## Search

Now showing items 1-10 of 53

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Preprint
No Chaos in Dixon's System

(2020)

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behaviour, if its two parameters take their value in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into sixteen different regions in each of which the system exhibits qualitatively the same behaviour. In particular, we prove that ...

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Preprint
Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads

(2019-09-25)

We study a rate-independent system with non-convex energy and in the
case of a time-discontinuous loading. We prove existence of
the rate-dependent viscous regularization by time-incremental problems, while
the existence of the so called parameterized BV-solutions is obtained via
vanishing viscosity in a suitable parameterized setting. In addition, we prove
that the solution set is compact.

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Preprint
Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

(2005)

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that ...

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Preprint
Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity

(Universität Kassel, FB 17, Mathematik/Informatik, 2005)

Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v^infinity, p^infinity to the problems in the unbounded domain Omega the error v^infinity - v^R, p^infinity - p^R is estimated in H^1(Omega_R) and L^2(Omega_R), respectively. Here v^R, p^R are the approximating solutions on the truncated domain Omega_R, the parameter R controls the exhausting of Omega. ...

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Preprint
A crack on the interface of piezo-electric bodies

(Universität Kassel, FB 17, Mathematik/Informatik, 2005)

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezo-electric media under various boundary conditions on the crack surfaces. The Griffith formulae are obtained for increments of energy functionals due to growth of the crack and the notion of the energy release matrix is introduced. Normalization conditions for bases of singular solution are proposed to adapt them to the energy, stress, and deformation fracture criteria. Connections between these ...

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Preprint
Stability of preconditioned finite volume schemes at low Mach numbers

(Universität Kassel, FB 17, Mathematik/Informatik, 2004)

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an ...

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Preprint
An algebraic proof of Iitaka's conjecture C2,1

(Universität Kassel, FB 17, Mathematik/Informatik, 2002)

We give a proof of Iitaka's conjecture C2,1 using only elementary methods from algebraic geometry.

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Preprint
A generalization of Student’s t-distribution from the viewpoint of special functions

(2005)

Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the ...

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Preprint
An approximation method using approximate approximations

(Universität Kassel, FB 17, Mathematik/Informatik, 2005)

The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential ...

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Preprint
Anisotropic adaptive resolution of boundary layers for heat conduction problems

(Universität Kassel, FB 17, Mathematik/Informatik, 2005)

We deal with the numerical solution of heat conduction problems featuring steep gradients. In order to solve the associated partial differential equation a finite volume technique is used and unstructured grids are employed. A discrete maximum principle for triangulations of a Delaunay type is developed. To capture thin boundary layers incorporating steep gradients an anisotropic mesh adaptation technique is implemented. Computational tests are performed for an academic problem where the exact solution is known as ...