# Multiscale methods for the solution of the Helmholtz and Laplace equations

*Authors*

- Dahmen, Wolfgang
- Kleemann, Bernd
- Prößdorf, Siegfried
- Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 65N12 65N22 65N35 65N38 65R20

*Keywords*

- Multiscale methods, compression, fast solution, Helmholtz equation, Laplace equation, collocation, preconditioning, scattering

*DOI*

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# Convex Analysis of the Energy Model of Semiconductor Devices

*Authors*

- Albinus, Günter

*2010 Mathematics Subject Classification*

- 35K22 35K60 47H05 47H19 47N99 65J15 65M20 80A20 82D99

*Keywords*

- Convex analysis, monotone operators, evolution equation, Lyapunov function, energy model, semiconductor devices, convex thermodynamic potentials, time discretization, Rothe's method

*DOI*

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# On the solution of the generalized airfoil equation

*Authors*

- Okada, Susumu
- Prößdorf, Siegfried

*2010 Mathematics Subject Classification*

- 45E05 45E10 65R20

*Keywords*

- Generalized airfoil equation, singular integral equation, Chebyshev polynomials, Lagrange interpolation, weighted Sobolev-type spaces

*DOI*

*Appeared in*

- J. Integral Equations Appl. 9 (1997) pp. 71-98

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# Splitting of some nonlocalized solutions of the Korteweg-de Vries equation into solitons

*Authors*

- Khruslov, Evgenii Ya.
- Stephan, Holger

*Keywords*

- asymptotic solitons, KdV equation, inverse scattering method

*DOI*

*Appeared in*

- Matematicheskaya fizika, analiz, geometriya, Charkov, 5 (1998), No. 1/2, pp. 49-67

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# Hopfield models as generalized random mean field models

*Authors*

- Bovier, Anton
- Gayrard, Véronique

*2010 Mathematics Subject Classification*

- 60K35 82B44

*Keywords*

- Hopfield model, mean field theory, spin glasses, neural networks, Gibbs measures, large deviations, concentration of measure, random matrices, replica symmetry

*DOI*

*Appeared in*

- Mathematical Aspects of Spin glasses and Neuronal Networks, A. Bovier, P. Picco Eds., Progress in Probability 41, Birkhaeuser, 1998, pp. 3-89

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# Automatic control via thermostats of a hyperbolic Stefan problem with memory

*Authors*

- Colli, Pierluigi
- Grasselli, Maurizio
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35R35 35R70 45K05 93C20

*Keywords*

- Feedback control, Stefan problems, memory kernels, hyperbolic heat conduction

*DOI*

*Abstract*

A hyperbolic Stefan problem based on the linearized Gurtin-Pipkin heat conduction law is considered. Temperature and free boundary are controlled by a thermostat acting on the boundary. This feedback control is based on temperature measurements performed by real thermal sensors located into the domain containing the two-phase system and/or at its boundary. Three different types of thermostats are analyzed: ideal switch, relay switch, and Preisach hysteresis operator. The resulting models lead to formulate integrodifferential hyperbolic Stefan problems with nonlinear and nonlocal boundary conditions. In all the cases, existence results are proved. Uniqueness is also shown, unless in the situation corresponding to the ideal switch.

*Appeared in*

- Appl. Math. Optimiz., 39 (1999), pp. 229 - 255

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# Stability Index for Invariant Manifolds of Stochastic Systems

*Authors*

- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 60H10 93E15 34C30 34D08 34F05

*Keywords*

- stochastic stability, Lyapunov exponents, moment Lyapunov exponents, stability index

*DOI*

*Abstract*

A lot of works has been devoted to stability analysis of a stationary point for linear and non-linear systems of stochastic differential equations. Here we consider the stability of an invariant compact manifold of a non-linear system. To this end we derive a linearized system for orthogonal displacements of a solution from the manifold. For this system, we introduce notions of Lyapunov exponents, moment Lyapunov exponents, and stability index. The stability index controls the asymptotic behavior of solutions of the input system in a neighborhood of the manifold. Most extensively we study these problems in the case when the invariant manifold is an orbit.

*Appeared in*

- Random Comput. Dynamics 5 (1997) no. 4

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# Numerical study of a stochastic weighted particle method for a model kinetic equation

*Authors*

- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- nonlinear integral equations, nonlinear kinetic equations, stochastic weighted particle method, Boltzmann equation

*DOI*

*Abstract*

A stochastic weighted particle method is applied to a model nonlinear kinetic equation. A detailed study of various numerical approximations is presented. The main effect achieved by the new method is an artificial increase of the relative number of simulation particles with prescribed velocities.

*Appeared in*

- J. Comput. Phys., 128 (1996), pp. 351-362

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# A special system of reaction equations

*Authors*

- Zacharias, Klaus

*2010 Mathematics Subject Classification*

- 34A34 34A05 92E20 80A30

*Keywords*

- Nonlinear equations and systems, explicit solutions and reductions, chemical reactions, chemical kinetics

*DOI*

*Abstract*

A system of reaction equations describing polymer degradation is shown to be integrable in analytic terms.

*Appeared in*

- Z. Angew. Math. Mech., 78 (1998), pp. 271-275

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# Harmonic crystal on the wall: a microscopic approach

*Authors*

- Bolthausen, Erwin
- Ioffe, Dmitry

*2010 Mathematics Subject Classification*

- 60K35 82B24

*Keywords*

- 2D Gaussian SOS model, Winterbottom construction, surface tension, random walk representation, torsional rigidity

*DOI*

*Abstract*

A three dimensional Winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic Gaussian SOS droplets under the fixed volume constraint. The proof is based on a coarse graining of the random microscopic region "wetted" by the crystal, random walk representation of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem.

*Appeared in*

- Comm. Math. Phys., 187 (1997), pp. 523-566

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# Dynamics of spiral waves on unbounded domains using center-manifold reductions

*Authors*

- Sandstede, Björn
- Scheel, Arnd
- Wulff, Claudia

*2010 Mathematics Subject Classification*

- 35B32 35K57 57S20 57S30

*Keywords*

- spiral waves, non-compact groups, center manifolds, Hopf bifurcation

*DOI*

*Abstract*

An equivariant center-manifold reduction near relative equilibria of G-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie group G is possibly non-compact. Moreover, it is not required that G induces a strongly continuous group action on the underlying function space. In fact, G may act discontinuously. The results are applied to bifurcations of stable patterns arising in reaction-diffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov-Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidly-rotating spiral waves to meandering or drifting waves, and from twisted scroll rings are investigated.

*Appeared in*

- J. Differential Equations, 141 (1997), pp. 122-149

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# The lifting line equation for a curved wing in oscillatory motion

*Authors*

- Chiocchia, Gianfranco
- Prößdorf, Siegfried
- Tordella, Daniela

*2010 Mathematics Subject Classification*

- 76Bxx 65R20 45E05 45E10

*Keywords*

- Cauchy singularity, logarithmic singularity, Prandtl singular integral equation, operator method, Pistolesi-Weissinger 3/4-chord method, Posio´s theory, integrodifferential equation, Gaussian quadrature method, Chebyshev polynomials, circulation distribution

*DOI*

*Abstract*

An unsteady linear lifting line method for the determination of the circulation and lift distribution along the span of a curved wing subject to harmonic small amplitude oscillations is presented. The method relies on the Pistolesi-Weissinger 3/4-chord steady lifting line theory and couples it to the unsteady theory developed by Possio for the motion of lifting surfaces. It leads to an integro-differential equation of a modified Prandtl's type, where the unknown is the circulation. This equation has been carefully analysed in order to evidence all the singularities and to treat them in the most convenient way. The numerical procedure consists of a gaussian quadrature technique based on Chebyshev's polynomial approximation of the unknown function. The method has been appraised through the comparison of a number of solutions, pertaining to different wing configurations, with existing solutions based on lifting surface theory.

*Appeared in*

- Z. Angew. Math. Mech., 77 (1997) 4, pp. 295-315

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# Shift spaces and attractors in non invertible horse shoes

*Authors*

- Bothe, Hans Günter

*2010 Mathematics Subject Classification*

- 58F12 58F15 58F03

*Keywords*

- Horse shoes, non invertible maps, shift spaces, attractors

*DOI*

*Abstract*

As well known, a horse shoe map, i.e. a special injective reimbedding of the unit square I^{2} in ℝ^{2} (or more general, of the cube I^{m} in ℝ^{m}) as considered first by S. Smale [4], defines a shift dynamics on the maximal invariant subset of I^{2} (or I^{m}). It is shown that this remains true almost surely for non injective maps provided the contraction of the mapping in the stable direction in sufficiently strong.

*Appeared in*

- Fund. Math. 152 (1997) no. 3, pp. 267--289.

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# Entwicklung einer Schnittstelle für einen DAE-Solver in der chemischen Verfahrenstechnik

*Authors*

- Horn, Dietmar

*2010 Mathematics Subject Classification*

- 68U30 65C99

*Keywords*

- software engineering, interfaces

*DOI*

*Abstract*

Bei der Entwicklung numerischer Verfahren zur Lösung großer strukturierter Systeme von Algebro-Differentialgleichungen, die in der chemischen Prozeßsimulation vorkommen, ist eine möglichst automatische Bereitstellung der Beispieldaten notwendig. Dazu wurde eine Schnittstelle definiert, die die notwendigen Informationen enthält. Zur automatischen Schnittstellenerzeugung wurde ein Programm implementiert, das ausgehend von den Protokollen des Prozeßsimulators SPEEDUP diese Schnittstelle erzeugt.

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# A mean field approximation for hopping transport in disordered materials

*Authors*

- Brehmer, Ludwig
- Liemant, Alfred

*2010 Mathematics Subject Classification*

- 60K35 82B44

*Keywords*

- Mean field approximation, charge carriers transport, hopping processes

*DOI*

*Abstract*

By a mean field aproximation a macroscopic charge transport equation is derived from a hopping model with long range interaction. The transport coefficients are calculated from microscopic hopping characteristics of the disordered medium.

