Existence of periodic travelling waves to reaction-diffusion equations with excitable-oscillatory kinetics
Authors
- Haaf, Hermann
DOI
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WIAS Preprint No. 1297, (1994)On some estimations of Weyl sums
Authors
- Pustyl´nikov, Lev D.
- Schmeling, Jörg
ORCID: 0000-0001-6956-9463
DOI
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WIAS Preprint No. 1297, (1994)Waveform iteration and one-sided Lipschitz conditions
Authors
- Bremer, Ingo
2010 Mathematics Subject Classification
- 65L05
Keywords
- Initial value problem, ordinary differential equation, waveform-iteration, one-sided Lipschitz condition
DOI
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WIAS Preprint No. 1297, (1994)Secondary Euler characteristics of locally symmetric spaces. Results and Conjectures
Authors
- Juhl, Andreas
ORCID: 0000-0002-0097-9760
2010 Mathematics Subject Classification
- 22E40 22E46 58F17 58F18 58F20
DOI
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WIAS Preprint No. 1297, (1994)Estimates of Weyl sums over subsequences of natural numbers
Authors
- Schmeling, Jörg
ORCID: 0000-0001-6956-9463
2010 Mathematics Subject Classification
- 11L15
Keywords
- estimates, pseudo-ergodicity, skew rotations, Weyl sum
DOI
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WIAS Preprint No. 1297, (1994)On the approximation of singular integral equations by equations with smooth kernels
Authors
- Duduchava, Roland
ORCID: 0000-0002-2067-7655 - Prößdorf, Siegfried
2010 Mathematics Subject Classification
- 45E05 47G10 65N30 65N35 65R20
Keywords
- Singular integral equations, approximate solutions, regularization, approximation by smooth kernels
DOI
Appeared in
- Integral Equations Operator Theory, 21 (1995), pp. 224--237
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WIAS Preprint No. 1297, (1994)Bifurcation analysis for spherically symmetric systems using invariant theory
Authors
- Lauterbach, Reiner
ORCID: 0000-0002-9310-3177 - Sanders, Jan A.
2010 Mathematics Subject Classification
- 58E09 34C23 58F14
Keywords
- Bifurcation, equivariance, heteroclinic cycle
DOI
Appeared in
- J. Dynamics Differential Equations, 9 (1997), pp. 535-560
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WIAS Preprint No. 1297, (1994)A quadrature method for the hypersingular integral equation on an interval
Authors
- Bühring, Kathrin
2010 Mathematics Subject Classification
- 65R20 45E05
Keywords
- quadrature methods, hypersingular integral equation, stability, error estimates, cosine transformation
DOI
Appeared in
- J. Integral Equations Appl., 7 (1995), pp. 263--301
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WIAS Preprint No. 1297, (1994)Controllability near Takens-Bogdanov points
Authors
- Häckl, Gerhard
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 93B05 93C73 58F14
Keywords
- Controllability, nonlinear time-invariant systems, bifurcation
DOI
Appeared in
- J. of Dyn. and Control Systems, 2 (1996), pp. 583-598
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WIAS Preprint No. 1297, (1994)Recurrence of ancestral lines and offspring trees in time stationary branching populations
Authors
- Matthes, Klaus
- Siegmund-Schultze, Rainer
- Wakolbinger, Anton
2010 Mathematics Subject Classification
- 60J80 60G55 92D25 82B99
Keywords
- Markov branching populations, genealogy of branching processes, recurrence of ancestral lines
DOI
Appeared in
- Math. Nachr. 185 (1997), pp. 163--211.
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WIAS Preprint No. 1297, (1994)Reaction-diffusion processes of electrically charged species
Authors
- Gajewski, Herbert
- Gröger, Konrad
2010 Mathematics Subject Classification
- 35K45 35K57 35B40 78A35
Keywords
- Initial boundary value problem, drift-diffusion processes, a priori estimates, Lyapunov function, equilibria, asymptotic behaviour
DOI
Appeared in
- Math. Nachr., 177 (1996), pp. 109-130.
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WIAS Preprint No. 1297, (1994)Nonlinear Galerkin methods for evolution equations with Lipschitz continuous strongly monotone operators
Authors
- Albinus, Günter
2010 Mathematics Subject Classification
- 47H05 65M55 47H20 65M60
Keywords
- nonlinear Galerkin methods, hierarchical bases, incremental unknowns
DOI
Appeared in
- Proceedings of the 6th International Colloquium on Differential Equations (D. Bainov, ed.), vol. 1, Sofia, pp. 1-14.
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WIAS Preprint No. 1297, (1994)Stability of bifurcating periodic solutions of differential inequalities in ℝ3
Authors
- Kučera, Milan
2010 Mathematics Subject Classification
- 34A40 58F14 34C25 58F10
Keywords
- Ordinary differential inequality, bifurcation of periodic solutions, stability, attractivity
DOI
Abstract
A bifurcation problem for the inequality U (t) ∈ K ⋀ (U̇ (t) - AλU(t) - G(λ,U(t)), V - U(t)) ≥ 0 for all V ∈ K, a. a. t ∈[0,T) is considered, where K is a closed convex cone in ℝ3 , Aλ a real 3 x 3 matrix, λ a real parameter, G a small perturbation. We investigate small periodic solutions bifurcating at λ0 from the branch of trivial solutions and corresponding to parameters λ for which the trivial solution is unstable. It is proved that these solutions are stable or they are contained in a certain attracting set Aλ if zero is stable as the solution of our inequality with λ = λ0.
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WIAS Preprint No. 1297, (1994)A stochastic weighted particle method for the Boltzmann equation
Authors
- Rjasanow, Sergej
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 65R20 45K05 82C40
Keywords
- particle collisions, stochastic weighted particle method, algorithms, Boltzmann equation, Monte Carlo method
DOI
Abstract
A class of algorithms for the numerical treatment of the Boltzmann equation is introduced. The basic idea is a more general procedure of modelling collisions between particles. This procedure is based on a random weight transfer from the particles with the pre-collision velocities to the particles with the post-collision velocities.
