Complex Systems

Methodological Research

Adaptive tomography

Data analysis on the sphere

Systemic models for bacterial interaction

Estimators for the transfer of in information within an ecological system

Characteristic quantities of an ecological system change with time and form a network.
There is a transfer of information among such quantities. In 2000 Thomas Schreiber describes a methode measuring such an information transfer within the framework of information theory. Starting from time series data of two quantities X and Y of an ecological system the aim is to give proper estimators for the transfer of information from X to Y.

Summation methods with respect to orthogonal polynomials

To approximate a function based on a set of natural functions like sine and cosine is a fundamental task of applied mathematics. In some applications orthogonal polynomials are the fundamental building blocks for approximation. In general on has to perform a special summation method to guarantee convergence and positivity of the approximation process. The goal is to design proper summation methods based on orthogonal polynomials.

Obermaier, J.
Powers of the Dirichlet kernel with respect to orthogonal polynomials and related operators.
Acta Sci. Math. 83, 539-549 (2017)

Determination of complex systems' maturity

We develop a method which allows us to determine the maturity of a general complex system in the sense of Gunderson's and Holling's adaptive cycle metaphor on the base of longitudinal abundance data. Starting from this type of data, we determine Schreiber's transfer entropy between two components each, facing the challenge of finding proper estimators for this measure. This step allows us to consider the corresponding system as a network of information transfer. On the base of this network, we want to compute three systems' properties which Gunderson and Holling consider significant for its development. Therefor, we developed definitions based on Ulanowicz's information theoretical approach to systems theory and definitions from the area of spectral graph theory. We verify the reasonableness of our method by means of several simulations and case studies, including microbiological, economic and environmental systems.