Iterated index and the mean Euler characteristic

The Euler-Poincaré characteristic of index maps∗

We apply the concept of the Euler-Poincaré characteristic and the periodicity number to the index map of an isolated invariant set in order to obtain a new criterion for the existence of periodic points of a continuous map in a given set.

On the Mean Euler Characteristic and Mean Betti's Numbers of the Ising Model with Arbitrary Spin *

The behaviour of the mean Euler–Poincaré characteristic and mean Betti’s numbers in the Ising model with arbitrary spin on Z2 as functions of the temperature is investigated through intensive Monte Carlo simulations. We also consider these quantities for each color a in the state space SQ = {−Q,−Q + 2, . . . , Q} of the model. We find that these topological invariants show a sharp transition at...

On the mean Euler characteristic and mean Betti numbers of the Griffiths’ model

The behaviour of the mean Euler–Poincaré characteristic and mean Betti’s numbers in the Griffiths’ model on Z2 (the Q-states Ising model) as functions of the temperature is investigated through intensive Monte Carlo simulations. We also consider these quantities for each color a in the state space SQ = {−Q,−Q+2, . . . , Q} of the model. We find that these topological invariants show a sharp tra...

The Euler Characteristic

I will describe a few basic properties of the Euler characteristic and then I use them to prove special case of a cute formula due to Bernstein-Khovanskii-Koushnirenko. 1. Basic properties of the Euler characteristic The Euler characteristic is a function χ which associates to each reasonable topological space X an integer χ(X). For us a reasonable space would be a space which admits a finite s...

Exponentiation and Euler Characteristic