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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 ...
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 ...
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.
Preprint
Image compression predicated on recurrent iterated function systems
(2008)
Recurrent iterated function systems (RIFSs) are improvements of iterated function systems (IFSs) using elements of the theory of Marcovian stochastic processes which can produce more natural looking images. We construct new RIFSs consisting substantially of a vertical contraction factor function and nonlinear transformations. These RIFSs are applied to image compression.
Preprint
Statistical Analysis of Diabetes Mellitus
(2009)
Diabetes mellitus is a disease where the glucosis-content of the blood does not automatically decrease to a ”normal” value between 70 mg/dl and 120 mg/dl (3,89 mmol/l and 6,67 mmol/l) between perhaps one hour (or two hours) after eating. Several instruments can be used to arrive at a relative low increase of the glucosis-content. Besides drugs (oral antidiabetica, insulin) the blood-sugar content can mainly be influenced by (i) eating, i.e., consumption of the right amount of food at the right time (ii) physical ...
Preprint
Duplication coefficients via generating functions
(2006)
In this paper, we solve the duplication problem P_n(ax) = sum_{m=0}^{n}C_m(n,a)P_m(x) where {P_n}_{n>=0} belongs to a wide class of polynomials, including the classical orthogonal polynomials (Hermite, Laguerre, Jacobi) as well as the classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouk) for the specific case a = −1. We give closed-form expressions as well as recurrence relations satisfied by the duplication coefficients.
Preprint
Approximate solutions and error estimates for a Stokes boundary value problem
(2009)
The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we 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 system ...
Preprint
Error estimates for approximate approximations on compact intervals
(2005)
The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in-tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.
In the present ...
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 ...
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 one is ...