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# On criticality for competing influences of boundary and external field in the Ising model

*Authors*

- Greenwood, Priscilla E.
- Sun, Jiaming

*2010 Mathematics Subject Classification*

- 60K35 82B27

*Keywords*

- Competing influences, Ising model, Gibbs measures

*DOI*

*Abstract*

Consider the Gibb's measures μ_{Λ(1/h),-,s} (defined below) of the Ising model, in a box Λ(l/h) in Z^{d} with side length 1/h, with external field s and negative boundary condition at a temperature T < T_{c}. It is well known that when s = 0, namely no external field, μ_{Λ(l/h),-,0} converges weakly to the pure (－)-phase μ_ as h ↘ 0. And when s ≠ 0 is fixed, μ_{Λ(l/h),-,s} converges weakly to a measure μ_{s} which does not depend on the boundary conditons (Ellis (1985)). But if one lets the external field s decrease as h goes to zero in such a way that it competes with the negative boundary (in particular, if s = Bh), then one may obtain different limits in different ranges of B. This phenomenon of competing influences has been investigated by several authors. Martirosyan (1987) first proved that at low temperature T and with large B, the Gibb's measure μ_{Λ(l/h),-,Bh} converges weakly to the pure (＋)-phase μ_{+}. Schonmann (1994) (referred to in the sequel as [Sch]) showed that at low temperature T, there are values B_{1}(T) ≤ B_{2}(T) such that when B < B_{1}(T), μ_{Λ(l/h),-,Bh} converges weakly to μ_ and when B > B_{2}(T), the limit is μ_{+}. This says that the negative boundary conditon dominates in the limit when B < B_{1}(T) whereas the small external field dominates when B > B_{2}(T). The question, then, is whether there exists a critical value B_{0} = B_{0}(T) = B_{1}(T) = B_{2}(T) for all T < T_{c} such that μ_{Λ(l/h),-,Bh} converges to μ_ when B < B_{0} and to μ_{+} when B > B_{0}. In the case of d = 2, this question was completely solved by Schonmann and Shlosman (1996), using large deviation results and techniques. For higher dimensions, Greenwood and Sun (1997) ([GS] hereafter) proved the criticality of a certain value B_{0} for all T < T_{c}, but only in terms of the convergence of average spins rather than in terms of weak convergence. This paper extends these results by showing that for low temperature and the same critical value B_{0}, μ_{Λ(l/h),-,Bh} converges weakly to μ_ when B < B_{0} and to μ_{+} when B > B_{0}. In [Sch], the main results are about the relaxation time of a stochastic Ising model in relation to an external field h. He shows that the relaxation time blows up when h ↘ 0 as exp(λ/h^{d-1}). In fact he obtains upper and lower bounds for λ = λ(T), which are derived from his B_{1}(T), B_{2} (T) and his estimate of the spectral gap of the generator of the evolution. One might hope to obtain a critical value of λ using Schonmann's methods and the critical value B_{0}. This indeed again gives bounds for λ but not a critical value. A reason is that estimation of the spectral gap is involved.

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# Decomposition and diagonalization in solving large systems

*Authors*

- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 65J05 65J15 65Y05

*Keywords*

- Decomposition, diagonalization, large scale systems

*DOI*

*Abstract*

Consider the nonlinear equation (*) x = Tx + ƒ with a strictly contractive operator T in some real separable Hilbert space. A well-known procedure to approximate the unique solution x* (ƒ) of (*) is the projection-iteration method which can be characterized as a method of diagonalization. In case that (*) is a large system which can be represented as a system of weakly coupled subsystems, an efficient method to approximate x* (ƒ) is the decomposition method which is a block iteration scheme. One realization of this method is the waveform relaxation method. In this note we combine the diagonalization technique with the decomposition method and derive conditions for the convergence of the resulting iteration scheme.

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# Evaluation of moment Lyapunov exponents for second order linear autonomous SDE

*Authors*

- Milstein, Gregori N.

*Keywords*

- Lyapunov exponents, moment Lyapunov exponents, stability index

*DOI*

*Abstract*

Deterministic methods for evaluation of moment Lyapunov exponents are derived for two-dimensional systems with non-degenerate noise.

*Appeared in*

- Random Comput. Dynamics Vol. 4 (1996) No. 4 pp. 301--315

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# Portabilität und Adaption von Software der linearen Algebra für Distributed Memory Systeme

*Authors*

- Hebermehl, Georg
- Hübner, Friedrich-Karl

*2010 Mathematics Subject Classification*

- 65Y05 65Y10

*Keywords*

- Linear algebra, communication routines, message passing, portability

*DOI*

*Abstract*

Durch die Verwendung anerkannter Grundbausteine für elementare Operationen der linearen Algebra und von Kommunikationsroutinen sowie üblicher blockzyklischer Datenverteilungen können Algorithmen höheren Levels weitgehend portabel und optimal auf Distributed Memory Computern adaptiert werden. Insbesondere wird über die Bereitstellung der Kommunikationsbibliothek BLACS für PARSYTEC-Rechner berichtet.

*Appeared in*

- Software Engineering im Scientific Computing, W. Mackens, S. M. Rump, eds., Universitaet Hamburg, 1995, pp. 31--33

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# Idealkristalle als Abelsche Varietäten

*Authors*

- Krause, Udo

*2010 Mathematics Subject Classification*

- 14K25 22E70 35Q40 81Q10 82D25

*DOI*

*Abstract*

Es wird gezeigt, daß sich Idealkristalle in natürlicher Weise als hauptpolarisierte Abelsche Varietäten darstellen lassen. Es existiert demzufolge auf Λ × Λ eine ganzzahlige schiefsymmetrische Matrix E: Λ × Λ → ℤ, deren Elementarteiler sämtlich gleich 1 sind, wobei Λ das Translationsgitter des Idealkristalls im Phasenraum V ist. Bezüglich derartiger Gitter kann die Gitterdarstellung der Heisenberggruppe definiert werden oder mit anderen Worten: Auf einer hauptpolarisierten Abelschen Varietät kann nichtrelativistische Quantenmechanik betrieben werden. Die Gitterdarstellung wird detailliert betrachtet. Der harmonische Oszillator wird in der Gitterdarstellung berechnet und illustriert. Es zeigt sich, daß die Eigenfunktionen des harmonischen Oszillators in der Gitterdarstellung systematisch aus der mit einem Exponentialfaktor multiplizierten Riemannschen Thetafunktion für die zugrundeliegende hauptpolarisierte Abelsche Varietät hervorgehen. Die vollständigen Eigenfunktionensysteme der Impuls- sowie der Ortsoperatoren werden in der Gitterdarstellung aufgestellt und führen zu einer deutlichen Verbesserung des Galerkinverfahrens für die Berechnung der Bandstruktur des Kristallelektrons.

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# An efficient ADI-solver for scattered data problems with global smoothing

*Authors*

- Arge, Erlend
- Kunoth, Angela

*2010 Mathematics Subject Classification*

- 65D17, 65N06 73N20.

*Keywords*

- Scattered data approximation, fourth order elliptic problems, difference methods, preconditioned conjugate gradient methods, ADI methods

*DOI*

*Abstract*

For the approximate representation of large data sets over a parameter domain in ℝ^{2}, many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth order elliptic partial differential equation. Its discretization by a difference scheme makes it particularly easy to incorporate (appropriate approximations of) known data points. Because of the solution of the resulting symmetric linear system being the most CPU-demanding step, we investigate first the performance of a preconditioned conjugate gradient method with an SSOR and a RILU preconditioner. However, since the partial differential operator does not contain mixed derivatives, using an alternating-direction-implicit scheme (ADI method) which has been employed successfully in the past for second order problems, together with Cholesky factorization of the corresponding one-dimensional operators provides a fast and effective method for the problem at hand. The computational studies show that an instationary ADI method is superior to the above mentioned preconditioned conjugate gradient solvers both with respect to work load and accuracy of the solution. For the fourth order model problem considered in this paper, the instationary ADI method with Wachspress parameters results in a number of iterations that is essentially independent of the number of variables.

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# A cascadic multigrid algorithm in the finite element method for the plane elasticity problem

*Authors*

- Gilyova, Lida V.
- Shaidurov, Vladimir V.

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- Elasticity problem, multigrid, cascadic algorithm, finite element method, conjugate-gradient method, Jacobi-type method

*DOI*

*Abstract*

For the plane elasticity problem a standard scheme of the finite element method with the use of piecewise linear elements on triangles is discussed. For its solution on a sequence of embedded triangulations, a cascadic arrangement of two iterative algorithms is used, which gives the simplest version of multigrid methods without preconditioning and restriction onto a coarser grid. The cascadic algorithm begins on the coarsest grid where the grid problem is solved by direct method. To obtain approximate solutions on finer grids, the iterative method is used; interpolation of the approximate solution from the preceding coarser grid is taken as the initial guess. It is proved that the convergence rate of this algorithm does not depend on the number of unknowns and grids.

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# Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders

*Authors*

- Peterhof, Daniela
- Sandstede, Björn
- Scheel, Arnd

*2010 Mathematics Subject Classification*

- 35J65 35B32 35B40

*Keywords*

- dynamical system treatment of elliptic equations

*DOI*

*Abstract*

In applications, solitary-wave solutions of semilinear elliptic equations Δu + g(u,∇u) = 0 (x,y) ∈ ℝ × Ω in infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems in x has proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov-Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbation µ h(x,u,∇u) periodic in x is added.

*Appeared in*

- J. Differential Equations, 140 (1997), pp. 266-308

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# Convergence to a non-trivial equilibrium for two-dimensional catalytic super-Brownian motion

*Authors*

- Fleischmann, Klaus
- Klenke, Achim

*2010 Mathematics Subject Classification*

- 60J80 60G57 60K35

*Keywords*

- catalytic super-Brownian medium, catalyst, superprocess, measure-valued branching, non-extinction, persistence, two-dimensional process, equilibrium state, absolutely continuous states, self-similarity, time-space gaps of super-Brownian motion, asymptotic density, local L^2-Lipschitz continuity

*DOI*

*Abstract*

In contrast to the classical super-Brownian motion (SBM), the SBM (X^{ϱ}_{t})_{ t ≥ 0} in a super-Brownian medium ϱ (constructed in [DF96a]) is known to be persistent in all three dimensions of its non-trivial existence: The full intensity is carried also by all longtime limit points ([DF96a, DF96b, EF96]). Uniqueness of the accumulation point, however, has been shown so far only in dimensions d=1 and d=3 ([DF96a, DF96b]). Here we fill this gap and show that convergence also holds in the critical dimension d=2. We identify the limit as a random multiple of Lebesgue measure.

Our main tools are a self-similarity of X^{ϱ} in d=2 and the fact that the medium has "gaps" in the space-time picture. The self-similarity implies that persistent convergence of X^{ϱ}_{t} as t → ∞ is equivalent to the absolute continuity of X^{ϱ}_{t} at a fixed time t > 0. Absolute continuity however will be obtained via the fact that in absence of the catalytic medium, X^{ϱ} is smoothed according to the heat flow.

*Appeared in*

- Ann. Appl. Probab., 9(2) (2000), pp. 298-318, under new title: Smooth density field of catalytic super-Brownian motion.

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# Free energy, free entropy, and a gradient structure for thermoplasticity

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35Q79 37D35 74C10 74F05 82B35

*Keywords*

- Generalized gradient systems, GENERIC, variational formulations, incremental minimization, entropy as driving functional, primal and dual, entropy-production potential, thermodynamic modeling, viscoplasticity

*DOI*

*Abstract*

In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.

Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature.

We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (*J. Mech. Physics Solids*, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

*Appeared in*

- Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. In Honor of Michael Ortiz's 60th Birthday., K. Weinberg, A. Pandolfi, eds., vol. 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, 2016, pp. 135--160

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# Multiscale Methods for Boundary Integral Equations and their Application to Boundary Value Problems in Scattering Theory and Geodesy

*Authors*

- Kleemann, Bernd H.
- Rathsfeld, Andreas
- Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 45L10 65R20 65N38

*Keywords*

- Multiscale methods, wavelet algorithm, pseudodifferential equations, Galerkin method, collocation

*DOI*

*Abstract*

In the present paper we give an overview on multiscale algorithms for the solution of boundary integral equations which are based on the use of wavelets. These methods have been introduced first by Beylkin, Coifman, and Rokhlin [5]. They have been developed and thoroughly investigated in the work of Alpert [1], Dahmen, Proessdorf, Schneider [16-19], Harten, Yad-Shalom [25], v.Petersdorff, Schwab [33-35], and Rathsfeld [39-40]. We describe the wavelet algorithm and the theoretical results on its stability, convergence, and complexity. Moreover, we discuss the application of the method to the solution of a two-dimensional scattering problem of acoustic or electromagnetic waves and to the solution of a fixed geodetic boundary value problem for the gravity field of the earth. The computational tests confirm the high compression rates and the saving of computation time predicted by the theory.

*Appeared in*

- Notes on Numerical Fluid Mechanics Vol. 54, Proceedings of the 12th GAMM-Seminar Kiel on Boundary Elements: Implementation and Analysis of Advanced Algorithms, eds.: W.Hackbusch, G. Wittum, Vieweg-Verlag, Braunschweig, Wiesbaden, 1996

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# Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation

*Authors*

- Sandstede, Björn

*2010 Mathematics Subject Classification*

- 34C37 35B35 58F14

*Keywords*

- Heteroclinic orbits, Stability, FitzHugh-Nagumo equation

*DOI*

*Abstract*

In this article existence and stability of N-front travelling wave solutions of partial differential equations on the real line is investigated. The N-fronts considered here arise as heteroclinic orbits bifurcating from a twisted heteroclinic loop in the underlying ordinary differential equation describing travelling wave solutions. It is proved that the N-front solutions are linearly stable provided the fronts building the twisted heteroclinic loop are linearly stable. The result is applied to travelling waves arising in the FitzHugh-Nagumo equation.

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# Weak solution to some Penrose-Fife phase-field systems with temperature-dependent memory

*Authors*

- Colli, Pierluigi
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K50 80A20 80A22

*Keywords*

- Penrose-Fife model, phase transitions, memory effects, nonlinear heat conduction, phase-field systems, nonlinear parabolic equations

*DOI*

*Abstract*

In this paper a phase-field model of Penrose-Fife type is considered for a diffusive phase transition in a material in which the heat flux is a superposition of two different contributions: one part is proportional to the spatial gradient of the inverse temperature, while the other is of the form of the Gurtin-Pipkin law introduced in the theory of materials with thermal memory. It is shown that an initial-boundary value problem for the resulting state equations has a unique solution, thereby generalizing a number of recent results.

*Appeared in*

- J. Differ. Equations 142, no. 1, (1998), pp. 54-77

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# On effects of discretization on estimators of drift parameters for diffusion processes

*Authors*

- Kloeden, Peter E.
- Platen, Eckhard
- Schurz, Henri
- Sørensen, Michael

*2010 Mathematics Subject Classification*

- 60H10 62F12 65U05

*Keywords*

- Discrete time sampling, Inference for stochastic processes, Maximum likelihood estimation, Numerical methods, Simulation, Stochastic differential equations, Stochastic Taylor expansions

*DOI*

*Abstract*

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

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# Global solutions to a coupled parabolic-hyperbolic system with hysteresis in 1-d magnetoelasticity

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35G25 73R05 82D40.

*Keywords*

- Magnetic hysteresis, ferromagnetism, Preisach operator, magnetoelasticity, PDE's with hysteresis, parabolic-hyperbolic systems

*DOI*

*Abstract*

In this paper the system of field equations governing the one-dimensional magnetoelastic evolution in a ferromagnet, which is immersed in an electromagnetic field and subjected to mechanical loads at a constant temperature below the Curie point, is considered. It is assumed that displacement currents are negligible and that all field quantities depend on one space variable only. The hysteretic relation between the applied magnetic field and the magnetization in the ferromagnet are modeled using the notion of hysteresis operators; in particular, hysteresis operators of Preisach type are included. It is shown that an initial-boundary value problem for the system admits global solutions for arbitrary initial data, if viscosity is present in the material, and for small initial data, if not. The considered field equations may be regarded as a model for the effect of magnetostriction in ferromagnets.

*Appeared in*

- Nonlin. Anal., 33 (1998), pp. 341-358

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# A Wavelet Algorithm for the Solution of a Singular Integral Equation over a Smooth Two-dimensional Manifold

*Authors*

- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 45L10 65R20 65N38

*Keywords*

- singular integral equation, collocation, wavelet algorithm

*DOI*

*Abstract*

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a smooth surface. Using a fixed set of parametrizations, we introduce special wavelet bases for the spaces of test and trial functions. The trial wavelets have two vanishing moments only if their supports do not intersect the lines belonging to the common boundary of two subsurfaces defined by different parameter representations. Nevertheless, analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be compressed to sparse matrices such that the iterative solution of the matrix equations becomes fast. Finally, we present a fast quadrature algorithm for the computation of the compressed stiffness matrix.

*Appeared in*

- J. Integral Equations Appl., 10 (1998), No. 4, pp. 445-501

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# A Wavelet Algorithm for the Boundary Element Solution of a Geodetic Boundary Value Problem

*Authors*

- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 45L10 65R20 65N38

*Keywords*

- singular integral equation, collocation, wavelet algorithm.

*DOI*

*Abstract*

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a part of the surface of the earth. This singular equation is the boundary integral equation corresponding to the oblique derivative boundary problem for Laplace's equation. We introduce special wavelet bases for the spaces of test and trial functions. Analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be reduced to sparse matrices such that the assembling of the matrices and the iterative solution of the matrix equations become fast. Though the theoretical results apply only to integral equations with "smooth" solutions over "smooth" manifolds, we present numerical tests for a geometry as difficult as the surface of the earth.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg. 157 (1998) pp. 267-287

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# Fast computations with the harmonic Poincaré-Steklov operators on nested refined meshes

*Authors*

- Khoromskij, Boris N.
- Prößdorf, Siegfried

*DOI*

*Abstract*

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré-Steklov operators in presence of nested mesh refinement. For both interior and exterior problems the matrix-vector multiplication for the finite element approximations to the Poincaré-Steklov operators is shown to have a complexity of the order O(N_{ref}log^{3}N) where N_{ref} is the number of degrees of freedom on the polygonal boundary under consideration and N = ^{2-p0} · N_{ref}, p_{0} ≥ 1, is the dimension of a finest quasi-uniform level. The corresponding memory needs are estimated by O(N_{ref}log^{2}N). The approach is based on the multilevel interface solver (as in the case of quasi-uniform meshes, see [20]) applied to the Schur complement reduction onto the nested refined interface associated with nonmatching decomposition of a polygon by rectangular substructures.

*Appeared in*

- Adv. Comput. Methods, 8 (1998), pp. 111-135

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# On the asymptotic analysis of singularly perturbed systems with sliding mode

*Authors*

- Fridman, Leonid M.
- Rumpel, Rainer J.

*2010 Mathematics Subject Classification*

- 34C15 34E10

*Keywords*

- sliding mode, discontinuous, singularly perturbed, nonlinear oscillations, dry friction, relay control

*DOI*

*Abstract*

In this paper we study singularly perturbed systems with discontinuity surfaces. This means that we have a system of ordinary differential equations with a small parameter and a piecewise smooth vector field. The state where the trajectory moves on the discontinuity surface is called *sliding mode*. We present an asymptotic representation for trajectories with temporary sliding and apply the result to stick-slip vibrations.

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# Generalized heteroclinic cycles in spherically invariant systems and their pertubations

*Authors*

- Chossat, Pascal
- Guyard, Frédéric
- Lauterbach, Reiner

ORCID: 0000-0002-9310-3177

*DOI*

*Abstract*

In this paper we want to investigate the effects of forced symmetry breaking perturbations, see LAUTERBACH & ROBERTS [29], as well as [28, 31], on the heteroclinic cycle which was found in the ℓ = 1, ℓ = 2 mode interaction by ARMBRUSTER & CHOSSAT [1, 12] and generalized by CHOSSAT and GUYARD [25, 14]. We show that this cycle is embedded in a larger class of cycles, which we call a generalized heteroclinic cycle (GHC). After describing the structure of this set we discuss its stability. The main problem is to find a selection principle, that is to give a mechanism which enables the physical system to select one particular heteroclinic cycle on this generalized heteroclinic cycle. After that the persistence under symmetry breaking perturbations is investigated. We will discuss also the application to the spherical Bénard problem, which was the initial motivation for this work.

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# On a System of Nonlinear PDE's with Temperature-Dependent Hysteresis in One-Dimensional Thermoplasticity

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35G25 73B30 73E60 73B05

*Keywords*

- Thermoplasticity, hysteresis, Prandtl-Ishlinskii operator, weak solutions, PDE's with hysteresis

*DOI*

*Abstract*

In this paper, we develop a thermodynamically consistent description of the uniaxial behaviour of thermoelastoplastic materials that are characterized by a constitutive law of the form σ(x,t)＝ 𝒫[εθ(x,t)](x,t), where ε, σ, θ denote the fields of strain, stress and absolute temperature, respectively, and where {𝒫[·, θ]}_{θ>0} denotes a family of(rate-independent) hysteresis operators of Prandtl-Ishlinskii type, parametrized by the absolute temperature. The system of state equations governing the space-time evolution of the material are derived. It turns out that the resulting system of two nonlinearly coupled partial differential equations involves partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a global weak solution. The paper can be regarded as a first step towards a thermodynamic theory of rate-independent hysteresis operators depending on temperature.

*Appeared in*

- J. Math. Anal. Appl. 209 (1997) pp. 25-46

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# Asymptotic Behavior of the Solutions to a Landau-Ginzburg System with Viscosity for Martensitic Phase Transitions in Shape Memory Alloys

*Authors*

- Sprekels, Jürgen
- Zheng, Songmu
- Zhu, Peicheng

*2010 Mathematics Subject Classification*

- 35Q72 73B30 35B40

*Keywords*

- Nonlinear thermoviscoelasticity, shape memory alloys, phase transitions asymptotic behaviour, compact orbits, Landau-Ginzburg theory

*DOI*

*Abstract*

In this paper, we investigate the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a (possibly small) viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as function of the order parameter. Results concerning the asymptotic behavior of the solution as time tends to infinity are proved, and the compactness of the orbit is shown.

