## 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
Convergence analysis of time-discretization schemes for rate-independent systems

(2017-12-21)

It is well known that rate-independent systems involving nonconvex energy
functionals in general do not allow for time-continuous solutions even if the
given data are smooth. In the last years, several solution concepts were
proposed that include discontinuities in the notion of solution, among them
the class of global energetic solutions and the class of BV-solutions.
In general, these solution concepts are not equivalent and numerical schemes
are needed that reliably approximate that type of solutions ...

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Preprint
A free boundary approach to the Rosensweig instability of ferrofluids

(2017-04-18)

We establish the existence of saddle points for a free boundary problem describing the two-dimensional free surface of a ferrofluid which undergoes normal field instability (also known as Rosensweig instability). The starting point consists in the ferro-hydrostatic equations for the magnetic potentials in the ferrofluid and air, and the function describing their interface. The former constitute the strong form for the Euler-Lagrange equations of a convex-concave functional. We extend this functional in order to ...

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Preprint
On the existence of symmetric minimizers

(2018-01-23)

In this note we revisit a less known symmetrization method for functions with respect to a topological group, which we call G-averaging. We note that, although quite non-technical in nature, this method yields G-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstract theorem and show how it can be applied to the p-Laplace and polyharmonic Poisson problem in order to construct symmetric solutions. We also pose some open problems and explore further possibilities where the ...

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Preprint
On Oseen Resolvent Estimates: A Negative Result

(2009)

We consider the resolvent problem for the scalar Oseen equation in the whole space R^3. We show that for small values of the resolvent parameter it is impossible to obtain an L^2-estimate analogous to
the one which is valid for the Stokes resolvent, even if the resolvent
parameter has positive real part.

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Preprint
Mathematical problem solving, modelling, applications, and links to other subjects

(1989)

The paper will consist of three parts. In part I we shall present some background
considerations which are necessary as a basis for what follows. We
shall try to clarify some basic concepts and notions, and we shall collect
the most important arguments (and related goals) in favour of problem solving,
modelling and applications to other subjects in mathematics instruction.
In the main part II we shall review the present state, recent trends, and
prospective lines of development, both in empirical or theoretical ...

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Preprint
Modellierung und numerische Simulation der Thermoregulation von Früh- und Neugeborenen

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

Bei frühgeborenen Säuglingen spielt die Thermoregulation zur Aufrechterhaltung einer überlebenswichtigen Körpertemperatur durch Wärmeproduktion, -abgabe bzw. -aufnahme eine entscheidende Rolle. Der Einsatz moderner Inkubatoren soll die körpereigenen Thermoregulatoren unterstützen, und es ist im Hinblick auf verschiedene medizinische Fragestellungen wünschenswert, diesen Prozess modellieren zu können. Wir stellen ein einfaches Modell auf der Basis von partiellen Differentialgleichungen vor und beschreiben detailliert ...

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Preprint
Negative Größen bei Diophant?

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

In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki ...

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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 ...