Minkowski content and fractal Euler characteristic for conformal graph directed systems

Multifractal Analysis for Conformal Graph Directed Markov Systems

Abstract. We derive the multifractal analysis of the conformal measure (or equivalently, the invariant measure) associated to a family of weights imposed upon a (multi-dimensional) graph directed Markov system (GDMS) using balls as the filtration. This analysis is done over a subset of J which is often large. In particular, it coincides with the limit set when the GDMS under scrutiny satisfies ...

The Minkowski and conformal superspaces

We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, ba...

The Euler characteristic of an even-dimensional graph

We write the Euler characteristic χ(G) of a four dimensional finite simple geometric graph G = (V,E) in terms of the Euler characteristic χ(G(ω)) of two-dimensional geometric subgraphs G(ω). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation ∑ x∈V K(x) = χ(G) can so be rewritten as an average 1 − E[K(x, f)]/2 over a collection two dimensional “sectional gr...

Exponentiation and Euler Characteristic

The number of Euler tours of a random directed graph

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random d-in/d-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler...