} interaction potentials and external potentials given by sixth-order polynomials.

*Appeared in*

- Journal of Statistical Physics 100 (2000), pp. 779-790

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# A scaling limit theorem for a class of superdiffusions

*Authors*

- Engländer, János
- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 60J80 60J60 60G57 37L25 35B40 35K55

*Keywords*

- measure-valued process, superprocess, super-Brownian motion, scaling limit, single point source, invariant curve, recurrent diffusion, lambda-lemma, elliptic equation, parabolic PDE

*DOI*

*Abstract*

We consider the σ-finite measure-valued diffusion corresponding to the evolution equation u_{t} = Lu + β(x)u - ƒ(x,u), where

ƒ (x,u) = α (x)u^{2} + ∫^{∞}_{0} (e^{-ku}-1+ku)n(x,dk)

and n is a smooth kernel satisfying an integrability condition. We assume that β,α ∈ C^{η}(ℝ^{d}) with η∈(0,1], and α > 0.

Under appropriate spectral theoretical assumptions we prove the existence of the random measure

lim e^{-λct} X_{t} (dx)

t↑∞

(with respect to the vague topology), where λ_{c} is the principal eigenvalue of L + β on ℝ^{d} and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β.

When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we prove the existence of the previous limit with respect to the weak topology. As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then

lim e^{-βt} X_{t} (dx)

t↑∞

exists and equals to a nonnegative nondegenerate random multiple of the invariant measure for Y.

Taking L = ½ Δ on ℝ and replacing β by δ_{0} (super-Brownian motion with a single point source), we prove a similar result with λ_{c} replaced by ½ and with the deterministic measure e^{-IxI}dx. The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.

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# On the Gibbsian nature of the random field Kac model under block-averaging

*Authors*

- Külske, Christof

*2010 Mathematics Subject Classification*

- 82B44 82B28 82B20

*Keywords*

- Kac-model, random field model, Gibbs-measures, renormalization group transformations

*DOI*

*Abstract*

We consider the Kac-Ising model in an arbitrary configuration of local magnetic fields η = (η_{i})_{i∈𝐙d}, in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed η, with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration η and decays exponentially, uniformly in η, for which we give explicit bounds.

*Appeared in*

- J. statist. Phys. 104 (2001), pp. 991-1012

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# Structural tests in additive regression

*Authors*

- Härdle, Wolfgang
- Sperlich, Stefan
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62H25 62G10

*Keywords*

- Additive models, Component analysis, Haar basis, Hypothesis testing, Nonparametric alternative, Regression

*DOI*

*Abstract*

We consider the component analysis problem for a regression model with an additive structure. The problem is to test if some of the additive components is of polynomial structure, e.g. linear, without specifying the structure of the remaining components. A particular case is the problem of selecting the significant covariates. The presented method is based on the wavelet transform using the Haar basis, which allows for applications under mild conditions on the design and smoothness of the regression function. The results demonstrate that each component of the model can be tested with the rate corresponding to the case if all the remaining components were known. The proposed procedure is also computationally straightforward. Simulation results and a real data example about female labor supply demonstrate the good performance of the test.

*Appeared in*

- J. Amer.Stat. Acc., 96, No. 456, pp. 1333-1347, 2001

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# Fluctuations of the free energy in the REM and the $p$-spin SK models

*Authors*

- Bovier, Anton
- Kurkova, Irina
- Löwe, Matthias

*2010 Mathematics Subject Classification*

- 82C44 60K35

*Keywords*

- spin glasses, Sherrington-Kirkpatrick model, $p$-spin model, random energy model, Central Limit Theorem, extreme values, martingales

*DOI*

*Abstract*

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that (for $p$ even) the random corrections to the free energy are on a scale $N^-(p-2)/4$ only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, $b$, smaller than a critical $b_p$. We also show that $b_prightarrow sqrt2ln 2$ as $puparrow +infty$. Additionally we study the formal $puparrow +infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at it all temperatures. For $b< sqrt2ln2$, fluctuations are found at an it exponentially small scale, with two distinct limit laws above and below a second critical value $sqrtln 2/2$: For $b$ up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For $b$ larger than the critical $sqrt2ln 2$, the fluctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to $1/2$.