Appeared in
- J. Comput. Phys., 124 (1996), pp. 243-253
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WIAS Preprint No. 1297, (1994)Time-space analysis of the cluster-formation in interacting diffusions
Authors
- Fleischmann, Klaus
- Greven, Andreas
2010 Mathematics Subject Classification
- 60K35 60J60 60J15
Keywords
- interacting diffusion, clustering, infinite particle system, delayed coalescing random walk with immigration, transformed Fisher-Wright tree, low dimensional systems, ensemble of log-coalescents
DOI
Abstract
A countable system of linearly interacting diffusions on the interval [0, 1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coeffcient and occur for a large class of initial configurations (universality).
Appeared in
- Electronic Journal of Probability, 1 (1996), pp. 1-46
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WIAS Preprint No. 1297, (1994)A method of constructing of dynamical systems with bounded nonperiodic trajectories
Authors
- Leonov, Gennadii A.
DOI
Abstract
A fifth-order system is considered for which the existence of a set of bounded trajectories that are neither periodic nor almost periodic is proven by means of analytical methods. The set is situated in the region of dissipation and has a positive Lebesgue measure.
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WIAS Preprint No. 1297, (1994)Comparison of interacting diffusions and an application to their ergodic theory
Authors
- Cox, J. Theodore
- Fleischmann, Klaus
- Greven, Andreas
2010 Mathematics Subject Classification
- 60K35 60J60 60J15
Keywords
- comparison arguments, interacting diffusion, interacting particle system, clustering
DOI
Abstract
A general comparison argument for expectations of certain multi-time functionals of infnite systems of linearly interacting diffusions differing in the diffusion coeffcient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.
Appeared in
- Probab. Theory Relat. Fields, 105 (1996), pp. 513-528
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WIAS Preprint No. 1297, (1994)A functional law of large numbers for Boltzmann type stochastic particle systems
Authors
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K35 76P05 82C40
Keywords
- Interacting particle systems, empirical measures, law of large numbers, Boltzmann type equation
DOI
Abstract
A large system of particles is studied. Its time evolution is determined as the superposition of two components. The first component is the independent motion of each particle. The second component is the random interaction mechanism between pairs of particles. The intensity of the interaction depends on the state of the system and is assumed to be bounded. Convergence of the empirical measures is proved as the number of particles tends to infinity. The limiting deterministic measure-valued function is characterized as the unique solution of a nonlinear equation of the Boltzmann type.
Appeared in
- Stochastic Anal. Appl., 14 (1996), pp. 591--636
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WIAS Preprint No. 1297, (1994)A new approach to the single point catalytic super-Brownian motion
Authors
- Fleischmann, Klaus
- Le Gall, Jean-Francois
2010 Mathematics Subject Classification
- 60J80 60J55 60G57
Keywords
- Super-stable subordinator, sample path smoothness, Campbell measure formula, canonical measures, backward measurability, point-catalytic medium, critical branching, super-Brownian motion, superprocess, measure-valued branching process, total extinction
DOI
Abstract
A new approach is provided to the (critical continuous) super-Brownian motion X in R with a single point-catalyst δc as branching rate. We start from a superprocess U with constant branching rate and spatial motion given by the stable subordinator with index 1/2. We prove that the total occupation time measure ∫0∞ ds Us of U is distributed as the occupation density measure λc of X at the catalyst c. This result is a superprocess analogue of the classical fact that the set of zeros of a linear Brownian motion is the range of a stable subordinator with index 1/2. We then show that the value Xt of the process X at time t is determined from the measure λc by an explicit representation formula. On a heuristic level, this formula says that a mass λc(ds) of ''particles'' leaves the catalyst at time s and then evolves according to the Itô measure of Brownian excursions. This representation formula has important applications. First of all, with probability one, the density field x of X satisfies the heat equation outside of c with the noisy boundary condition at c given by the singularly continuous random measure λc. In particular, x is C∞ outside the catalyst. This property is in sharp contrast to the constant branching rate case. Another consequence is that the total mass Xt(R) is always strictly positive but dies out in probability as t → ∞. As a final application a new derivation of the singularity of the measure λc is provided.
Appeared in
- Probab. Theory Relat. Fields 102, 63-82 (1995)
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WIAS Preprint No. 1297, (1994)Pendulum with positive and negative dry friction. Continuum of homoclinic orbits
Authors
- Leonov, Gennadii A.
DOI
Abstract
A two-order differential equation of pendulum with dry friction is considered. The existence of a continuum of homoclinic orbits with various homotopic properties on the cylinder is proven.
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WIAS Preprint No. 1297, (1994)Zum Einfluß der Wärmeleitung und der Ladungsträgerdiffusion auf das Verhalten eines Halbleiterlasers
Authors
- Förste, Joachim
2010 Mathematics Subject Classification
- 35Q60 78A60
Keywords
- Diffusion, Stabilität, Wärmeleitung, Laser
DOI
Abstract
Bekanntlich wird eine katastrophenartige Schädigung von GaxAl1-xAs Doppel-heterostrukturlasern durch Überhitzung herbeigeführt, deren Ursache eine hohe nichtradiative Rekombinationsrate an der Spiegelfacette ist. Auf der Grundlage zweidimensionaler Bilanzgleichungen wird der Einfluß der lateralen Diffusion diskutiert.
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WIAS Preprint No. 1297, (1994)Asymptotische Input-Output-Linearisierung und Störgrößenkompensation in nichtlinearen Reaktionssystemen
Authors
- Müller, Wolfdietrich
2010 Mathematics Subject Classification
- 93D15 93A13 93A25 93B07 93B11 93B18 93B29 93B30 93C10 93C15 93C73 93C83 93C95 92E20
DOI
Abstract
Entsprechend einem von E.D. Gilles und Mitarbeitern am Institut für Systemdynamik und Regelungstechnik der Universität Stuttgart vorgeschlagenen Konzept wird eine reproduzierbare Qualität der Endprodukte in komplexen verfahrenstechnischen Produktionsprozessen mit Hilfe einer Prozeßführung gewährleistet, die in einer hierarchischen Struktur nach zentral vorgegebenen Kriterien durch lokale Feedback-Steuerungen in den Teilprozessen eine Linearisierung des Input-Output-Verhaltens der Teilprozesse erzwingt. Da in der Regel nicht alle Zustandsgrößen einer Messung zugänglich sind, kann diese Strategie nur asymptotisch realisiert werden, und zwar mit Hilfe eines Beobachters, der Schätzungen für die nicht meßbaren Zustandsgrößen berechnet. Neue, von H.W. Knobloch und Mitarbeitern an der Universität Würzburg gewonnene Ergebnisse zur Theorie nichtlinearer Beobachter gestatten nun einerseits größere Freiheiten bei der Konstruktion des Beobachters und bieten andererseits zusätzlich die Möglichkeit, gewisse Klassen von Störtermen in den Prozeßgleichungen zu identifizieren und damit ihrem Einfluß entgegenzuwirken. An Hand eines konkreten Modellproblems aus der chemischen Verfahrenstechnik, einer exothermen Folgereaktion A → B → C in einem kontinuierlich durchflossenen Rührkesselreaktor, werden die verschiedenen Möglichkeiten dieses Zugangs im Detail diskutiert sowie Strategien zur effektiven Konstruktion der nichtlinearen Beobachter vorgeschlagen.
