1,256 Works

Surface energy and boundary layers for a chain of atoms at low temperature

Sabine Jansen, Wolfgang König, Bernd Schmidt & Theil, ##!Error: Attribute Unknown!##
the temperature goes to zero and at fixed positive pressure, the Gibbs measures for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The...

Abstract forced symmetry breaking

Daniela Peterhof & Lutz Recke
We consider abstract forced symmetry breaking problems of the type F(x,λ) = y, x ≈ O(x0), λ ≈ λ0, 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(x0, λ0) = 0 and, hence, that F(x,λ0) = 0 for all elements x of the group orbit O(x0) of x0. We look for solutions x which bifurcate from the...

The link between coherence echoes and mode locking

Sebastian Eydam & Wolfrum, ##!Error: Attribute Unknown!##
We investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally- coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self- organized periodic pulsations of the mean field, called mode locking, have been described re- cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu- ral frequencies are randomly chosen...

Center manifolds for homoclinic solutions

Björn Sandstede
In this article, center-manifold theory for homoclinic solutions of ordinary differential equations or semilinear parabolic equations is developed. Here, a center manifold along a homoclinic orbit q(t) is a locally invariant manifold containing all solutions which stay close to q(t) in phase space for all times. Therefore, as usual, the low-dimensional center manifold contains the interesting recurrent dynamics nearby the homoclinic orbit and a considerable reduction of dimension is achieved. The manifold is of class...

Rigorous results on some simple spin glass models

Anton Bovier & Irina Kurkova
In this paper we review some recent rigorous results that provide an essentially complete solution of a class of spin glass models introduced by Derrida in the 1980ies. These models are based on Gaussian random processes on {-1,1}N whose covariance is a function of a ultrametric distance on that set. We prove the convergence of the free energy as well as the Gibbs measures in an appropriate sense. These results confirm fully the predictions of...

Non-intrusive tensor reconstruction for high dimensional random PDEs

Martin Eigel, Johannes Neumann, Reinhold Schneider & Sebastian Wolf
This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high dimensional parametric random PDEs which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling.

Asymptotic expansions of the contact angle in nonlocal capillarity problems

Serena Dipierro, Francesco Maggi & Enrico Valdinoci
We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting a family of fractional interaction kernels The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion...

Hysteresis filtering in the space of bounded measurable functions

Pavel Krejčí & Philippe Laurençot
We define a mapping which with each function 푢 ∈ L∞(0,T) and an admissible value of 푟 > 0 associates the function ξ with a prescribed initial condition ξ0 which minimizes the total variation in the r-neighborhood of 푢 in each subinterval [0,t] of [0,T]. We show that this mapping is non-expansive with respect to 푢, r and ξ0, and coincides with the so-called play operator if 푢 is a regulated function.

Optimal choice of observation window for Poisson observations

Yury Kutoyants & Vladimir Spokoiny
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.

A stochastic log-Laplace equation

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

Hysteresis and phase transition in many-particle storage systems

Wolfgang Dreyer, Clemens Guhlke & Michael Herrmann
We study the behavior of systems consisting of ensembles of interconnected storage particles. Our examples concern the storage of lithium in many-particle electrodes of rechargeable lithium-ion batteries and the storage of air in a system of interconnected rubber balloons. We are particularly interested in those storage systems whose constituents exhibit non-monotone material behavior leading to transitions between two coexisting phases and to hysteresis. In the current study we consider the case that the time to...

Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices

Mindaugas Radziunas, Anissa Zeghuzi, Jürgen Fuhrmann, Thomas Koprucki, Hans-Jürgen Wünsche, Hans Wenzel & Uwe Bandelow
We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

Heterogeneous dynamic process flowsheet simulation of chemical plants

Friedrich Grund, Klaus Ehrhardt, Jürgen Borchardt & Dietmar Horn
For large-scale dynamic simulation problems in chemical process engineering, a heterogeneous simulation concept is described which allows to distribute the solution of the models of coupled dynamic subprocesses to a computer network. The main principle of such a technique is to solve the submodels of an overall model independently of each other on subsequent time intervals. This is done by estimating the vector of input variables of the submodels, calculating the corresponding time behaviour of...

Rate-independent evolution of sets

Riccarda Rossi, Ulisse Stefanelli & Marita Thomas
The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish...

Linearized plasticity is the evolutionary Γ-limit of finite plasticity

Alexander Mielke & Ulisse Stefanelli
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Γ-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity

On convergence rates of suprema in the presence of non-negligible trends

Valentin Konakov
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...

Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

Marek Biskup, Ryoki Fukushima & Wolfgang König
We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the...

Optimal stopping via deeply boosted backward regression

Denis Belomestny, John G.M. Schoenmakers, Vladimir Spokoiny & Yuri Tavyrikov
In this note we propose a new approach towards solving numerically optimal stopping problems via boosted regression based Monte Carlo algorithms. The main idea of the method is to boost standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from finance.

The asymptotic behavior of semi-invariants for linear stochastic systems

Grigori N. Milstein
The asymptotic behavior of semi-invariants of the random variable ln |X(t, 푥)|, where X(t,푥) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(푝). Namely, it is obtained that the 푛-th semi-invariant is asymptotically proportional to the time 헍 with the coefficient of proportionallity g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the...

Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations

Etienne Emmrich & Robert Lasarzik
A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model...

Orthogonality of fluxes in general nonlinear reaction networks

D. R. Michiel Renger & Johannes Zimmer
We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

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

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

Monte Carlo difference schemes for the wave equation

Sergej M. Ermakov & Wolfgang Wagner
The paper is concerned with Monte Carlo algorithms for iteration processes. A recurrent procedure is introduced, where information on various iteration levels is stored. Stability in the sense of boundedness of the correlation matrix of the component estimators is studied. The theory is applied to difference schemes for the wave equation. The results are illustrated by numerical examples.

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

Mikhail Shashkov & Dmitry Turaev
Dynamical systems with a homoclinic loop to a saddle equilibrium state are considered. Andronov and Leontovich have shown (see [1939], [1959]) that a generic bifurcation of a two-dimensional C1-smooth dynamical system with a homoclinic loop leads to appearance of a unique periodic orbit. This result holds true in the multi-dimensional setting if some additional conditions are satisfied, which was proved by Shilnikov [1962, 1963, 1968] for the case of dynamical systems of sufficiently high smoothness....

Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

Robert Lasarzik, Elisabetta Rocca & Giulio Schimperna
In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that...

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