05.11.2019 um 14:15 Uhr in Raum 69/E23
Harsha Kumar (TU München)
Dynamics of the consensus problem on directed graphs
A fundamental issue in the area of complex network dynamics and multi-agent systems is the consensus (or agreement) problem, which requires a unanimous decision among processes (or agents) for a data value. In this talk, I will explain consensus dynamics on directed graphs (digraphs) with a non-linear communication protocol (interaction function) on unweighted edges. I will show the existence of bifurcations arising from the stated nonlinearity for strongly connected digraphs. I will also demonstrate how to combine the above result with techniques from fast-slow systems to get dynamic bifurcations, using the van-der-Pol-type nonlinearity as an example. If time permits, I will talk about a conjecture regarding a weaker criterion on digraphs that also show bifurcations as mentioned earlier. In the second part of the talk, I will explain a symmetrization algorithm that creates undirected graphs from digraphs. Finally, I will present a result on the topological equivalence of linear consensus dynamics on the input and output graphs for this algorithm. This constructs a bridge between dynamics on digraphs and signed undirected graphs.
08.11.2019 um 13:30 Uhr in Raum 69/E18
Timothy Nadhomi (University of Silesia in Katowice, Poland)
Properties of the Sugeno Integral
Fuzzy measure theory is a generalization of classical measure theory. This generalization is obtained by replacing the additivity axiom of classic measure with weaker axioms of monotonicity. The development of fuzzy measure theory has been motivated by the increasing apprehensiveness that the additivity property of classical measures is in some applications context too restrictive and consequently unrealistic. Jensen inequality is one of the most important tools in actuarial mathematics, and in the mathematics of finance in general. The Jensen inequality makes in particular any insurance policy possible. Recently there are many results extending the Jensen inequality for other aggregation operators and one of the is the so called Sugeno Integral. Sugeno integral is one of the most important fuzzy integrals, which has many applications in various fields. The thesis presents the notion of Sugeno integral. In particular we look at the Jensen type inequality for Sugeno integral, Conditions to the Jensen inequality for the generalized Sugeno integral, utility theory and Sugeno integral as an aggregation function