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WIAS Preprint No. 1297, (1994)On the Existence of Unbiased Monte Carlo Estimators
Authors
- Mathé, Peter
ORCID: 0000-0002-1208-1421
2010 Mathematics Subject Classification
- 65C05 47B10
Keywords
- unbiased Monte Carlo estimators, operator ideals
DOI
Abstract
For many typical instances where Monte Carlo methods are applied attempts were made to find unbiased estimators, since for them the Monte Carlo error reduces to the statistical error. These problems usually take values in the scalar field. If we study vector valued Monte Carlo methods, then we are confronted with the question whether there can exist unbiased estimators. This problem is apparently new. Below it is settled precisely. Partial answers are given, indicating relations to several classes of linear operators in Banach spaces.
Appeared in
- J. Approx. Theory 85 (1996) pp. 1--15
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WIAS Preprint No. 1297, (2008)Holomorphic transforms with application to affine processes
Authors
- Belomestny, Denis
- Kampen, Jörg
- Schoenmakers, John G. M.
ORCID: 0000-0002-4389-8266
2010 Mathematics Subject Classification
- 60J25 91B28
Keywords
- Itô-Lévy processes, holomorphic transforms, affine processes
DOI
Abstract
In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.
Appeared in
- Journal of Functional Analysis, Vol. 257, 4, (2009) pp. 1222-1250
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WIAS Preprint No. 1297, (1994)Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series
Authors
- Neumann, Michael H.
2010 Mathematics Subject Classification
- 62M15 62M10 62G07
Keywords
- Spectral density estimation, wavelet estimators, asymptotic normality, large deviations
DOI
Abstract
In the present paper we consider nonlinear wavelet estimators of the spectral density ƒ of a zero mean stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of estimators of coefficients that arise from a Fourier series expansion according to a wavelet basis. The main goal of this paper is to prepare the ground for the application of these methods to spectral density estimation, which is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. For that we derive upper estimates for their cumulants, which yield the asymptotic normality in terms of probabilities of large deviations. Using these results we can conclude the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. Hence, we obtain estimators of ƒ, which keep all interesting properties like high spatial adaptivity that are already known from wavelet estimators in the case of Gaussian regression. It turns out that optimally tuned versions of these estimators attain the optimal uniform rate of convergence of their L2-risk in a wide variety of Besov smoothness classes.
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WIAS Preprint No. 1297, (1994)Bootstrap confidence bands in nonparametric regression
Authors
- Neumann, Michael H.
2010 Mathematics Subject Classification
- 62G07 62G09 62G15
Keywords
- Nonparametric regression, confidence bands, bootstrap, local linear estimator
DOI
Abstract
In the present paper we construct asymptotic confidence bands in nonparametric regression. Our assumptions admit unequal variances of the observations and nonuniform, possibly considerably clustered design. The confidence band is based on an undersmoothed local linear estimator, and an appropriate quantile is obtained via the wild bootstrap made popular by Härdle and Mammen (1990). We derive certain rates (in the sample size n) for the error in coverage probability, which is an improvement of existing results for methods that rely on the asymptotic distribution of the maximum of some Gaussian process. We propose a practicable rule for a data-dependent choice of the bandwidth.
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WIAS Preprint No. 1297, (1994)On the stabilization of trigonometric collocation methods for a class of ill-posed first kind equations
Authors
- Bruckner, Gottfried
2010 Mathematics Subject Classification
- 65R30
Keywords
- Regularization-discretization procedures, moderately ill-posed, boundary integral equations, convergence rates
DOI
Abstract
In this paper regularization-discretization procedures are developed for the numerical solution of moderately ill-posed linear first kind equations appearing as boundary integral equations for Dirichlet boundary value problems, e.g. the Dirichlet-Laplace problem. The method consists in firstly regularizing the noisy right-hand side by trigonometric interpolation and then applying a trigonometric collocation procedure to the regularized data. Convergence rates are obtained in Sobolev spaces, Hölder-Zygmund spaces or Hölder spaces according to the error analysis of the used procedures for exact data. The method can be generalized to other kinds of equations and approximation procedures.
Appeared in
- J Inverse Ill-posed Problems 5 (1997), pp. 117-127.
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WIAS Preprint No. 1297, (1994)A wavelet algorithm for the solution of the double layer potential equation over polygonal boundaries
Authors
- Rathsfeld, Andreas
ORCID: 0000-0002-2029-5761
2010 Mathematics Subject Classification
- 45L10 65R20
Keywords
- potential equation, collocation, wavelet algorithm
DOI
Abstract
In this paper we consider a piecewise linear collocation method for the solution of the double layer potential equation corresponding to Laplace's equation over polygonal domains. We give a wavelet algorithm for the computation of the corresponding stiffness matrix and for the solution of the arising matrix equation with no more than O(N·[log N]8) arithmetic operations. The error of the resulting approximate solution is of order O(N-2·[log N]6).Finally, we give some remarks on the generalization of the algorithm to the piecewise cubic collocation and present numerical tests.
Appeared in
- J. Integral Equations Appl. 7 (1995) pp. 47--98
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WIAS Preprint No. 1297, (1994)Error estimates and extrapolation for the numerical solution of Mellin convolution equations
Authors
- Rathsfeld, Andreas
ORCID: 0000-0002-2029-5761
2010 Mathematics Subject Classification
- 45L10 65R20
Keywords
- Mellin convolution, potential equation, quadrature method, extrapolation
DOI
Abstract
In this paper we consider a quadrature method for the numerical solution of a second kind integral equation over the interval, where the integral operator is a compact perturbation of a Mellin convolution operator. This quadrature method relies upon singularity subtraction and transformation technique. Stability and convergence order of the approximate solution are well known. We shall derive the first term in the asymptotics of the error which shows that, in the interior of the interval, the approximate solution converges with higher order than over the whole interval. This implies higher orders of convergence for the numerical calculation of smooth functionals to the exact solution. Moreover, the asymptotics allows us to define a new approximate solution extrapolated from the dilated solutions of the quadrature method over meshes with different mesh sizes. This extrapolated solution is designed to improve the low convergence order caused by the non-smoothness of the exact solution even when the transformation technique corresponds to slightly graded meshes. Finally, we discuss the application to the double layer integral equation over the boundary of polygonal domains and report numerical results.
Appeared in
- IMA J. Numer. Anal. 16 (1996) pp. 217--255
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WIAS Preprint No. 1297, (1994)On a Penrose-Fife Model with Zero Interfacial Energy Leading to a Phase-field System of Relaxed Stefan Type
Authors
- Colli, Pierluigi
ORCID: 0000-0002-7921-5041 - Sprekels, Jürgen
ORCID: 0009-0000-0618-8604
2010 Mathematics Subject Classification
- 35R35 35K50 80A22
Keywords
- Stefan problems, phase transitions, phase-fieldmodels, singular parabolic systems.
DOI
Abstract
In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arises naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with non-conserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.
Appeared in
- Ann. Math. Pura Appl. 169 (1995) pp. 269-285
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WIAS Preprint No. 1297, (1994)Boundary integral equations for the biharmonic Dirichlet problem on nonsmooth domains
Authors
- Khoromskij, Boris N.
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 31A30 47G10 65N38
Keywords
- biharmonic equation, nonsmooth curve, boundary integral operators, boundary integral equations
DOI
Abstract
In this paper we study boundary integral formulations of the interior and exterior Dirichlet problem for the bi-Laplacian in a plane domain with a piecewise smooth boundary having corner points. The mapping properties of single and double layer biharmonic potentials, of the Calderon projections and the Poincaré-Steklov operators for such domains are analysed. We derive direct boundary integral equations equivalent to the variational formulation of the problem.
Appeared in
- J. Integral Equations Appl., 11 (1999), No. 2, pp. 217-253.
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WIAS Preprint No. 1297, (1994)Global solutions to a Penrose-Fife phase-field model under flux boundary conditions for the inverse temperature
Authors
- Horn, Werner
- Laurençot, Philippe
- Sprekels, Jürgen
ORCID: 0009-0000-0618-8604
2010 Mathematics Subject Classification
- 35K50
Keywords
- Phase transitions, phase-field models, nonlinear parabolic systems
DOI
Abstract
In this paper, we study an initial-boundary value problem for a system of phase-field equations arising from the Penrose-Fife approach to model the kinetics of phase transitions. In contrast to other recent works in this field, the correct form of the boundary condition for the temperature field is assumed which leads to additional difficulties in the mathematical treatment. It is demonstrated that global existence and, in the case of only one or two space dimensions, also uniqueness of strong solutions can be shown under essentially the same assumptions on the data as in the previous papers where a simplified boundary condition for the heat exchange with the surrounding medium has been used.
Appeared in
- Math. Meth. Appl. Sci. 19 (1996) pp. 1053-1072
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WIAS Preprint No. 1297, (1994)Feedback stabilization of nonlinear discrete-time systems
Authors
- Müller, Wolfdietrich
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 93D15 93C55 34H05
Keywords
- Discrete-time control system, Smooth feedback stabilization, Center manifold
DOI
Abstract
It is the merit of D. Aeyels [4] to have shown a way in which center manifold theory can be used in a constructive manner to find a smooth feedback control for stabilizing an equilibrium of a continuous-time system described by a nonlinear ordinary differential eqution ẋ = ƒ(x,u). In this paper we are going to extend Aeyels' approach to nonlinear discrete-time systems described by equations of the type
x(k + 1)=ƒ(x(k),u(k)), k = 0, 1, 2, ... , where we assume that ƒ is sufficiently smooth and satisfies ƒ(0,0) = 0. In critical cases, i.e. in situations where the linearization of the system in the neighborhood of the equilibrium includes non-controllable modes, under some non-resonance conditions we derive sufficient conditions for the existence of a smooth nonlinear stabilizing feedback.
Appeared in
- J. of Difference Equations and Applications, 4 (1998), pp. 579-596
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WIAS Preprint No. 1297, (1994)On moderate deviations for martingales
Authors
- Grama, Ion G.
2010 Mathematics Subject Classification
- 60F10 60G44 62E17
Keywords
- Martingale, central limit theorem, rate of convergence, moderate deviation
DOI
Abstract
Let Xn = (Xnt,Fnt)0 ≤ t ≤ 1 be the square integrable martingales with the quadratic characteristics ⟨Xn⟩, n = 1, 2, .... We have proved that the large deviation relation P(Xn1 ≥ r)/(1 - Φ(r)) → 1 is valid with r growing to infinity at some rate depending on Ln2δ = E ∑0 ≤ t ≤1 |Δ Xnt |2+2δ and Nn2δ = E|⟨Xn⟩1 -1|1+δ, where δ > 0 and Ln2δ → 0, Nn2δ → 0 as n → ∞. The exact bound for the remainder is obtained too.
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WIAS Preprint No. 1297, (1994)Mean-square approximation for stochastic differential equations with small noises
Authors
- Milstein, Grigori N.
- Tret´yakov, Michael V.
2010 Mathematics Subject Classification
- 60H10
Keywords
- Stochastic differential equations, small noises, numerical methods
DOI
Abstract
New approach to construction of mean-square numerical methods for solution of stochastic differential equations with small noises is proposed. The approach is based on expanding of the exact solution of the system with small noises by powers of time increment and regrouping of expansion terms according to powers of time increment and small parameter. The theorem on mean-square estimate of method errors is proved. Various efficient numerical schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The proposed methods are tested by calculation of Lyapunov exponents and simulation of a laser Langevin equation with multiplicative noises.
