- Úvod
- Ústav
- Lidé
- Výzkum
- Aplikace
- Semináře a akce
- Knihovna
- Doktorské studium
- Kariéra
It is known that electrical networks with resistors can be considered as weighted graphs. Then effective resistance of a network can be defined and it is closely related to properties of a corresponding random walk on the graph. A natural generalization of a network with resistors is given by an electrical network with resistors, capacitors, and inductors (so-called network with impedances). It is more convenient for us to work with the admittance (i.e. the inverse of impedance). We consider admittances as rational functions of λ ( λ =iω , where ω>0 is the frequency of alternating voltage). Then we consider two approaches. In the first approach, we allow λ to be any complex number and define an effective admittance of a network as a complex-valued function on the complex parameter λ. In the second approach, we uniquely define effective admittance as a rational function on λ . Moreover, we elaborate on connections between the two approaches.
In this talk we look at a higher-dimensional generalization of the Erdős--Rényi random graph model to random simplicial complexes, namely the Linial--Meshulam model. In this high-dimensional setting there will often be many nonequivalent ways to topologically generalize common graph properties. Several of these will be surveyed and then new work will be discussed establishing an anticollapsibility threshold in the Linial--Meshulam model as a generalization of path connectivity of random graphs. This work is motivated by questions about high-dimensional trees. This talk will assume familiarity with random graphs but all relevant topological notions will be defined. This is joint work with Davide Lofano.