In this third talk we will discuss properties related to edge expansion. In particular, we will define the Cheeger constant (which measures how easy it is to cut off a large piece of the graph) and state the Cheeger inequalities. We also will define and discuss discrepancy for undirected and directed graphs. We also state the Perron-Frobenius Theorem which is a useful tool in spectral graph the...

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time. We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman–Kac evolution operator. We show that for the resulting ...

We present a new extension of the well-known Perron-Frobenius theorem to regular matrix pairs (E, A). The new extension is based on projector chains and is motivated from the solution of positive differential-algebraic systems or descriptor systems. We present several examples where the new condition holds, whereas conditions in previous literature are not satisfied.

Let Kn ̄2x `2n :x i & 0 for 1% i% n ́ and suppose that f :Kn MNKn is nonexpansive with respect to the F " -norm and f(0) ̄ 0. It is known that for every x `Kn there exists a periodic point ξ ̄ ξ x `Kn (so f p(ξ ) ̄ ξ for some minimal positive integer p ̄ pξ) and f k(x) approaches 2 f j(ξ ) :0% j! p ́ as k approaches infinity. What can be said about P*(n), the set of positive integers p for which there...