Appeared in
- SIAM J. on Scientific Computing, vol. 18 (1997), no. 4, pp.1067-1087, under new title: mean-square numerical methods for stochastic differential equations.
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WIAS Preprint No. 1297, (1994)Weak approximation for stochastic differential equations with small noises
Authors
- Milstein, Grigori N.
- Tret´yakov, Michael V.
2010 Mathematics Subject Classification
- 60H10
Keywords
- Stochastic differential equations, small noises, numerical methods, Monte-Carlo technique
DOI
Abstract
New approach to construction of weak numerical methods, which are intended for Monte-Carlo technique, is proposed for a stochastic system with small noises. The theorem on estimate of method error in terms of product hiε j (h is a time increment, ε is a small parameter) is proved. Various efficient weak schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The Talay-Tubaro expansion of the global error is considered for such systems. The efficient approach to reduction of the Monte-Carlo error is proposed. The derived methods are tested by calculation of Lyapunov exponents and by simulation of a bistable dynamical system for which multiplicative stochastic resonance is observed.
Appeared in
- SIAM J. on Numerical Analysis, vol. 34 (1997), no. 6, pp. 2142-2167, under new title: Numerical methods in the weak sense for stochastic differential equations with small noise.
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WIAS Preprint No. 1297, (1994)Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise
Authors
- Schurz, Henri
2010 Mathematics Subject Classification
- 60H10 65C20 65L20 65U05
Keywords
- Stochastic differential equations, numerical methods, implicit Euler, Mil'shtein and Balanced methods, equilibrium solution, asymptotical mean square stability
DOI
Abstract
Several results concerning asymptotical mean square stability of an equilibrium point (here the null solution) of specific linear stochastic systems given at discrete time-points are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monograph of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Mil 'shtein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be verified for autonomous systems, while the principle of 'monotonic nesting' of the mean square stability domains holds for linear systems. The Euler method taking any integration step size with drift-implicitness 0.5 is able to indicate mean square stability of any equilibrium point of the continuous time system. As a practicable alternative for controlling the temporal mean square evolution, the class of Balanced methods with deterministic, positive scalar correction provides the most mean square stable numerical solution known under 'low smoothness conditions' so far. The paper summarizes and continues the stability examinations of Schurz (1993). The results can also be used to deduce recommendations for the practical implementation of numerical methods solving nonlinear systems in terms of their linearization. Finally, effects of the presented mean square calculus are shown by the Kubo oscillator perturbed by white noise and a simplified system of noisy Brusselator equations.
Appeared in
- J. Stoch. Anal. Appl., 14 (1996), pp. 313--354
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WIAS Preprint No. 1297, (1994)Jumping behavior in singularly perturbed systems modelling bimolecular reactions
Authors
- Nefedov, Nikolai N.
- Schneider, Klaus R.
- Schuppert, Andreas
2010 Mathematics Subject Classification
- 34D15 34E05 92E20
Keywords
- Singular perturbation, asymptotic methods, upper and lower solution, jump behavior of reaction rates
DOI
Abstract
Singular singularly perturbed systems of ordinary differential equations modelling the dynamics of fast bimolecular reactions are considered. The asymptotic behavior of the solution of the initial value problem on a finite time interval is studied under conditions (change of stability) which are not treated in the usual standard theory. The application of the obtained results to the model under consideration yields conditions under which the reaction rate jumps. This behavior has to. be taken into account for identification problems in chemical process modelling.
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WIAS Preprint No. 1297, (1994)Effects of distributed delays on the stability of structures under seismic excitation and multiplicative noise
Authors
- Karmeshu, Prof.
- Schurz, Henri
2010 Mathematics Subject Classification
- 74H55 74H50 26A15
Keywords
- Stochastic stability, weak and strong time delay, random oscillations, seismic excitation, stochastic differential equations, numerical methods, implicit Euler, Mil'shtein and Balanced methods
DOI
Abstract
The effects of seismic excitation and multiplicative noise (arising from environmental fluctuations) on the stability of a single degree of freedom system with distributed delays are investigated. The system is modelled in the form of a stochastic integro-differential equation interpreted in Stratonovich sense. Both deterministic stability and stochastic moment stability are examined for the system in the absence of seismic excitation. The model is also extended to incorporate effects of symmetric nonlinearity. The simulation of stochastic linear and nonlinear systems are carried out by resorting to numerical techniques for the solution of stochastic differential equations.
Appeared in
- SADHANA, 20 (1995), pp. 451--474
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WIAS Preprint No. 1297, (1994)Simulation of stochastic auto-oscillating systems through variable stepsize algorithms with small noise
Authors
- Artemiev, Sergey S.
- Averina, Tatjana A.
- Schurz, Henri
2010 Mathematics Subject Classification
- 60H10 65C20 65L20 65U05
Keywords
- Stochastic differential equations, auto-oscillating systems, limit cycle, strange attractor, bifurcation, numerical methods, mean square error, variable stepsize algorithm, simulation studies
DOI
Abstract
The paper considers some questions of the numerical analysis of stochastic auto-oscillating systems and their simulation on computers. A low computer costs, variable stepsize algorithm based on local error estimation of stochastic Runge-Kutta- Fehlberg methods is stated for solving nonlinear stochastic differential equations. In particular, it turns out to be very efficient for dynamical systems with small noise intensity. Results of numerical experiments for a plenty of well-known examples from Physics, Chemistry, Biology and Ecology are illustrated with the help of the dialogue system 'Dynamics and Control'.
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WIAS Preprint No. 1297, (1994)An inverse model problem in kinetic theory
Authors
- Babovsky, Hans
2010 Mathematics Subject Classification
- 35R30 78A40 65R30
Keywords
- Inverse problem, kinetic theory
DOI
Abstract
The paper deals with an inverse model problem in linear kinetic theory: the identification of a density profile of a scattering medium in a slab geometry from measurement of the reflected portion of a particle flux entering the medium. We prove well-posedness of the problem and present a robust algorithm for the identification.
Appeared in
- Inverse Problems, 11 (1995), pp. 555--570
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WIAS Preprint No. 1297, (1994)Local adaptivity to inhomogeneous smoothness. 1. Resolution level
Authors
- Lepskii, Oleg V.