*Appeared in*

- SIAM J. Math. Anal. 29 (1998) pp. 69 - 84

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# Modellierung und Simulation von Bauelementen der Nano- und Optoelektronik

*Authors*

- Gajewski, Herbert
- Glitzky, Annegret
- Griepentrog, Jens André
- Hünlich, Rolf
- Kaiser, Hans-Christoph
- Rehberg, Joachim
- Röpke, Wilfried
- Stephan, Holger
- Wenzel, Hans

*DOI*

*Abstract*

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

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

*Appeared in*

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

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# Multivariate wavelet thresholding: a remedy against the curse of dimensionality?

*Authors*

- Neumann, Michael H.

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- Nonparametric curve estimation, multivariate wavelet estimators, nonlinear thresholding, curse of dimensionality, anisotropic wavelet basis, anisotropic smoothness classes, smoothness classes with dominating mixed derivatives, optimal rate of convergence

*DOI*

*Abstract*

It is well-known that multivariate curve estimation suffers from the "curse of dimensionality". However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit typical restrictions on the curve, which require a local adaptation to different degrees of smoothness in the different directions. It turns out that the application of a anisotropic multivariate basis, which has in contrast to the conventional multivariate resolution scheme a multidimensional scale parameter, is essential. Some simulations indicate the possible gains by this new method over thresholded estimators based on the multiresolution basis with a one-dimensional scale index.

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# On an information-type inequality for the Hellinger process

*Authors*

- Gushchin, Alexander A.

*2010 Mathematics Subject Classification*

- 60G07 60G48

*Keywords*

- semimartingale, Hellinger process, Cramér-Rao inequality, linear regression

*DOI*

*Abstract*

Let (Ω,𝔉, 𝔽) be a filtered space with two probability measures P and P' on (Ω,𝔉). Let X be a d-dimensional locally square-integrable semimartingale relative to P and P' with the canonical decomposition X = X_{0} + M + A and X = X_{0} + M' + A' respectively. We give a lower bound for the Hellinger process h(1⁄2; P, P') of order 1/2 between P and P' in terms of A, A' and the quadratic characteristic of M and M'. This result implies simple sufficient conditions for the entire separation of measures in a linear regression model with martingale errors.

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# On estimation of non-smooth functionals

*Authors*

- Lepskii, Oleg V.
- Nemirovski, Arkadi
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- non-smooth functionals, integral norm, rate of estimation

*DOI*

*Abstract*

Let a function ƒ be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of ƒ, more precisely, L_{r} -norm ∥ƒ∥_{r} of ƒ. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: estimation of a smooth (differentiable in L_{2}) functional or estimation of a singular functional like the value of ƒ at a certain point or the maximum of ƒ. In the first case, the rate of estimation is typically n^{-1/2} , n being the number of observations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function ƒ in the corresponding norm. We show that the case of estimation of ∥ƒ∥_{r} is in some sense intermediate between the above extreme two. The optimal rate of estimation is worse than n^{-1/2} but better than the usual nonparametric rate. The results depend on the value of r . For r even integer, the rate occurs to be n^{-β/(2β+1-1/r)} where β is the degree of smoothness. If r is not even integer, then the nonparametric rate n ^{-β/(2β+1)} can be improved only by some logarithmic factor.

*Appeared in*

- Probability Theory and Related Fields, 113 (1999), 221-253. under the title: On estimation of the Lr-norm of a regression functionals.

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# On distinguishability of two nonparametric sets of hypothesis

*Authors*

- Ermakov, Mikhail S.

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Hypothesis testing, asymptotic efficiency, signal detection, hypothesis testing about density, nonparametric hypothesis testing

*DOI*

*Abstract*

Let we observe a signal S(t), t ∈ (0, 1) in Gaussian white noise ∈ dw(t). The problem is to test a hypothesis S ∈ Θ_{1} ⊂ L_{2} (0, 1) versus alternatives S ∈ Θ_{2} ⊂ L_{2}(0, 1). The sets Θ_{1}, Θ_{2} are closed and bounded. We show that there exists a statistical procedure allowing to make a true solution S ∈ Θ_{1} or S ∈ Θ_{2} with probability tending to one as ∈ → 0 ( i.e. to distinguish two nonparametric sets Θ_{1} and Θ_{2}) iff there exists a finite-dimensional subspace H ⊂ L_{2} (0, 1) such that the projections Θ_{1} and Θ_{2} on H have no common points. A similar result is also obtained for the problems of testing hypotheses about density.

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# Mean Square Stability Analysis of Some Linear Stochastic Systems

*Authors*

- Ryashko, Lev B.
- Schurz, Henri

*2010 Mathematics Subject Classification*

- 60H10 65C05 65C20 65U05

*Keywords*

- Stochastic systems, Mean square stability, Positive linear operators, Spectral radius, Stochastic differential equations, Numerical methods, theta-methods

*DOI*

*Abstract*

Mean square stability analysis of some continuous and discrete time stochastic systems is carried out in this paper. We present a general approach to mean square stability investigation of systems with multiplicative noise and apply presented theory to discretized linear oscillators as often met in Mechanical Engineering. The analysis relies on the spectral theory of positive operators. As one of the results one obtains a simple and efficient criterion to decide the question of stability of equilibria of linear systems. Conclusions for practical usage and preference of numerical methods solving stochastic differential equations (SDEs) with white noise can be drawn too. For illustration and practical meaningfulness, we describe stability domains of stochastic θ-methods in terms of parametric restrictions.

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# Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice

*Authors*

- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- change-point, local polynomial fit local structure, nonparametric regression, pointwise adaptive estimation

*DOI*

*Abstract*

New method of adaptive estimation of a regression function is proposed. The resulting estimator achieves near optimal rate of estimation in the classical sense of mean integrated squared error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function ƒ or its derivatives. For instance, in the case of a jump of a regression function, beyond the interval of length (in order) n^{-1} log n around change-points the quality of estimation is essentially the same as if the location of this jump were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a non-asymptotic way and can be therefore applied for an arbitrary sample size.

*Appeared in*

- Ann. Statist., 26 (1998), No. 4, pp. 1356-1378

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# Numerische Lösung großer strukturierter DAE-Systeme der chemischen Prozeßsimulation

*Authors*

- Borchardt, Jürgen
- Bruell, Ludger
- Grund, Friedrich
- Horn, Dietmar
- Hubbuch, Frank
- Michael, Tino
- Sandmann, Horst
- Zeller, Robert

*2010 Mathematics Subject Classification*

- 65Y05 80A30 65H10 65F50 80A30 92E20

*Keywords*

- Algebraic-differential equations, systems of partitioned nonlinear equations, systems of sparse linear algebraic equations, parallelization of numerical methods, simulation of chemical processes in chemical plants

*DOI*

*Abstract*

Parallelizable numerical methods for solving large scale DAE systems are developed at the level of differential, nonlinear and linear equations. For this the subsystem-wise structure of the DAE systems based on unit-oriented modelling is explored. Partitionings are used to parallelize waveform relaxation and structured Newton methods. Initial values are computed with a modified Newton method. To solve large sparse systems of linear equations a special Gaussian elimination method is used. The algorithms were implemented on a CRAY C90 vector computer, as well as on both, moderately parallel CRAY J90 vector computers and massively parallel CRAY T3D machines. The methods were tested using several real life examples.

*Appeared in*

- Mathematik Schluesseltechnologie fuer die Zukunft (Hoffmann, K.-H., Jaeger, W., Lohmann, T., and Schunk, H., eds.), Springer-Verlag Berlin Heidelberg New York, 1997, pp. 91-103.

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# Phase Transitions of Shape Memory Alloys in Soft and Hard Loading Devices

*Authors*

- Schwarz, Michael

*2010 Mathematics Subject Classification*

- 34A09 34A50 82D25

*Keywords*

- Phase Transitions, Shape Memory Alloys, Rate Laws

*DOI*

*Abstract*

Shape memory alloys exhibit a complex load deformation temperature behaviour (especially e.g. hysteresis and "inner" loops) which is due to the occurence of a first order phase transition.

By load deformation diagrams the so-called shape memory effect can be made visible.

We study a model of I. Müller et al. [1], based on statistical mechanics, which is applicable to biaxial loading of polycrystalline bodies and incorporates the rotational part of a deformation. Recently, I. Müller et al. [3] have proposed a second model incorporating the coherence energy for solid phase mixtures. For two principle variants of experiments (soft and hard loading devices) we present numerical simulations of load deformation curves for either of the two models. Comparing these with experimental results of so-called "inner loops" in a hystersis the second model shows its superiority.

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# Instability of localised buckling modes in a one-dimensional strut model

*Authors*

- Sandstede, Björn

*2010 Mathematics Subject Classification*

- 73H10 73H05 35B35 34C37

*Keywords*

- stability, multiple pulses, strut

*DOI*

*Abstract*

Stability of localised equilibria arising in a fourth-order partial differential equation modelling struts is investigated. It was shown in Buffoni, Champneys & Toland (1996) that the model exhibits many multi-modal buckling states bifurcating from a primary buckling mode. In this article, using analytical and numerical techniques, the primary mode is shown to be unstable under dead loading in a large range of parameter values, while is likely to be stable under rigid loading for small axial loads. Furthermore, for general reversible or Hamiltonian systems, stability of the multi-modal solutions is established assuming stability of the primary state. As this hypothesis is not satisfied for the buckling mode arising in the strut model, any multi-modal buckling state will be unstable for both loading devices.

*Appeared in*

- Philos. Trans. Roy. Soc. London Ser. A, 355 (1997), pp. 2083-2097

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# Stochastic interacting particle systems as a numerical tool

*Authors*

- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- Nonlinear Boltzmann equation, stochastic particle method, variance reduction, collision mechanism, partial weight transfer

*DOI*

*Abstract*

Stochastic particle methods for the numerical treatment of the nonlinear Boltzmann equation are considered. An approach to the problem of variance reduction is discussed, and results of some numerical experiments are presented.

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# Forced frequency locking in S1-equivariant differential equations

*Authors*

- Peterhof, Daniela
- Recke, Lutz

*2010 Mathematics Subject Classification*

- 58F35 34A47 34C25 34C27 34D35

*Keywords*

- Forced symmetry breaking, forced frequency locking, bifurcation from solution orbits, rotating waves, modulated waves

*DOI*

*Abstract*

The aim of this paper is to present a simple analytic stategy for predicting, or engineering, two frequency locking phenomena for S^{1}-equivariant ordinary differential equations. First we consider the forced frequency locking of a rotating wave solution of the unforced equation with a forcing of "rotating wave type", and we describe the creation of modulated wave solutions which is connected with this locking phenomenon. And second, we consider the forced frequency locking of a modulated wave solution with a forcing of "modulated wave type". Especially, we describe the sets of all control parameters and of all forcings such that frequency locking occures, the dynamic stability and the asymptotic behavior (for the forcing intensity tending to zero) of the locked solutions and the structural stability of all the phenomena. This paper is essentially founded on results from our previous work [41] concerning abstract forced symmetry breaking. The equations considered in the present paper are finite dimensional prototypes of certain infinite dimensional models describing the behavior of continuous wave operated or self-pulsating multisection DFB lasers under continuous or pulsating light injection, respectively.

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# Convergence criteria for waveform iteration methods applied to partitioned DAE systems in chemical process simulation

*Authors*

- Borchardt, Jürgen
- Michael, Tino

*2010 Mathematics Subject Classification*

- 65L05 80A30 65F50 65Y05 80A30 92E20

*Keywords*

- Algebraic-differential equations, waveform iteration, partitioned systems, parallelization of numerical methods, chemical process simulation

*DOI*

*Abstract*

The application of block waveform iteration methods to initial value problems for implicit DAE systems of index 1 arising in chemical process simulation is considered. These methods permit the concurrent treatment of blocks of subsystems of the entire system gaining a coarse granular parallelism. Their convergence properties strongly depend on the assignment of variables to equations and the partitioning of the system into subsystem blocks.

The convergence of block waveform iteration methods applied to semiexplicit DAE sytems of index 1 is proved. The convergence conditions are given in such a way that only the single blocksystems have to satisfy some corresponding conditions. It is shown that these conditions are fulfilled for a simplified modeling of distillation columns. Resulting from the convergence considerations an assignment and partitioning algorithm is given. A prototype of a waveform-iteration code has been implemented and tested by means of examples included in the user package of the chemical process simulator SPEEDUP.

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# Simulation of Monolithic Microwave Integrated Circuits

*Authors*

- Hebermehl, Georg
- Heinrich, Wolfgang
- Schlundt, Rainer

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

*2010 Mathematics Subject Classification*

- 35Q60 35L20 65N22 65F10 65F15

*Keywords*

- Microwave device, three-dimensional simulation, scattering matrix, Maxwellian equations, finite-volume method, finite-difference method, eigenvalue problem, system of simultaneous linear equations.

*DOI*

*Abstract*

The electric properties of monolithic microwave integrated circuits can be described in terms of their scattering matrix using Maxwellian equations. The corresponding three-dimensional boundary value problem of the Maxwellian equations can be solved by means of a finite-volume scheme in the frequency domain. This results in a two-step procedure: a time and memory consuming eigenvalue problem for nonsymmetric matrices and the solution of a large-scale system of linear equations with indefinite symmetric matrices.

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# Improved Numerical Solutions for the Simulation of Monolithic Microwave Integrated Circuits

*Authors*

- Hebermehl, Georg
- Heinrich, Wolfgang
- Schlundt, Rainer

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

*2010 Mathematics Subject Classification*

- 35Q60 35L20 65N22 65F10 65F15

*Keywords*

- Microwave device, three-dimensional simulation, scattering matrix, orthogonality relation, Maxwellian equations, finite-volume method, finite-difference method, eigenvalue problem, system of simultaneous linear equations

*DOI*

*Abstract*

The electrical properties of the circuits are described in terms of their scattering matrix using Maxwellian equations. Using a finite-volume scheme a three-dimensional boundary value problem for the Maxwellian equations in the frequency domain can be solved. This results in a two-step procedure: a time and memory consuming eigenvalue problem for nonsymmetric matrices and the solution of a large-scale system of linear equations with indefinite symmetric matrices. Improved numerical solutions for these two linear algebraic problems, the computation of the scattering matrix and of the used orthogonality relation are treated in this paper. The numerical effort could be reduced considerably.

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# Longtime behavior of a branching process controlled by branching catalysts

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60J80 60J55 60G57

*Keywords*

- catalytic reaction diffusion equation, super-Brownian motion, superprocess, branching functional, critical branching, measure-valued branching, persistence, super-Brownian medium, random medium, catalyst process, catalytic medium, Brownian collision local time, self-similarity, random ergodic limit

*DOI*

*Abstract*

The model under consideration is a catalytic branching model constructed in [DF96], where the catalysts themselves suffer a spatial branching mechanism. Main attention is paid to dimension d=3. The key result is a convergence theorem towards a limit with full intensity (persistence), which in a sense is comparable with the situation for the "classical" continuous super-Brownian motion. As by-products, strong laws of large numbers are derived for the Brownian collision local time controlling the branching of reactants, and for the catalytic occupation time process. Also, the occupation measures are shown to be absolutely continuous.

*Appeared in*

- Stochastic Process. Appl., 71(2) (1997), pp. 241-257

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# Approximate wavelets and the approximation of pseudodifferential operators

*Authors*

- Maz´ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 41A30 41A63 65D30

*Keywords*

- Cubature formulas, approximate multiresolution, multivariate wavelets

*DOI*

*Abstract*

The paper studies an approximate multiresolution analysis for spaces generated by smooth functions which provide high order cubature formulas for integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy methods of the wavelet theory can be applied. We obtain a decomposition of the finest scale space into almost orthogonal wavelet spaces. For one example we study some properties of the analytic prewavelets, describe the projection operators onto the wavelet spaces and consider some applications to the cubature of integral operators.

*Appeared in*

- Applied and Computational Harmonic Analysis, 6 (1999), No. 3, pp. 287-313.

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# Oscillations and Dynamical Systems: Normalization Procedures and Averaging

*Authors*

- Lopatin, Alexey K.

*DOI*

*Abstract*

The present lecture deals with the development of new normalization procedures and averaging algorithms in problems of nonlinear vibrations. Namely, the development of asymptotic methods of perturbation theory is considered, making wide use of group theoretical techniques. Various assumptions about specific group properties are investigated, and are shown to lead to modifications of existing methods, such as the Bogoliubov averaging method and the Poincaré-Birkhoff normal form, as well as to the formulation of new ones. The development of normalization techniques on Lie groups is also treated.

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# Adaptive and spatially adaptive testing of a nonparametric hypothesis

*Authors*

- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Signal detection, minimax hypothesis testing, nonparametric alternative, error probabilities, adaptive test

*DOI*

*Abstract*

The present paper continues studying the problem of nonparametric hypothesis testing started in Lepski and Spokoiny, 1995 and Spokoiny, 1995. Let a function ƒ be observed with noise. A null simple hypothesis ƒ ≡ ƒ_{0} is tested against a composite alternative of the form ƒ- ƒ_{0} _{r} ≥ ᵨ. Additionally it is assumed that the underlying function ƒ possesses some smoothness properties, namely, that ƒ belongs to some Besov (or Sobolev) ball B_{s,p,q}(M) = {ƒ : ƒ B_{s,p,q} ≤ M}. The aim is to evaluate the fastest rate of decay of the radius g to zero as the noise level tends to zero (or, equivalently, as the number of observations tends to infinity) for which testing with prescribed error probabilities is still possible. The ealier results show that the answer depends heavily on the smoothness parameters s,p, q, M. Below we consider the problem of adaptive (assumption free) testing if these parameters are unknown. A test Φ* is proposed which is near minimax and adaptive at the same time. Compared with the optimal (minimax) rate, this test has a performance which is worse within a log log-factor that is inessential but unavoidable payment for adaptation.

*Appeared in*

- Math. Methods Statist., 7 (1998), No. 3, pp. 245-273

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# Optimal Pointwise Adaptive Methods in Nonparametric Estimation

*Authors*

- Lepski, Oleg V.
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- pointwise adaptive estimation, bandwidth selection, Hölder type constraints

*DOI*

*Abstract*

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered.

Two procedures are proved to be asymptotically optimal for different settings.

First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel.

We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable kernel. The resulting estimator is optimal among all feasible estimators.

The important feature of this procedure is that no prior information is used about smoothness properties of the estimated function i.e. the procedure is completely adaptive and "works" for the class of all functions. With it the attainable accuracy of estimation depends on the function itself and it is expressed in terms of "ideal" bandwidth corresponding to this function.

The second procedure can be considered as a specification of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class Σ(β,L) with unknown parameters β, L.

This assumption allows to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense.

*Appeared in*

- Ann. Statist., 25 (1997), no. 6, pp. 2512-2546

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# Existence and Stability of N-Pulses on Optical Fibers with Phase-Sensitive Amplifiers

*Authors*

- Alexander, James C.
- Jones, Christopher K. R. T.
- Sandstede, Björn

*2010 Mathematics Subject Classification*

- 34C37 35B35 35Q55 78A60

*Keywords*

- homoclinic orbits, bifurcation, stability, optical fibers

*DOI*

*Abstract*

The propagation of pulses in optical communication systems in which attenuation is compensated by phase-sensitive amplifiers is investigated. A central issue is whether optical fibers are capable of carrying several pieces of information at the same time. In this paper, multiple pulses are shown to exist for a fourth-order nonlinear diffusion model due to Kutz and co-workers [10]. Moreover, criteria are derived for determining which of these pulses are stable. The pulses arise in a reversible orbit-flip, a homoclinic bifurcation investigated here for the first time. Numerical simulations are used to study multiple pulses far away from the actual bifurcation point. They confirm that properties of the multiple pulses including their stability are surprisingly well predicted by the analysis carried out near the bifurcation.

*Appeared in*

- Phys. D, 106 (1997), pp. 167-206

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# Energetic Systems and Global Attractors for the 3D Navier-Stokes Equations

*Authors*

- Bondarevsky, Vadim G.

*2010 Mathematics Subject Classification*

- 35Q30 47H20 34C35 58F39 76F20

*Keywords*

- Evolutionary equations, uniqueness and global regularity problem, long-time behaviour

*DOI*

*Abstract*

The purpose of this paper is to address the question of whether or not there exists a linear space of initial data generating global in time solutions of the initial-boundary value problem for 3-dimensional Navier-Stokes equations (3D NSE), and to establish unconditional results about global attractors for 3D NSE by bypassing the problem of uniqueness and global regularity. For this goal, the author introduces the notion of energetic system which is capable of both incorporating 3D NSE and adequate modelling of properties of 3D NSE.

