Embedding properties of sets with finite box-counting dimension

The box-counting dimension for geometrically finite Kleinian groups

We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the “global measure formula” for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we ...

Analysis of Resting-State fMRI Topological Graph Theory Properties in Methamphetamine Drug Users Applying Box-Counting Fractal Dimension

Introduction: Graph theoretical analysis of functional Magnetic Resonance Imaging (fMRI) data has provided new measures of mapping human brain in vivo. Of all methods to measure the functional connectivity between regions, Linear Correlation (LC) calculation of activity time series of the brain regions as a linear measure is considered the most ubiquitous one. The strength of the dependence obl...

Finite Sets and Counting

These notes present an approach to obtaining the basic properties of the natural numbers in terms of the properties of finite sets (which are introduced independently of the natural numbers). The role played by the usual recursion theorem is taken over by a construction which can be regarded as a recursion theorem for finite sets.

Notes on counting finite sets

Embedding Finite Metric Spaces in Low Dimension

This paper presents novel techniques that allow the solution to several open problems regarding embedding of finite metric spaces into Lp. We focus on proving near optimal bounds on the dimension with which arbitrary metric spaces embed into Lp. The dimension of the embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no ...