Condenser capacity and hyperbolic diameter

On the Capacity of a Condenser

On the Diameter of Hyperbolic Random Graphs

Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguná, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as...

Counting Hyperbolic Manifolds with Bounded Diameter

Let ρn(V ) be the number of complete hyperbolic manifolds of dimension n with volume less than V . Burger, Gelander, Lubotzky, and Moses[2] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV logV ≤ log ρn(V ) ≤ bV logV, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number gro...

Capacity of a condenser whose plates are circular arcs

We find an asymptotic formula for the conformal capacity of a plane condenser both plate of which are concentric circular arcs as the distance between them vanishes. This result generalizes the formula for the capacity of parallel linear plate condenser found by Simonenko and Chekulaeva in 1972 and sheds light on the problem of finding an asymptotic formula for the capacity of condenser whose p...

A Bound for the Diameter of Random Hyperbolic Graphs

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPK10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for α > 1 2 , C ∈ R, n ∈ N, set R = 2 lnn + C and build the graph G = (V,E) with |V | = n as follows: For each v ∈ V , generate i.i.d. polar coordinates (rv, ...