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# The invariance of asymptotic laws of stochastic systems under discretization

*Authors*

- Schurz, Henri

*2010 Mathematics Subject Classification*

- 60H10 65C05 65C20 65L20 65U05

*Keywords*

- Linear stochastic systems, Mean square stability, Stationarity, Stochastic differential equations, Implicit numerical methods, Positive linear operators, Spectral radius, Damped harmonic oscillator

*DOI*

*Abstract*

The stochastic trapezoidal rule provides the only discretization scheme from the family of implicit Euler methods (see [11]) which possesses the same asymptotic (stationary) law as underlying linear continuous time stochastic systems with white or coloured noise. This identity is shown for systems with multiplicative (para- metric) and additive noise using fixed point principles and the theory of positive operators. The key result is useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in Mechanical Engineering.

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# Persistence of a two-dimensional super-Brownian motion in a catalytic medium

*Authors*

- Etheridge, Alison M.
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60J80 60G57 60K35

*Keywords*

- catalytic super-Brownian medium, catalyst, superprocess, measure-valued branching, non-extinction, persistence

*DOI*

*Abstract*

The super-Brownian motion X^{ϱ} in a catalytic medium ϱ constructed in [DF96a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF96a]) and three ([DF96b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d ≤ 3), given the catalyst, the variance of the process is finite 'uniformly in time'. This is in contrast to the 'classical' super-Brownian motion where this holds only in high dimensions (d ≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction.

*Appeared in*

- Probab. Theory Related Fields, 110 (1998), pp. 1-12

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# Preservation of probabilistic laws through Euler methods for Ornstein-Uhlenbeck process

*Authors*

- Schurz, Henri

*2010 Mathematics Subject Classification*

- 60H10 65C20 65L20 65U05

*Keywords*

- Stochastic differential equations, Stationary Ornstein-Uhlenbeckprocess, Numerical methods, Family of implicit Euler methods, Preservation of asymptotic laws, Stability

*DOI*

*Abstract*

There is a lack of appropriate replication of the asymptotical behaviour of stationary stochastic differential equations solved by numerical methods. The paper illustrates this fact with the stationary Ornstein-Uhlenbeck process and family of implicit Euler methods. For description of occuring bias, notions of asymptotical p-th mean, mean, mean square and equilibrium preservation are introduced, due to stochasticity of stationary law. Only the trapezoidal formula among these methods is optimal in the sense of replication of exact asymptotical behaviour. We also discuss the general probabilistic law of linear Euler methods. The results can be useful for implementation of stochastic-numerical algorithms (e.g. for linear-implicit methods) in several disciplines of Natural and Environmental Sciences.

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# Biorthogonal Spline-Wavelets on the Interval - Stability and Moment Conditions

*Authors*

- Dahmen, Wolfgang
- Kunoth, Angela
- Urban, Karsten

*2010 Mathematics Subject Classification*

- 15A12 35Q30 65F35 65N30 41A17 41A63

*Keywords*

- Multiresolution analysis on the interval, biorthogonal wavelets, moment conditions, Riesz bases, discrete Sobolev norms

*DOI*

*Abstract*

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0,1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on ℝ developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal as well as on the dual side have finitely supported masks the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases for L_{2}([0,1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0,1] within a range depending on the parameters in the Jackson and Bernstein estimates.

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# On the identification of soil transmissivity from measurements of the groundwater level

*Authors*

- Bruckner, Gottfried
- Handrock-Meyer, Sybille
- Langmach, Hartmut

*2010 Mathematics Subject Classification*

- 35R30 86A05 76S05 65N30

*Keywords*

- Inverse problems, direct methods, finite elements, linear boundary value problem

*DOI*

*Abstract*

This paper is devoted to the inverse problem of identifying a spatially varying coefficient in a linear elliptic differential equation describing the filtration of groundwater. Practice suggests that the gradient of the piezometric head, i.e., Darcy's velocity, may have discontinuities and the transmissivity coefficient is a piecewise constant function.

For solving this problem we have used a direct method of G. Vainikko. Starting at a weak formulation of the problem a suitable discretization is obtained by the method of minimal error. If necessary this method can be combined with Tikhonov's regularization.

The main difficulty consists in generating distributed state observations from measurements of the ground water level. For this step we propose an optimized data preparation procedure using additional information like knowledge of the sought parameter values at some points and lower and upper bounds for the parameter.

First numerical tests show that locally sufficiently many measurements provide locally satisfactory results.

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# Global behaviour of reaction-diffusion system modelling chemotaxis

*Authors*

- Gajewski, Herbert
- Zacharias, Klaus

*2010 Mathematics Subject Classification*

- 35K45 35K57 35B40 92C15 92D25

*Keywords*

- Initial boundary value problem, reaction-diffusion equations, a priori estimates, Lyapunov function, equilibria, asymptotic behaviour, population dynamics, chemotaxis

*DOI*

*Abstract*

Using Lyapunov functionals the global behaviour of the solutions of a reaction-diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given that this dynamical system tends to a (not necassarity trivial) stationary state.

*Appeared in*

- Math. nachr., 195 (1998), pp. 77-114.

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# Forced symmetry breaking perturbations for periodic solutions

*Authors*

- Guyard, Fréderic
- Lauterbach, Reiner

ORCID: 0000-0002-9310-3177

*2010 Mathematics Subject Classification*

- 34C25 34C37 58F35

*Keywords*

- Forced symmetry breaking, periodic solution, orbit space, heteroclinic cycles

*DOI*

*Abstract*

Using the formalism defined by R. Lauterbach and M. Roberts [21], we develop a geometric approach for the problem of forced symmetry breaking for periodic orbits in G-equivariant systems of ODE's. We show that this problem can be studied as the perturbation of the identity mapping on the double coset space LG/K where K is the maximal subgroup of G acting on the periodic orbit and L the symmetry of the perturbation. We exhibit some example where this kind of symmetry breaking allows to show the existence of heteroclinic cycles between periodic solutions.

*Appeared in*

- Nonlinearity, 10 (1997), pp. 291-310

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# The Low-Temperature Phase of Kac-Ising Models

*Authors*

- Bovier, Anton
- Zahradník, Miloš

*2010 Mathematics Subject Classification*

- 60K35 82B20 82B26

*Keywords*

- Ising models, Kac potentials, low temperature Gibbs states, contours, Peierls argument

*DOI*

*Abstract*

We analyse the low temperature phase of ferromagnetic Kac-Ising models in dimensions d ≥ 2. We show that if the range of interactions is γ^{-1}, then two disjoint translation invariant Gibbs states exist, if the inverse temperature β satisfies β - 1 ≥ γ^{κ}, where κ = d(1-ε) ⁄ (2d+2)(d+1), for any ε > 0. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument.

*Appeared in*

- J. Statist. Phys. 87 (1997) pp. 311-333

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# Efficient mixing of product walks on product groups

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421

*2010 Mathematics Subject Classification*

- 60J15

*Keywords*

- Product random walk, mixing time

*DOI*

*Abstract*

We are going to study the mixing behavior of product-type random walks on product groups. This study is inspired by the investigation of the relaxation of random walks on d-dimensional grids with possibly direction dependent mesh size. Typically such walks are designed to randomly visit a coordinate direction and then to carry out a random step within the chosen component according to some random walk in this direction. We will derive a dependence of the mixing times of such random walks in terms of the component mixing times. If we are free to optimize the random visiting scheme, then we can speed up mixing in case the component mixing times vary much. In more homogeneous situations the overall mixing time is bounded by a multiple of the sum of the single ones times the logarithm of the number of components.

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# Delayed exchange of stabilities in singularly perturbed systems

*Authors*

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

*2010 Mathematics Subject Classification*

- 34D15 34El5

*Keywords*

- Singular perturbation, delayed loss of stability, delayed exchange of stabilities, upper and lower solution

*DOI*

*Abstract*

We consider a scalar nonautonomous singularly perturbed differential equation whose degenerate equation has two solutions which intersect at some point. These solutions represent families of equilibria of the associated equation where at least one of these families loses its stability at the intersection point. We study the behavior of the solution of an initial value problem of the singularly perturbed equation in dependence on the small parameter. We assume that the solution stays at the beginning near a stable branch of equilibria of the associated system where this branch loses its stability at some critical time t_{c}. By means of the method of upper and lower solutions we determine the asymptotic delay t* of the solution for leaving the unstable branch. The obtained result holds for the case of transcritical bifurcation as well as for the case of pitchfork bifurcation. We consider some examples where we prove that a well-known result due to N.R. Lebovitz and R.J. Schaar about an immediate exchange of stabilities cannot be applied to singularly perturbed systems whose right hand side depends on ε.

*Appeared in*

- Z. Angew. Math. Mech., 78 (1998), Suppl. 1, pp. S199-S202

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# Abstract forced symmetry breaking

*Authors*

- Peterhof, Daniela
- Recke, Lutz

*2010 Mathematics Subject Classification*

- 58E07 58E09 58F35

*Keywords*

- Forced symmetry breaking, bifurcation from solution orbits, G-invariant implicit function theorem, locking cones, principle of reduced stability

*DOI*

*Abstract*

We consider abstract forced symmetry breaking problems of the type F(x,λ) = y, x ≈ O(x_{0}), λ ≈ λ_{0}, y ≈ O. It is supposed that for all λ the maps F(·,λ) are equivariant with respect to representations of a given compact Lie group, that F(x_{0}, λ_{0}) = 0 and, hence, that F(x,λ_{0}) = 0 for all elements x of the group orbit O(x_{0}) of x_{0}. We look for solutions x which bifurcate from the solution family O(x_{0}) as λ and y move away from λ_{0} and zero, respectively. Especially, we describe the number of different solutions x (for fixed control parameters λ and y), their dynamic stability, their asymptotic behavior for y tending to zero and the structural stability of all these results. Further, generalizations are given to problems of the type F(x,λ) = y(x,λ), x ≈ O(x_{0}), λ ≈ λ_{0}, y(x,λ) ≈ 0. This work is a generalization of results of J. K. HALE, P. TÁBOAS , A. VANDERBAUWHEDE and E. DANCER to such extend that the conclusions are applicable to forced frequency locking problems for rotating and modulated wave solutions of certain S^{1}-equivariant evolution equations which arise in laser modeling.

*Appeared in*

- J. Differential Equations, 144 (1998) pp. 233--262.

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# Optimal Control of Martensitic Phase Transitions in a Deformation-Driven Experiment on Shape Memory Alloys

*Authors*

- Bubner, Nikolaus
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 49K20 93C20

*Keywords*

- Shape memory alloys, phase transitions, systems of nonlinear PDE's, optimal control, necessary conditions of optimality

*DOI*

*Abstract*

We consider an optimal control problem for first order martensitic phase transitions in a deformation-driven experiment on shape memory alloys. For the resulting system of nonlinear partial differential equations using Falk's Landau-Ginzburg model we show the existence and uniqueness of a classical and a weak solution, respectively. The control problem is stated, and the necessary conditions of optimality are derived.

