1,258 Works

A discrete collocation method for Symm's integral equation on curves with corners

Johannes Elschner & Ernst P. Stephan
This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a closed polygonal boundary in ℝ2. We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at...

An analogue of grad-div stabilization in nonconforming methods for incompressible flows

Mine Akbas, Alexander Linke, Leo G. Rebholz & Philipp W. Schroeder
Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spacial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization is presented for nonconforming flow discretizations of Discontinuous Galerkin...

Gibbsian representation for point processes via hyperedge potentials

Benedikt Jahnel & Christof Külske
We consider marked point processes on the $d$-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

Global--in--time solvability of thermodynamically motivated parabolic systems

Pierre-Étienne Druet
In this paper, doubly non linear parabolic systems in divergence form are investigated form the point of view of their global?in?time weak solvability. The non?linearity under the time derivative is given by the gradient of a strictly convex, globally Lipschitz continuous potential, multiplied by a position?dependent weight. This weight admits singular values. The flux under the spatial divergence is also of monotone gradient type, but it is not restricted to polynomial growth. It is assumed...

Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks

Pascal Gussmann & Alexander Mielke
The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Gamma converge to the linearized elastic energy with a local constraint of noninterpenetrability along the crack.

Classical solutions of quasilinear parabolic systems on two-dimensional domains

Hans-Christoph Kaiser, Hagen Neidhardt & Joachim Rehberg
Using a classical theorem of Sobolevskii on equations of parabolic type in a Banach space and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems in diagonal form admit a local, classical solution in the space of 푝-integrable functions, for some 푝 > 1, over a bounded two dimensional space domain. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck's...

Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures

Vaios Laschos & Alexander Mielke
By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and...

A method of constructing of dynamical systems with bounded nonperiodic trajectories

Gennadii A. Leonov
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.

Efficient all-optical control of solitons

Sabrina Pickartz, Uwe Bandelow & Shalva Amiranashvili
We consider the phenomenon of an optical soliton controlled (eg. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.

Another phase transition in the Axelrod model

Alex Stivala & Paul Keeler
Axelrod's model of cultural dissemination, despite its apparent simplicity, demonstrates complex behavior that has been of much interest in statistical physics. Despite the many variations and extensions of the model that have been investigated, a systematic investigation of the effects of changing the size of the neighborhood on the lattice in which interactions can occur has not been made. Here we investigate the effect of varying the radius R of the von Neumann neighborhood in...

Tricomi's composition formula and the analysis of multiwavelet approximation methods for boundary integral equations

Siegfried Prößdorf
The present paper is mainly concerned with the convergence analysis of Galerkin-Petrov methods for the numerical solution of periodic pseudodifferential equations using wavelets and multiwavelets as trial functions and test functionals. Section 2 gives an overview on the symbol calculus of multidimensional singular integrals using Tricomi's composition formula. In Section 3 we formulate necessary and sufficient stability conditions in terms of the so-called numerical symbols and demonstrate applications to the Dirchlet problem for the Laplace...

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

Vadim V. Strygin
The well-known classic feedback and feedforward techniques are the main tools for investigations of the control problems. Unlike these strategies, the vibrational control technique, introduced by S.M. Meerkov [1], has proven to be a viable alternative to conventional feedback and feedforward strategies in stabilization problems when the outputs, states and disturbances are difficult to access. Mathematical modelling of such systems is closely connected with nonlinear singularly perturbed systems under parametric excitations. In this paper a...

Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes

Hongjun Luo, Dietmar Kolb, Heinz-Jürgen Flad, Wolfgang Hackbusch & Thomas Koprucki
We have studied various aspects concerning the use of hyperbolic wavelets and adaptive approximation schemes for wavelet expansions of correlated wavefunctions. In order to analyze the consequences of reduced regularity of the wavefunction at the electron-electron cusp, we first considered a realistic exactly solvable many-particle model in one dimension. Convergence rates of wavelet expansions, with respect to L2 and H1 norms and the energy, were established for this model. We compare the performance of hyperbolic...

Correct voltage distribution for axisymmetric sinusoidal modeling of induction heating with prescribed current, voltage, or power

Olaf Klein & Peter Philip
We consider the problem of determining the voltage in coil rings, which arise as an axisymmetric approximation of a single connected induction coil during modeling of induction heating. Assuming axisymmetric electromagnetic fields with sinusoidal time dependence, the voltages are computed from the condition that the total current must be equal in each ring. Depending on which of the quantities total current, total voltage, or total power is to be prescribed, the ring voltages are given...

Extreme Value Behavior in the Hopfield Model

Anton Bovier & David M. Mason
We study a Hopfield model whose number of patterns M grows to infinity with the system size N, in such a way that M(N)2 log M(N)/N tends to zero. In this model the unbiased Gibbs state in volume N can essentially be decomposed into M(N) pairs of disjoint measures. We investigate the distributions of the corresponding weights, and show, in particular, that these weights concentrate for any given N very closely to one of the...

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

Markus Kraft & Wolfgang Wagner
In this paper we study a stochastic particle system that describes homogeneous gas phase reactions of a number of chemical species. First we introduce the system as a Markov jump process and discuss how relevant physical quantities are represented in terms of appropriate random variables. Then, we show how various deterministic equations, used in the literature, are derived from the stochastic system in the limit when the number of particles goes to infinity. Finally, we...

The thermodynamics of the Curie-Weiss model with random couplings.

Anton Bovier & Veronique Gayrard
We study the Curie-Weiss version of an Ising spin system with random, positively biased, couplings. In particular the case where the couplings ∈ij take the values one with probability p and zero with probability 1 - p which describes the Ising model on a random graph is considered. We prove that if p is allowed to decrease with the system size N in such a way that Np(N) ↑ ∞ as N ↑ ∞, then...

On an information-type inequality for the Hellinger process

Alexander A. Gushchin
Let (Ω,퓕,픽) be a filtered space with two probability measures P and P' on (Ω,퓕). Let X be a d-dimensional locally square-integrable semimartingale relative to P and P' with the canonical decomposition X = X0 + M + A and X = X0 + 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...

Landau-Ginzburg model for a deformation-driven experiment on shape memory alloys

Nikolaus Bubner
A Landau-Ginzburg model describing first order martensitic phase transitions in shape memory alloys is considered. The model developed by Falk is transformed in order to simulate deformation-driven experiments done by I. Müller and his co-workers. In these experiments, they do not only observe load-deformation hysteresis loops but also small loops inside these hysteresis loops. Numerical simulations for a CuZnAl single crystal show good agreement with the experiment. We find, for example, nucleation processes, moving phase...

The shock location for a class of sensitive boundary value problems

Adriana Bohé
The sensitive internal layer behaviour in an autonomous nonlinear singularly perturbed boundary value problem is investigated. For this problem we show that the internal layers solutions exhibit either an exponential or an algebraic sensitivity in reponse to small changes in the boundary conditions as well as in the coefficients of the equation and we derive a geometric method to determine the shock location as a function of the perturbations. The results are then applied to...

Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series

Michael H. Neumann
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...

A mathematical model for induction hardening including mechanical effects

Dietmar Hömberg
In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the...

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