## Search

Now showing items 1-10 of 10

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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
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
Characterization theorem for classical orthogonal polynomials on non-uniform lattices: The functional approach

(2010)

Using the functional approach, we state and prove a characterization theorem for classical orthogonal polynomials on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable) including the Askey-Wilson polynomials. This theorem proves the equivalence between seven characterization properties, namely the Pearson equation for the linear functional, the second-order divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations,and the ...

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Preprint
On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices

(2010)

The main aim of this paper is the development of suitable bases (replacing the power basis x^n (n\in\IN_\le 0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type,
the first of which allows to write solutions of arbitrary divided-difference equations in terms of series representations extending results given in [16] for the q-case. Furthermore it enables the representation ...

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Preprint
On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

(2010)

The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt
orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

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Preprint
On Solution Sets of Information Inequalities

(2011)

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.

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Preprint
Balanced Viscosity solutions to a rate-independent system for damage

(2017-05-02)

This article is the third one in a series of papers by the authors on
vanishing-viscosity solutions to rate-independent damage systems. While in the
first two papers [KRZ13, KRZ15] the assumptions on the spatial domain $\Omega$
were kept as general as possible (i.e. nonsmooth domain with mixed boundary conditions), we
assume here that $\partial\Omega$ is smooth and that the type of boundary
conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for ...

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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 ...