*Appeared in*

- Adv. Math. Sci. Appl. 8, (1998), pp. 299-325

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# Random walk for elliptic equations and boundary layer

*Authors*

- Milstein, Grigori N.

*2010 Mathematics Subject Classification*

- 35J25 60J15 65N99

*Keywords*

- Boundary value problem, weak methods of numerical integration of SDE, random walk, boundary layer

*DOI*

*Abstract*

We consider the Dirichlet problem for equations of elliptic type in a domain G with a boundary ∂G. A probabilistic representation of solutions to the problem is connected with a system of stochastic differential equations (SDE). Unlike usual approximation of SDE when a time-discretization is exploited, here a space-discretization is recommended. We construct weak approximations for which an estimate of their errors contains derivatives of the required solution to the Dirichlet problem only of lower order. In particular, it is important for problems with a boundary layer. We simulate a Markov chain in G on the basis of a one-step approximation using variable step in the space. The chain should be stopped entering a sufficiently small neighborhood of the boundary ∂G. We estimate the average number of steps before stopping and state some convergence theorems.

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# Optimal choice of observation window for Poisson observations

*Authors*

- Kutoyants, Yury
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62L05 62L12

*Keywords*

- Poisson process, observation window, sequential design, van Trees' inequality, information bound

*DOI*

*Abstract*

We consider the possibility of optimal choice of observation window in the problem of parameter estimation by the observations of an inhomogeneous Poisson process. A minimax lower bound is proposed for the risk of estimation under an arbitrary choice of observation window. Then the adaptive procedure is proposed which is asymptotically efficient in the sense of this bound.

*Appeared in*

- Stat. and Prob. Letters, 44 (1999) 291-298.

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# On the determination of point sources by boundary observations: uniqueness, stability and reconstruction

*Authors*

- Bruckner, Gottfried
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35L20 73D50

*Keywords*

- Determination of point sources, vibration of a string, boundary measurements, uniqueness, stability, reconstruction

*DOI*

*Abstract*

We consider the problem

u''(x,t) = u_{xx}(x,t) + λ(t) Σ^{N}_{k=1}α_{k}δ(x-ξ_{k}), 0 < x < 1,0 < t < T

u(x,0) = u'(x,0) = 0, 0 < x < 1

u(0,t) = u(1,t) = 0, 0 < t < T,

where u'(x,t) = ∂u ⁄ ∂t, (x,t), u'' (x,t) = ∂^{2}u ⁄ ∂t^{2} (x,t), and λ ∈ C^{1}[0,T], α_{k} ≠ 0, ∈ ℝ, ξ_{k} ∈ (0,1), and δ(·-ξ_{k}) is Dirac's delta function at ξ_{k}, 1 ≤ k ≤ n. Our task consists in the determination of N, α_{k}, ξ_{k}, 1 ≤ k ≤ N from the boundary observation ∂u ⁄ ∂x (0,t), 0 < t < T, where λ and T > 0 are given. We prove the uniqueness, give a stability estimate and provide a scheme for reconstructing α_{1}, α_{2}, ξ_{1}, ξ_{2} from ∂u ⁄ ∂x (0,t), 0 < t < T in the case N = 2.

*Appeared in*

- Inverse Problems 16, (2000), pp. 723-748.

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# Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature

*Authors*

- Bubner, Nikolaus
- Sokolowski, Jan
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 49K20 93C20

*Keywords*

- Shape memory alloys, phase transitions, state constraints, optimality conditions

*DOI*

*Abstract*

We consider two optimal control problems for first order martensitic phase transitions in a deformation-driven experiment on shape memory alloys including state constraints for the total stress and the temperature. We control by the elongation of a thin rod and by the outside temperature. The control problems are stated, and the necessary conditions of optimality are derived.

*Appeared in*

- Numer. Funct. Anal. Optim., 19 (1998), no. 5 & 6, pp. 489-498

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# Postrelativity - a paradigm for quantization with preferred Newtonian frame

*Authors*

- Schmelzer, Ilja

*DOI*

*Abstract*

We define a new paradigm - postrelativity - based on the hypothesis of a preferred hidden Newtonian frame in relativistic theories. It leads to a modification of general relativity with ether interpretation, without topological problems, black hole and big bang singularities. Semiclassical theory predicts Hawking radiation with evaporation before horizon formation. In quantum gravity there is no problem of time and topology. Configuration space and quasiclassical predictions are different from canonical quantization of general relativity. Uncertainty of the light cone or an atomic structure of the ether may solve ultraviolet problems. The similar concept for gauge fields leads to real, physical gauge potential without Faddeev-Popov ghost fields and Gribov copy problem.

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# Bootstrap confidence bands for the autoregression function

*Authors*

- Kreiss, Jens-Peter
- Neumann, Michael H.

*2010 Mathematics Subject Classification*

- 62G07 62M05 62G09 62G15

*Keywords*

- Nonparametric autoregression, nonparametric regression, strong approximation, bootstrap, wild bootstrap, confidence bands

*DOI*

*Abstract*

We derive a strong approximation of a local polynomial estimator (LPE) in nonparametric autoregression by an LPE in a corresponding nonparametric regression model. This generally suggests the application of regression-typical tools for statistical inference in nonparametric autoregressive models. It provides an important simplification for the bootstrap method to be used: It is enough to mimic the structure of a nonparametric regression model rather than to imitate the more complicated process structure in the autoregressive case. As an example we consider a simple wild bootstrap. Besides our particular application to simultaneous confidence bands, this suggests the validity of wild bootstrap for several other statistical purposes.

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# Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations

*Authors*

- Neumann, Michael H.

*2010 Mathematics Subject Classification*

- 62G07 62G09 62M07

*Keywords*

- Density estimation, strong approximation, bootstrap, weak dependence, mixing, whitening by windowing, simultaneous confidence bands, nonparametric tests

*DOI*

*Abstract*

We derive a useful approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors. We construct, on a sufficiently rich probability space, such a pairing of the random variables of both experiments that the set of observations {X_{1}, ..., X_{n}} from the time series model is nearly the same as the set of observations {Y_{1}, ... , Y_{n}} from the i.i.d. model. The set ({X_{1}, ... , X_{n}}Δ{Y_{1}, ... , Y_{n}})∩([a_{1}, b_{1}] x ... x [a_{d}, b_{d}]) has with a high probability at most O({[n^{1/2} ∏ (b_{i} - a_{i})] + 1} log(n)) elements. Although this does not imply very much for parametric problems, it has important implications in nonparametric statistics. It yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i.i.d. model. Moreover, we show that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap. Using these results we derive simultaneous confidence bands as well as supremum-type nonparametric tests based on reasoning for the i.i.d. model.

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# A duality-type method for the design of beams

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 49D37 49D05

*Keywords*

- variable domains, optimization, nonconvex duality

*DOI*

*Abstract*

We discuss the nonconvex optimal shape design problem of minimizing the weight of a loaded beam subject to deflection constraints. We associate to it a convex minimization problem which will play the role of a dual. The algorithm we propose has a global character and iterates between the two optimization problems via a so called "resizing rule".

*Appeared in*

- Adv. Math. Sci. Appl. 9 (1999) no. 1 pp. 89-102

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# Asymptotic equivalence for nonparametric generalized linear models

*Authors*

- Grama, Ion G.
- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62B15 62G07 62G20

*Keywords*

- Nonparametric regression, statistical experiment, deficiency distance, global white noise approximation, exponential family, variance stabilizing transform

*DOI*

*Abstract*

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value ƒ(t_{i}) of a regression function ƒ at a grid point t_{i} (nonparametric GLM). When ƒ is in a Hölder ball with exponent β > 1⁄2, we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

*Appeared in*

- Probab. Theory Related Fields, 111 (1998), pp. 167-214

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# Viscous Perturbations of Vorticity Conserving Flows and Separatrix Splitting

*Authors*

- Balasuriya, Sanjeeva
- Jones, Christopher K. R. T.
- Sandstede, Björn

*2010 Mathematics Subject Classification*

- 76B47 76D99 86A05 37J40

*Keywords*

- two-dimensional vortices, splitting of separatrices, perturbation methods, extended Melnikov theory, forcing, Lagrangian transport, Melnikov function, meandering ocean jet

*DOI*

*Abstract*

We examine the effect of the breaking of vorticity conservation by viscous dissipation on transport in the underlying fluid flow. The transport of interest is between regimes of different characteristic motion and is afforded by the splitting of separatrices. A base flow that is vorticity conserving is assumed therefore to have a separatrix that is either a homoclinic or a heteroclinic orbit. The corresponding vorticity dissipating flow, with small time-dependent forcing and viscous parameter ε, maintains an O(ε) closeness to the inviscid flow in a weak sense. An appropriate Melnikov theory that allows for such weak perturbations is then developed. A surprisingly simple expression for the leading order distance between perturbed invariant (stable and unstable) manifolds is derived which depends only on the inviscid flow. Finally, the implications for transport in barotropic jets are discussed.

*Appeared in*

- Nonlinearity, 11 (1998), pp. 47-77

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# On large and moderate large deviations of empirical bootstrap measure

*Authors*

- Ermakov, Mikhail S.

*2010 Mathematics Subject Classification*

- 60F10 62C12 62E25

*Keywords*

- Large deviations, bootstrap method, empirical measure.

*DOI*

*Abstract*

We find the asymptotics for the large and moderate large deviation probabilities of common distribution of the empirical measure and the empirical bootstrap measure (empirical measure obtaining by the bootstrap method). For the most widespread statistical functionals depending on empirical measure we compare their asymptotics of moderate large deviation probabilities with similar asymptotics given by the bootstrap procedure.

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# On the Storage Capacity of the Hopfield Model

*Authors*

- Löwe, Matthias

*2010 Mathematics Subject Classification*

- 60K40 82C32 82B20

*Keywords*

- Hopfield model, neural networks, storage capacity, Markov chains, large deviations

*DOI*

*Abstract*

We give a review on the rigorous results concerning the storage capacity of the Hopfield model. We distinguish between two different concepts of storage both of them guided by the idea that the retrieval dynamics is a Monte-Carlo dynamics (possibly at zero temperature). We recall the results of McEliece et al. [MPRV87] as well as those by Newman [N88] for the storage capacity of the Hopfield model with unbiased i.i.d. patterns and comprehend some recent development concerning the Hopfield model with semantically correlated or biased patterns.