- Spokoiny, Vladimir
ORCID: 0000-0002-2040-3427
2010 Mathematics Subject Classification
- 62G07 62G20
DOI
Abstract
The problem of nonparametric estimation of functions of inhomogeneous smoothness is considered. The goal is to define the notion of local smoothness of a function ƒ(·), to evaluate the optimal rate of convergence of estimators (depending on this local smoothness) and to construct an asymptotically efficient locally adaptive estimator. We treat local (or δ - local) smoothness properties of a function ƒ(·) at a point t as the corresponding characteristics of this function on the interval [t-δ,t+δ]. The value δ measures the ``locality'' of our procedure. The smaller this value is taken the more precise is our resolution analysis. But this value can not be taken arbitrary small since we should ce able to restore local smoothness properties of a function from the noisy data. The main result of the paper describes just the maximal rate of convergence of this parameter δ to zero as the noise level ε goes to zero. We call this value the resolution level. The value of this level strongly depends on the upper considered smoothness β* what we wish to attain. If κ*ε is the bandwidth corresponding to this smoothness β* then the resolution level δ*ε can not be chosen less (in order) than κ*ε. In particular, this yields that it is impossible to improve at the same time the accuracy of our procedure (which is measured by the upper smoothness β*) and its local adaptive properties. If we improve the accuracy of estimation at subintervals where a function is of high smoothness then we will have a low accuracy in a larger vicinity near a point with small smoothness. The main results claim that if the parameter of locality δ is taken less (in order) than the resolution level, then the corresponding risk is (asymptotically) infinite. After that we construct estimators with a finite asymptotic risk for the case of δ coinciding with the resolution level.
Appeared in
- Math. Methods Statistics, 4 (1995), pp. 239 -- 258
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WIAS Preprint No. 1297, (1994)Moment evolution of the outflow-rate from nonlinear conceptual reservoirs
Authors
- Karmeshu, [Prof.]
- Schurz, Henri
2010 Mathematics Subject Classification
- 60H10 60H30 65C20
Keywords
- Conceptual reservoirs, storage, outflow-rate, precipitation, Fokker-Planck equation, stochastic differential equations, white and coloured noise, numerical methods, simulation studies, moment evolution
DOI
Abstract
The temporal evolution of moments of outflow-rate is investigated in a stochastically perturbed nonlinear reservoir due to precipitation. The detailed stochastic behaviour of outflow is obtained from the numerical solution of a nonlinear stochastic differential equation with multiplicative noise. The timedevelopment of first two moments is studied for various choices of parameters. Using Stratonovich interpretation, it turns out that the mean outflow-rate is above that given by the deterministic solution. Based on the set of 9000 simulation runs, 90 % confidence intervals for the mean evolution of outflow-rate are computed. The effect of stochastic perturbations with finite correlation time is also investigated.
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WIAS Preprint No. 1297, (1994)A note on pathwise approximation of stationary Ornstein-Uhlenbeck processes with diagonalizable drift
Authors
- Schurz, Henri
2010 Mathematics Subject Classification
- 65U05 60H10 58G32 34F05
Keywords
- Stochastic differential equations, additive noise, implicit Euler methods
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 with real-diagonalizable drift and the familiy of implicit Euler methods. For the description of the occuring bias the notions of asymptotical p-th mean, mean, variance and equilibrium preservation are introduced. The main result can be useful for the implementation of numerical algorithms requiring more precise long-term runs, such as in discrete parametric estimation or in numerical computation of top Lyapunov exponents.
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WIAS Preprint No. 1297, (1994)On approximate approximations using Gaussian kernels
Authors
- Maz´ya, Vladimir
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 41A30 41A63 41A05 65D99
Keywords
- Multivariate approximation, quasi-interpolation, Gaussian kernels
DOI
Abstract
This paper discusses quasi-interpolation and interpolation with Gaussians from a new point of view concerning accuracy in numerical computations. Estimates are obtained showing a high order approximation up to some saturation error negligible in numerical applications. The construction of local high order quasi-interpolation formulas is given.
Appeared in
- IMA J. Numer. Anal., 16 (1996), pp. 13--29
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WIAS Preprint No. 1297, (1994)The qualocation method for Symm's integral equation on a polygon
Authors
- Elschner, Johannes
- Prössdorf, Siegfried
- Sloan, Ian H.
2010 Mathematics Subject Classification
- 65R20 45B05
Keywords
- First kind boundary integral equation, qualocation method, nonlinear parametrization, Mellin convolution operator, superconvergence
DOI
Abstract
This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in ℝ2 . Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O(hk ). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.
Appeared in
- Math. Nachr., 177 (1996), pp. 81--108
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WIAS Preprint No. 1297, (1994)Stefan problems and the Penrose-Fife phase field model
Authors
- Colli, Pierluigi
ORCID: 0000-0002-7921-5041 - Sprekels, Jürgen
ORCID: 0009-0000-0618-8604
2010 Mathematics Subject Classification
- 35R35 80A22
Keywords
- phase change process, time relaxation, space relaxation, initial energy, phase fraction, flux boundary condition, limits, unique weak solutions, compactness arguments
DOI
Abstract
This paper is concerned with singular Stefan problems in which the heat flux is proportional to the gradient of the inverse absolute temperature. Both the standard interphase equilibrium conditions and phase relaxations are considered. These problems turn out to be the natural limiting cases of a thermodynamically consistent model for diffusive phase transitions proposed by Penrose and Fife. By supplying the systems of equations with suitable initial and boundary conditions, a rigorous asymptotic analysis is performed, and the unique solutions to the different Stefan problems are derived as asymptotic limits of the solutions to the Penrose-Fife phase-field problem.
