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Using Fractal Dimensionality in the Search for Source Models of Ultra-High Energy Cosmic Rays

Although the existence of cosmic rays with energies extending well above $10^{19}$ eV has been confirmed, their origin remains one of the most important questions in astro-particle physics today. Several source models have been proposed for the observed set of Ultra High Energy Cosmic Rays (UHECRs). Yet none of these models have been conclusively identified as corresponding with all of the available data. One possible way of achieving a global test of anisotropy is through the measurement of the information dimension, $D_{\rm I}$, of the arrival directions of a sample of events. $D_{\rm I}$ is a measure of the intrinsic heterogeneity of a data sample. We will show how this method can be used to take into account the extreme asymmetric angular resolution and the highly irregular aperture of a monocular air-fluorescence detector. We will then use a simulated, isotropic event sample to show how this method can be used to place upper limits on any number of source models with no statistical penalty.