Amita Malik

Spatial Statistics of Apollonian Gaskets

ABSTRACT Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the nearest neighbor spacing, pair correlation, and electrostatic energy of centers of circles from Apollonian gaskets. Even though the centers of these circles are not uniformly distributed in any “ambient” space, after proper normalization, all these statistics seem to exhibit certain interesting limiting behaviors.

Extensive parity analysis of broken 5-diamond partitions

In this paper, we investigate congruences for Δ5(n) modulo 2 and characterize the parity of Δ5(4n + 1) and Δ5(4n + 2) according to the arithmetic property of n. As a consequence, we obtain various Ramanujan type congruences for Δ5(n). We also extend these results to several infinite families of congruences.