*Appeared in*

- Ann. Probab. 30 (2002), pp. 605-951

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# Singularly perturbed partly dissipative reaction-diffusion systems in case of exchange of stabilities

*Authors*

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

*2010 Mathematics Subject Classification*

- 35B25 35K57

*Keywords*

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

*DOI*

*Abstract*

We consider the singularly perturbed partly dissipative reaction-diffusion system ε^{2} (∂u ⁄ ∂t - ∂^{2}u ⁄ ∂x^{2} = g(u,v,x,t,ε), ∂v ⁄ ∂t = ƒ(u,v,x,t,ε) under the condition that the degenerate equation g(u,v,t,0) = 0 has two solutions u = φ_{i}(v,x,t), i = 1,2, that intersect (exchange of stabilities). Our main result concerns existence and asymptotic behavior in ε of the solution of the initial boundary value problem under consideration. The proof is based on the method of asymptotic lower and upper solutions.

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# Hysteresis filtering in the space of bounded measurable functions

*Authors*

- Krejčí, Pavel
- Laurençot, Philippe

*2010 Mathematics Subject Classification*

- 34C55 26A45

*Keywords*

- hysteresis, play operator, total variation

*DOI*

*Abstract*

We define a mapping which with each function 𝑢 ∈ L^{∞}(0,T) and an admissible value of 𝑟 > 0 associates the function ξ with a prescribed initial condition ξ^{0} which minimizes the total variation in the r-neighborhood of 𝑢 in each subinterval [0,t] of [0,T]. We show that this mapping is non-expansive with respect to 𝑢, r and ξ^{0}, and coincides with the so-called play operator if 𝑢 is a regulated function.

*Appeared in*

- Bollettino U. M. I. (8) 5-B (2002), pp. 755-772

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# Thermo-mechanical problems in induction heating of steel

*Authors*

- Pantelyat, Michael
- Uhle, Manfred

*2010 Mathematics Subject Classification*

- 58G11 83C50 73G25

*Keywords*

- induction heating, Maxwell's equations, heat transfer, mechanical deformations

*DOI*

*Abstract*

We discuss a 3D model that is capable for describing mechanical deformations of steel through induction hardening processes. It consists of a reduced system of Maxwell's equations, the heat transfer equation and a system of equations describing the mechanical state of the steel workpiece. In a first step the model is applied to simulation of an axisymmetrical induction hardening device, which is a wide-spread industrial equipment. We present numerical results obtained for a steel tube hardening.

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# Geometric approach to vibrational control of singularly perturbed systems

*Authors*

- Schneider, Klaus R.
- Fridman, Emilia

*2010 Mathematics Subject Classification*

- 34D05 34H05 93C15 93C70

*Keywords*

- vibrational, stabilization, singularly perturbed systems, normally hyperbolic invariant manifolds, averaging

*DOI*

*Abstract*

We extend the theory of vibrational stabilizability to systems with fast and slow variables. The mathematical tools for establishing corresponding results are the persistence theory of normally hyperbolic invariant manifolds, the averaging theory and appropriate transformations. At the same time we introduce modified concepts of vibrational stabilizability compared with the 'classical' definitions.

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# A model of a general elastic curved rod

*Authors*

- Ignat, Anca
- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 65N30 34B60 74B99

*Keywords*

- Deformation of elastic rods, low geometrical regularity, variable cross sections

*DOI*

*Abstract*

We indicate a new approach to the deformation of three-dimensional curved rods with variable cross section. The model consists of a system of nine ordinary differential equations for which we prove existence and uniqueness via the coercivity of the associated bilinear form. From the geometrical point of view, we are using the Darboux frame or a new local frame requiring just a once continuously differentiable parametrization of the curve. Our model also describes the deformation occurring in the cross sections of the rod.

*Appeared in*

- Math. Methods App. Sci. 25 (2002), no. 10, pp. 835-854

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# An equilibrium problem for a thermoelectroconductive body with the Signorini condition on the boundary

*Authors*

- Hömberg, Dietmar
- Khludnev, Aleksander M.

*2010 Mathematics Subject Classification*

- 35D05 35Q72 73F15

*Keywords*

- Resistance welding, Signorini condition, thermistor, thermo- viscoelasticity

*DOI*

*Abstract*

We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators.

*Appeared in*

- Math. Methods Appl. Sci., 24 (2001) pp. 233--244.

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# Asymptotics of solutions to Joukovskii-Kutta type problems at infinity

*Authors*

- Hinder, Rainer
- Meister, Erhard
- Nazarov, Sergueï A.

*2010 Mathematics Subject Classification*

- 76B05 35J05 35B40

*Keywords*

- Airfoil theory, Joukovskii-Kutta condition, Laplace equation, Helmholtz equation, asymptotic farfield behavior

*DOI*

*Abstract*

We investigate the behavior at infinity of solutions to Joukovskii-Kutta-type problems, arising in the linearized lifting surface theory. In these problems one looks for the perturbation velocity potential induced by the presence of a wing in a basic flow within the scope of a linearized theory and for the wing circulation. We consider at first the pure two-dimensional case, then the three-dimensional case, and finally we show in the case of a time-harmonically oscillating wing in ℝ^{3} in a weakly damping gas the exponential decay of solutions of the Joukovskii-Kutta problem.

*Appeared in*

- Appl. Anal., 76(2000), pp. 153-166

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# Regularity results for interface problems in 2D

*Authors*

- Petzoldt, Martin

*2010 Mathematics Subject Classification*

- 35B65 35J25 35J20 34L15

*Keywords*

- elliptic equations, regularity, interface problems, transmission problems, singularities, discontinuous diffusion coeffcients, Sturm-Liouville eigenvalue problem

*DOI*

*Abstract*

We investigate the regularity of solutions of interface problems in 2D. Our objective are regularity results which are independent of global bounds of the data (the diffusion). Therefore we introduce a criterion on the data,the quasi-monotonicity condition, which we show to be sufficient and necessary to provide regularity better then H^{1}. In the proof we use estimates of eigenvalues of a related Sturm-Liouville eigenvalue problem. This approach allows to derive sharp regularity results for quite a large class of configurations. Additionally we give a regularity result depending on the global bounds of the data.

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# Impact of gain dispersion on the spatio-temporal dynamics of multisection lasers

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Radziunas, Mindaugas
- Sieber, Jan
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 78A60 37N20 35B40

*Keywords*

- Multi-section DFB-Lasers, traveling wave equations, high frequency self-pulsations, gain dispersion, polarization

*DOI*

*Abstract*

We present a refined model for multi-section lasers, introducing an additional equation for material polarization in the well known travelling wave model. We investigate the polarization-induced changes in the spectral properties of the optical waveguide. Finally, we show the relevance of this model for a more realistic simulation of complicated dynamical behaviour of multi-section Distributed Feedback (DFB)-Lasers, such as fast self-pulsations, multi-stability, and hysteresis effects due to mode competition.

*Appeared in*

- IEEE Journal of Quantum Electronics, (2001), Vol.37, no.2, pp. 183-188

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# Structure adaptive approach for dimension reduction

*Authors*

- Hristache, Marian
- Juditsky, Anatoli
- Polzehl, Jörg

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

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62H40 62G20

*Keywords*

- dimension-reduction, multi-index model, index space, average derivative estimation, structural adaptation

*DOI*

*Abstract*

We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension m of the index space does not exceed 3, this space can be estimated with the rate n^{-1/2} under rather mild assumptions on the model.

*Appeared in*

- Ann. Statist., 29, no. 6 (2001), pp 1537 - 1566

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# An improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions

*Authors*

- Kraft, Markus

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

*2010 Mathematics Subject Classification*

- 65C35 60K40

*Keywords*

- Stochastic algorithm, gas phase reactions, temperature dependence, convergence, efficiency

*DOI*

*Abstract*

We propose an improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions. By combining forward and reverse reaction rates, a significant gain in computational efficiency is achieved. Two modifications of modelling the temperature dependence (with and without conservation of enthalpy) are introduced and studied quantitatively. The algorithm is tested for the combustion of n-heptane, which is a reference fuel component for internal combustion engines. The convergence of the algorithm is studied by a series of numerical experiments and the computational cost of the stochastic algorithm is compared with the DAE code DASSL. If less accuracy is needed the stochastic algorithm is faster on short simulation time intervals. The new stochastic algorithm is significantly faster than the original direct simulation algorithm in all cases considered.