Appeared in
- Adv. Math. Sci. Appl. 7, (1997), pp. 911-934
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WIAS Preprint No. 1297, (1994)How 1-dimensional hyperbolic attractors determine their basins
Authors
- Bothe, Hans Günter
2010 Mathematics Subject Classification
- 58F12 58F15
Keywords
- Hyperbolic attractors, basins
DOI
Abstract
Two attractors Λi (i=1,2) of diffeomorphisms ƒi : Mi → Mi will be called intrinsically equivalent if there is a homeomorphism h: Λ1 → Λ2 satisfying ƒ2h = hƒ1. If we can find a homeomorphism g: WsΛ1 → WsΛ2 of the basins WsΛi of Λi such that ƒ2g = gƒ1, then we say that Λ1, Λ2 are basin equivalent. Let Λ1, Λ2 be transversely tame 1-dimensional hyperbolic attractors which are intrinsically equivalent. Then, if WsΛ1, WsΛ2 are orientable and m = dim M1 = dim M2 ≥ 4, it is shown that Λ1, Λ2 are basin equivalent, provided these attractors are regarded, for some positive integer k, as attractors of ƒk1, ƒk2 instead of ƒ1, ƒ2, respectively. This conclusion implies that WsΛ1, WsΛ2 are homeomorphic under a homeomorphism which maps Λ1 to Λ2 and the stable foliation 𝔚sΛ1 of WsΛ1 to the stable foliation 𝔚sΛ2 of WsΛ2. (To be transversely tame is a weak restriction; hence, roughly, speaking, these facts hold for "almost all" 1-dimensional hyperbolic attractors.) If transverse tameness and m ≥ 4 is dropped from the assumption, then still the cartesian products WsΛi x ℝ are homeomorphic with a homeomorphism which maps Λ1 x {0} to Λ2 x {0} and 𝔚sΛ1 x ℝ to 𝔚sΛ2 x ℝ.
Appeared in
- Nonlinearity 9 (1996), No. 5, pp. 1173--1190
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WIAS Preprint No. 1297, (1994)Approximation of the Boltzmann equation by discrete velocity models
Authors
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K35 76P05 82C40
Keywords
- Boltzmann equation, discrete velocity models, weak convergence, random mass flow
DOI
Abstract
Two convergence results related to the approximation of the Boltzmann equation by discrete velocity models are presented. First we construct a sequence of deterministic discrete velocity models and prove convergence (as the number of discrete velocities tends to infinity) of their solutions to the solution of a spatially homogeneous Boltzmann equation. Second we introduce a sequence of Markov jump processes (interpreted as random discrete velocity models) and prove convergence (as the intensity of jumps tends to infinity) of these processes to the solution of a deterministic discrete velocity model.
Appeared in
- J. Statist. Phys., 78 (1995), pp. 1555--1570
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WIAS Preprint No. 1297, (1994)Remarks on the spectral properties of tight binding and Kronig-Penney models with substitution sequences
Authors
- Bovier, Anton
- Ghez, Jean-Michel
2010 Mathematics Subject Classification
- 82B44 82D30
DOI
Abstract
We comment on some recent investigations on the electronic properties of models associated to the Thue-Morse chain and point out that their conclusions are in contradiction with rigorously proven theorems and indicate some of the sources of these misinterpretations. We briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems.
Appeared in
- J. Phys. A, 28 (1995), pp. 2313--2324
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WIAS Preprint No. 1297, (1994)A mathematical model of emulsion polymerization
Authors
- Gajewski, Herbert
- Zacharias, Klaus
2010 Mathematics Subject Classification
- 35F25 35Q80 80A30 82D60
Keywords
- emulsion polymerization, nonlinear nonlocal first-order partial integro-differential equation
DOI
Abstract
We consider a mathematical model of polymerization which in the language of chemistry is called emulsion polymerization. Roughly speaking, polymerization is the formation of huge molecules (-the molecules of the polymer-) from smaller ones (-the molecules of the monomer). In the case of emulsion polymerization this process takes place in an aqueous medium in the presence of appropriate auxiliary substances. As an example we mention that the well-known polymer polyvinyl chloride (PVC) can be produced in this way starting from the monomer vinyl chloride. There are different possibilities to operate a polymerization reactor. One distinguishes batch (or discontinuous) and continuous reactors. The batch reactor is one where all ingredients are charged at the beginning of the polymerization and the reaction proceeds over a certain interval of time. Continuous reactors run with a continuous inflow and outflow of material. Mixed types of reactor operating are possible, e.g. the semibatch mode where part of the ingredients are added during the polymerization process. A general assumption is that the content of the reaction vessel is well stirred so that local inhomogeneities can be neglected. The mathematical model presented here was proposed in the seventies by Min and Ray ([12], [13], [14]). It has been modified and extended in the research group of Dr. Tauer ([20], [21] , [22]) at the (former) Institute of Polymer Chemistry (Teltow-Seehof). At the (former) Karl Weierstrass Institute of Mathematics (Berlin) the model has been investigated from the mathematical and numerical point of view during several years ([4], [5], [6], [7]).
Appeared in
- Scientific Computing in Chemical Engineering (F. Keil, W. Mackens, H. Voss, J. Werther, eds.), Springer-Verlag Berlin Heidelberg, 1996, pp. 60-67.
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WIAS Preprint No. 1297, (1994)Gibbs states of the Hopfield model with extensively many patterns
Authors
- Bovier, Anton
- Gayrard, Véronique
- Picco, Pierre
2010 Mathematics Subject Classification
- 82B44 82C32
Keywords
- Hopfield model, Gibbs states, self-averaging, spin glasses
DOI
Abstract
We consider the Hopfield model with M(N) = αN patterns, where N is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGPl]. The key new ingredient is a self averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.
Appeared in
- J. Statist. Phys., 79 (1995), pp. 395--414
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WIAS Preprint No. 1297, (1994)On asymptotic minimaxity of Kolmogorov and omega-square tests
Authors
- Ermakov, Mikhail S.
2010 Mathematics Subject Classification
- 62F05 62F12 62G20
Keywords
- Large deviations, nonparametric hypothesis testing, asymptotically minimax hypothesis testing, Bahadur efficiency, Hodges-Lehmann efficiency, Kolmogorov test, omegasquare test
DOI
Abstract
We consider the problem of hypothesis testing about a value of functional. For a given functional T the problem is to test a hypothesis T(P) = 0 versus alternatives T(P) > b0 > 0 where P is an arbitrary probability measure. Under the natural assumptions we show that the test statistics T(P̂n) depending on the empirical probability measures P̂n are asymptotically minimax. Since the sets of alternatives is fixed the asymptotic minimaxity is considered in the senses of Bahadur and Hodges-Lehmann efficiencies. In particular the functional T can be the functional corresponded to the test statistics of Kolmogorov and omega-square tests.
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WIAS Preprint No. 1297, (1994)On lower bounds of the moderate and cramer type large deviation probabilities in statistical inference
Authors
- Ermakov, Mikhail S.
2010 Mathematics Subject Classification
- 62F05 62F12 62G20
Keywords
- Moderate large deviations, Cramer type large deviations, asymptotic efficiency, asymptotically minimax estimation, asymptotically minimax hypothesis testing, Bahadur efficiency, Chernoff efficiency
DOI
Abstract
We indicate new simple assignments of the lower bounds for the probabilities of the moderate and Cramer type large deviations of type I and type II errors of statistical tests. These assignments are based on a one natural property of the normal distribution. Using these results we deduce easily the lower bounds for the probabilities of the moderate and Cramer type large deviations of estimators. The lower bounds were obtained under the more weak assumptions then in the previous papers. The lower bound for the probabilities of the Cramer type large deviations of estimators has not been proved earlier. The results are also extended on the problems of asymptotically minimax statistical inference about a value of functional.
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WIAS Preprint No. 1297, (1994)On convergence rates of suprema in the presence of non-negligible trends
Authors
- Konakov, Valentin
2010 Mathematics Subject Classification
- 62G07 62G20 62M40
Keywords
- Kernel estimation, smoothing parameter, Gaussian fields, Laplace type integral, Leray-Gel'fand differential form
DOI
Abstract
We investigate the convergence rates for the maximal deviation distribution of kernel estimates from the true density. The convergence rates for related Gaussian fields are also investigated. We consider the optimal choice of the smoothing parameter in the sense of Konakov and Piterbarg (1994) and in doing so we take into account a non-negligible trend. It is shown that the convergence rates depend on the asymptotic behaviour of the Laplace type integrals over a small neighbourhood of the manifold of points at which the trend attains its maximal value. Using integration over the level sets (Leray-Gel'fand differential forms) it is proved that the convergence rates are tipically logarithmically slow, even if the rates are to be uniform over as few as three points. Some improved approximations with power rates of convergence are also obtained.
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WIAS Preprint No. 1297, (1994)Irreversible phase transitions in steel
Authors
- Hömberg, Dietmar
ORCID: 0000-0001-9460-5729
2010 Mathematics Subject Classification
- 35Q72 35R35 35K55 80A22
Keywords
- mathematical model, phase changes in carbon steel, existence, numerical simulations
DOI
Abstract
We present a mathematical model for the austenite-pearlite and austenite- martensite phase transitions in eutectoid carbon steel. The austenite-pearlite phase change is described by the Additivity Rule. For the austenite-martensite phase change we propose a new rate law, which takes into account its irreversibility. We investigate questions of existence and uniqueness for the three-dimensional model and finally present numerical calculations of a continous cooling transformation diagram for the eutectoid carbon steel C1080.
Appeared in
- Math. Methods Appl. Sci., 20 (1997) pp. 59--77
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WIAS Preprint No. 1297, (1994)An analysis of crystal dissolution fronts in flows through porous media. Part I: Homogeneous charge distribution
Authors
- Hengst, Sabine
- Knabner, Peter
- van Duijn, Cornelius J.
2010 Mathematics Subject Classification
- 35K55 35K57 35R35 35B30 76S05
Keywords
- Transport, travelling wave, crystal dissolution, porous media, mathematical analysis
DOI
Abstract
We propose a model for transport of solutes in a porous medium participating in a dissolution/precipitation reaction, in general not in equilibrium. For an unbounded spatial domain, travelling wave solutions exists, if and only if the charge distribution is constant and we deal with a dissolution situation. The travelling wave in fact exhibits a sharp dissolution front. The wave is given in a nearly explicit manner. Also for the limit cases of equilibrium reaction or no dispersion, travelling waves are established under the same conditions, but with different qualitative properties.
Appeared in
- Adv. Water Resources, 18 (1995), pp. 171--185
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WIAS Preprint No. 1297, (1994)Self-averaging in a class of generalized Hopfield models
Authors
- Bovier, Anton
2010 Mathematics Subject Classification
- 60K35 82C32
Keywords
- Hopfield model, neural networks, self-averaging, law of large numbers
DOI
Abstract
We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1) Multi-spin interactions are permitted and 2) the random variables ξμi i describing the "patterns" can have arbitrary distributions with mean zero and finite 4+∈-th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven.
Appeared in
- J. Phys. A 27 (1994), No. 21, pp. 7069-7077.
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WIAS Preprint No. 1297, (1994)Large deviation principles for the Hopfield model and the Kac-Hopfield model
Authors
- Bovier, Anton
- Gayrard, Véronique
- Picco, Pierre
2010 Mathematics Subject Classification
- 60K35 82B44 82C32
DOI
Abstract
We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distributions of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.
Appeared in
- Probab. Theory Related Fields, 101 (1995), pp. 511--546
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WIAS Preprint No. 1297, (1994)Asymptotic properties of stochastic particle systems with Boltzmann type interaction
Authors
- Tribe, Roger
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K35 76P05 82C40
Keywords
- Interacting particle systems, empirical measures, law of large numbers, co vergence of moments, Boltzmann type equation
DOI
Abstract
We study the asymptotic behaviour of a stochastic particle system that is determined by an independent motion of each particle and by an interadion mechanism between pairs of particles. The limit of the empirical measures of the system is characterized by a nonlinear equation, which is related to the Boltzmann equation. Using a uniqueness result for the limiting equation, we establish a law of large numbers. We also investigate the convergence of moments of the empirical measures.
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WIAS Preprint No. 1297, (1994)Gibbs states of the Hopfield model in the regime of perfect memory
Authors
- Bovier, Anton
- Gayrard, Véronique
- Picco, Pierre
2010 Mathematics Subject Classification
- 60K35 82B44 82C32
Keywords
- thermodynamic properties of the Hopfield model, infinite number of infinite volume Gibbs measures, Bernoulli measures, convergent sequences of measures
DOI
Abstract
We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M ⁄N ↓ 0 as N ↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperatures T < 1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point. If M ⁄N → α, as N ↑ ∞ for a small enough, we show that for temperatures T smaller than some T(α) < 1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.
Appeared in
- Probab. Theor. Relat. Fields, 100 (1994), pp. 329--363
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