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# The Pinsker bound: a review

*Authors*

- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62G07 62G20 62C20

*Keywords*

- Signal estimation in Gaussian white noise, ellipsoid parameter spaces, asymptotic minimax risk, exact constants in nonparametric smoothing

*DOI*

*Abstract*

We give an account of the Pinsker bound describing the exact asymptotics of the minimax risk in a class of nonparametric smoothing problems. The parameter spaces are Sobolev classes or ellipsoids, and the loss is of squared L_{2}-type. The result from 1980 turned out to be a major step in the theory of nonparametric function estimation.

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# Dimension and invertibility of hyperbolic endomorphisms with singularities

*Authors*

- Schmeling, Jörg

ORCID: 0000-0001-6956-9463 - Troubetzkoy, Serge

*2010 Mathematics Subject Classification*

- 37D99

*Keywords*

- cocycles, transfer map, Anosov system, hyperbolic attractors, existence of SBR measures, Young dimension formula

*DOI*

*Abstract*

We introduce a class of endomorphisms which are piecewise smooth and have hyperbolic attractors. This class generalizes the class of piecewise smooth diffeomorphisms with hyperbolic attractors studied by Pesin [9], Sataev [13], and others [1]. Examples in our class are the fat Belykh map, projections of Solenoids onto cross-sections, and crossed horseshoes. We first develop the stable manifold theory, the existence of SBR measures and the ergodic theory of our class of maps. This theory mostly parallels the invertible case so we only sketch some of the important arguments. We generally follow the outline of [9], details can be found there. Our main results, theorem 5.2 hold in the two dimensional case: if the product of the Lyapunov exponents is less than one the mapping being invertible µ_{SBR}-a.e. on the attractor is equivalent to the Young formula holding. If the mapping is not invertible a.e. we can calculate the defect in the dimension formula 5.4. If the product of the Lyapunov exponents is greater than one then the attractor is two dimensional and the mapping restricted to the attractor is not invertible on a set of positive measure. Finally, we also give an easily checkable sufficient condition for a map to belong to the general class of maps we consider. This condition is easy to check for the systems we were motivated. In particular, in [15] this theorem is applied to fat Belykh maps where the entire picture of everywhere invertibility, invertibility on the attractor, almost everywhere invertibility on the attractor and noninvertibility almost surely is understood. Kaplan and Yorke have conjectured that for a broad class of systems the dimension of the attractor equals the Lyapunov dimension for most maps from the class. The results of [15] imply that for the Belykh family the Kaplan-Yorke conjecture holds for almost all parameter values.

*Appeared in*

- Ergodic Theory Dynam. Systems 18 (1998) no. 5, pp. 1257--1282.

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# On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity

*Authors*

- Barreira, Luis
- Pesin, Yakov
- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*2010 Mathematics Subject Classification*

- 58Fll 58F12 28D99 28C99

*Keywords*

- Conformal repeller, Gibbs measure, local entropy, Lyapunov exponents, multifractal analysis, multifractal rigidity, pointwise dimension

*DOI*

*Abstract*

We introduce the mathematical concept of multifracfality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several non-trivial examples. Among them are subshifts of finite type and one-dimensional Markov maps. An essential part of the paper is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a "physical" classification of dynamical systems. For a class of Markov maps, we show that if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent.

*Appeared in*

- Chaos 7 (1997) no. 1, pp. 27--38.

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# On some problems of hypothesis testing leading to infinitely divisible distributions

*Authors*

- Ingster, Yuri I.

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- Bayesian hypotheses testing, minimax hypotheses testing, asymptotics of error probabilities, infinitely divisible distributions

*DOI*

*Abstract*

We observe an n-dimensional Gaussian random vector x ＝ ξ + v where ξ is a standard n-dimensional Gaussian vector and v ∈ R^{n} is an unknown mean and we consider the hypothesis testing problem H_{0} : v ＝ 0 against two related types of alternatives: Bayesian: the coordinates of v may be equal to -b, 0 or +b only and the number of nonzero coordinates is random with binomial distribution Bi(h_{n},n); Minimax: the coordinates of v may be equal to -b, 0 or +b only and the number, k, of nonzero coordinates is nonrandom. The values b ＝ b_{n} > 0, h ＝ h_{n} ∈ (0, 1] or an integer k ＝ k_{n} ∈ [1,n] are given. These problems are of importance for many applications, for example for multi-channel detection and communication systems. We study the asymptotics of the log-likelihood distribution for Bayesian alternatives and show that they are either Gaussian or degenerate or belong to a special two-parametric class of infinitely divisible distributions. The latter corresponds to the case b_{n} ≍ √log n and h_{n} is small enough. We also show that randomization in the Bayesian alternative corresponds to asymptotically least favorable priors for minimax alrernative if nh_{n} ＝ k_{n} → ∞.

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# An almost sure central limit theorem for the Hopfield model

*Authors*

- Bovier, Anton
- Gayrard, Véronique

*2010 Mathematics Subject Classification*

- 60F05 60K35

*Keywords*

- Hopfield model, neural networks, central limit theorem, Brascamp-Lieb inequalities

*DOI*

*Abstract*

We prove a central limit theorem for the finite dimensional marginals of the Gibbs distribution of the macroscopic 'overlap'-parameters in the Hopfield model in the case where the number of random 'patterns', M, as a function of the system size N satisfies lim_{N↑∞}M(N)/N = 0, without any assumptions on the speed of convergence. The covariance matrix of the limiting gaussian distributions is diagonal and independent of the disorder for almost all realizations of the patterns.

*Appeared in*

- Markov Proc. Related Fields, 3 (1997), pp. 151-173

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# Dimension of hyperbolic measures - A proof of the Eckmann-Ruelle conjecture

*Authors*

- Barreira, Luis
- Pesin, Yakov
- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*2010 Mathematics Subject Classification*

- 58F11 28D05

*Keywords*

- Eckmann-Ruelle conjecture, hyperbolic measures, pointwise dimension

*DOI*

*Abstract*

We prove the long-standing Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems. Namely, we show that the pointwise dimension exists almost everywhere with respect to a Borel probability measure with non-zero Lyapunov exponents invariant under a C^{1+α} diffeomorphism of a smooth Riemannian manifold. This implies in particular that the Hausdorff dimension and box dimension of the measure as well as some other characteristics of dimension type of the measure coincide.

*Appeared in*

- ERA-AMS 2 (1996), No. 1

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# Any set of irregular points has full Hausdorff dimension and full topological entropy

*Authors*

- Barreira, Luis
- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*2010 Mathematics Subject Classification*

- 58F15 58F11

*Keywords*

- Birkhoff averages, irregular points, local entropies, Lyapunov exponents, pointwise dimensions

*DOI*

*Abstract*

We prove, for subshifts of finite type, conformal repellers, and two-dimensional horseshoes, that the set of points where both the pointwise dimension, local entropy, Lyapunov exponents, and Birkhoff averages do not exist carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute the first mathematical application of this theory.

*Appeared in*

- Israel J. Math. 116 (2000), pp. 29--70.

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# Metastates in disordered mean field models: Random field and Hopfield models

*Authors*

- Külske, Christof

*2010 Mathematics Subject Classification*

- 60K40 82B44 60G57

*Keywords*

- Disordered Systems, Size Dependence, Random Gibbs States, Metastates, Mean Field Models, Hopfield Model, Random Field Model

*DOI*

*Abstract*

We rigorously investigate the size dependence of disordered mean field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate 1/N ∑^{N}_{n=1} δμ_{n}(η) where μ_{n}(η) is the Gibbs measure in the finite volume {1,...,n} and the frozen disorder variable η is fixed. We treat explicitely the Hopfield model with finitely many patterns and the Curie Weiss Random Field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for large N. Moreover it does not converge for a.e. η but rather in distribution for whose limits we give explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.

*Appeared in*

- J. Statist. Phys., 88 (1997), pp. 1257-1293

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# Stationary particle systems approximating stationary solutions to the Boltzmann equation

*Authors*

- Caprino, Silvia
- Pulvirenti, Mario
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K35 76P05 82C40

*Keywords*

- Stationary Boltzmann equation, diffusive boundary condition, stochastic particle system, rate of convergence

*DOI*

*Abstract*

We show that a regularized stationary Boltzmann equation with diffusive boundary conditions can be rigorously derived from a suitable stochastic N-particle system.

*Appeared in*

- SIAM J. Math. Anal., 29 (1998), No. 4, pp. 913-934

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# Distribution of Overlap Profiles in the One-Dimensional Kac-Hopfield Model

*Authors*

- Bovier, Anton
- Gayrard, Véronique
- Picco, Pierre

*2010 Mathematics Subject Classification*

- 82B44 82C32 60K35

*Keywords*

- Hopfield model, Kac-potentials large deviations, mesoscopic scales

*DOI*

*Abstract*

We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature.In the thermodynamic limit we obtain large deviation estimates for the distribution of the "local" overlaps, the range of the interaction, γ^{-1}, being the large parameter. We show in particular that the local overlaps in a typical Gibbs configuration are constant and equal to one of the mean-field equilibrium values on a scale o(γ^{-2}). We also give estimates on the size of typical "jumps". i.e. the regions where transitions from one equilibrium value to another take place. Contrary to the situation in the ferromagnetic Kac-model, the structure of the profiles is found to be governed by the quenched disorder rather than by entropy.

*Appeared in*

- Comm. Math. Phys. 186 (1997) pp. 323-379

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# On the completeness of multifractal spectra

*Authors*

- Schmeling, Jörg

ORCID: 0000-0001-6956-9463

*2010 Mathematics Subject Classification*

- 37C45 37D35 28A80 37B25

*Keywords*

- multifractal analysis, Gibbs measures, depending conformal prellers, Lyapunov exponents, thermodynamic formalism, symbolic dynamics

*DOI*

*Abstract*

We study the behavior of multifractal spectra on the boundary of their domains of definition. In particular, we show that the dimension of the set of points having minimal (or maximal) pointwise dimension is not necessarily zero. However, this situation may be neglected in the sense of Baire category for Gibbs measures on expanding conformal repellers.

*Appeared in*

- Ergodic Theory Dynam. Systems 19 (1999( no. 6, pp. 1595--1616.

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# Reduced subcritical Galton-Watson processes in a random environment

*Authors*

- Fleischmann, Klaus
- Vatutin, Vladimir A.

*2010 Mathematics Subject Classification*

- 60J80 60J15

*Keywords*

- Branching process in a random environment, random walk in a random environment, reduced process, reduced tree, conditional limit theorem, age of the most recent common ancestor, source time, hybrid behavior

*DOI*

*Abstract*

We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (in time randomly varying) offspring generating functions are fractional linear. We show that this structure may differ significantly from that for the "classical" reduced subcritical Galton-Watson processes. In particular, it may look like a complex "hybrid" of classical reduced super- and subcritical processes. Some relations with random walks in a random environment are discussed.

*Appeared in*

- Adv. Appl. Probab. 31(1) (1999), pp. 88-111

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