*Appeared in*

- J. Comut. Phys. 185 (2003), pp. 139-157

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# Multi-pulse homoclinic loops in systems with a smooth first integral

*Authors*

- Turaev, Dmitry

*2010 Mathematics Subject Classification*

- 37J45 37G20 37D05 37J30 37G30 37G05 34C20 34C37

*Keywords*

- Hamiltonian dynamics, localized solution, orbit-flip, homoclinic bifurcation, hyperbolic set, superhomoclinic orbit

*DOI*

*Abstract*

We prove that the orbit-flip bifurcation in the systems with a smooth first integral (e.g. in the Hamiltonian ones) leads to appearance of infinitely many multi-pulse self-localized solutions. We give a complete description to this set in the language of symbolic dynamics and reveal the role played by special non-selflocalized solutions (e.g. periodic and heteroclinic ones) in the structure of the set of self-localized solutions. We pay a special attention to the superhomoclinic ("homoclinic to homoclinic") orbits whose presence leads to a particularly rich structure of this set.

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# Control variational methods for differential equations

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49K20 49K15 34B05 35J35

*Keywords*

- Differential systems, variational methods, optimal control

*DOI*

*Abstract*

We review recent results established in the literature via the optimal control approach to differential equations, and we show that a systematic study of general variational inequalities associated to fourth-order operators can be performed by similar methods.

*Appeared in*

- Optimal Control of Complex Structures (Hoffmann, K.-H., Lasiecka, I., Leugering, G., Sprekels, J., Troeltzsch, F., eds.), International Series of Numerical Mathematics, 139, 2002,Birkhaeuser, Basel [u.a.], pp. 245--257

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# Lipschitz continuity of polyhedral Skorokhod maps

*Authors*

- Krejčí, Pavel
- Vladimirov, Alexander A.

*2010 Mathematics Subject Classification*

- 47H30 52B70

*Keywords*

- polyhedral Skorokhod problem, oblique reflections, Lipschitz continuity

*DOI*

*Abstract*

We show that a special stability condition of the associated system of oblique projections (the so-called ℓ-paracontractivity) guarantees that the corresponding polyhedral Skorokhod problem in a Hilbert space X is solvable in the space of absolutely continuous functions with values in X. If moreover the oblique projections are transversal, the solution exists and is unique for each continuous input and the Skorokhod map is Lipschitz continuous in both C([0,T]; X) and W^{1,1}(0,T; X). An explicit upper bound for the Lipschitz constant is derived.

*Appeared in*

- Z. Anal. Anwendungen (J. Anal. Appl.) 20 (2001), pp. 817--844

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# Metastability and low lying spectra in reversible Markov chains

*Authors*

- Bovier, Anton
- Eckhoff, Michael
- Gayrard, Veronique
- Klein, Markus

*2010 Mathematics Subject Classification*

- 60J10 60K35

*Keywords*

- Markov chains, metastability, eigenvalue problems, exponential distribution

*DOI*

*Abstract*

We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of it metastable points as a subset of the state space $G_N$ such that (i) this set is reached from any point $xin G_N$ without return to $x$ with probability at least $b_N$, while (ii) for any two point $x,y$ in the metastable set, the probability $T^-1_x,y$ to reach $y$ from $x$ without return to $x$ is smaller than $a_N^-1ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: item(i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. item(ii) To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.

*Appeared in*

- Comm. Math. Phys. 228 (2002), pp. 219-255

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# Mutually catalytic branching in the plane: Finite measure states

*Authors*

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

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J80

*Keywords*

- Catalytic super-Brownian motion, catalytic super-random walk, collision local time, duality, martingale problem, segregation of types, stochastic pde

*DOI*

*Abstract*

We study a pair of populations in ℝ^{2} which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding ℤ^{2}-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model.

*Appeared in*

- Ann. Prob. 30 (2002), pp. 1681-1762

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# Kinetic schemes and initial boundary value problems for the Euler system

*Authors*

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

*2010 Mathematics Subject Classification*

- 82C40 76P05 65P05 76L05

*Keywords*

- Kinetic theory of gases, extended thermodynamics, Maximum Entropy Principle, shock waves

*DOI*

*Abstract*

We study kinetic solutions, including shocks, of initial and boundary value problems for the Euler equations of gases. In particular we consider moving adiabatic boundaries, which may be driven either by a given path or because they are subjected to forces.

In the latter case we consider a gas in a cylinder, and the boundary may represent a piston that suffers forces by the incoming and outgoing gas particles. Moreover, we will study periodic boundary conditions.

A kinetic scheme consists of three ingredients: (i) There are periods of free flight of duration τ_{M}, where the gas particles move according to the free transport equation. (ii) It is assumed that the distribution of the gas particles at the beginning of each of these periods is given by a MAXWELLian. (iii) The interaction of gas particles with a boundary is described by a so called extension law, that determines the phase density at the boundary, and provides additionally continuity conditions for the the fields at the boundary in order to achieve convergence.

The EULER equations result in the limit τ_{M} → 0.

We prove rigorous results for these kinetic schemes concerning (i) regularity, (ii) weak conservation laws, (iii) entropy inequality and (iv) continuity conditions for the fields at the boundaries. The study is supplemented by some numerical examples.

This approach is by no mean restricted to EULER equations or to adiabatic boundaries, but it holds also for other hyperbolic systems, namely those that rely on a kinetic formulation.

*Appeared in*

- Transport Theory Statist. Phys., 31 (2002), pp. 1-33, with new title: Kinetic schemes and initial boundary value problems

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# Sur les arches lipschitziennes

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49J40 34B10 93C15 34A05

*Keywords*

- Kirchhoff-Love model, variational approach, explicit solution

*DOI*

*Abstract*

We study the Kirchhoff-Love model in the case when the middle curve of the arch has corners. Our approach does not use the Dirichlet principle or the Korn inequality. We propose a variational formulation based on optimal control theory and we obtain explicit formulas for the deformation.

*Appeared in*

- C. R. Acad. Sci. Paris, t. 331 (2000), Serie I, pp. 179-184

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# Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Pereverzev, Sergei V.

*2010 Mathematics Subject Classification*

- 62G05 65J10

*Keywords*

- Ill-posed problems, inverse estimation, operator equations, Gaussian white noise, information complexity

*DOI*

*Abstract*

We study the efficiency of the approximate solution of ill-posed problems, based on discretized observations, which we assume to be given afore-hand. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self-regularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method, which achieves the best possible order of accuracy must use at least such amount of noisy observations. We accomplish our study with two specific problems, Abel's integral equation and the recovery of continuous functions from noisy coefficients with respect to a given orthonormal system, both classical ill-posed problems.

*Appeared in*

- SIAM J. Numer. Anal., 38 (2001) pp. 1999--2021.

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# Stable implied calibration of a multi-factor LIBOR model via a semi-parametric correlation structure.

*Authors*

- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Coffey, Brian

*2010 Mathematics Subject Classification*

- 60H05 60H10 90A09

*Keywords*

- Interest rate modelling, LIBOR models, calibration

*DOI*

*Abstract*

We will study the thorny issues around simultaneous calibration of LIBOR models to cap(let) and swaption prices in the markets. We will show in general that low factor market models calibrated to these prices tend to imply unrealistic instantaneous correlations between different forward LIBOR rates. Many-factor models, however, have in general a large parameter dimension and therefore tend to be unstable. In this paper we handle this problem by using a semi-parametric full rank correlation structure in a Brace-Gatarek-Musiela/Jamshidian framework, [1, 5] subject to certain natural constraints which enforce realistic behaviour of forward correlations. A LIBOR market model equipped with this correlation structure has essentially the same parameter dimension as a general two-factor model and we show that calibration of such a model to market swaption and cap(let) volatilities is very stable. Moreover, the implied instantaneous forward LIBOR correlation matrix is consistent with estimations from historical data. Further, application of principal component analysis to the thus obtained multi-factor model yields stably calibrated low-factor models.

*Appeared in*

- International Journal of Theoretical and Applied Finance Vol. 6, No. 4, 1-13 (2003) under new title: Systematic Generation of Correlation Structures for the Libor Market Model.

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