Formal Results Regarding Metric-Space Techniques for the Study of Astrophysical Maps

Abstract

We extend a newly developed formal system for the description of astrophysical maps. In this formalism, we consider the difference between maps to be the distance between elements of a pseudometric space (the space of all such maps). This ansatz allows us to measure quantitatively the difference between any two maps and to order the space of all maps. For each physical characteristic of interest, this technique assigns an ``output'' function to each map; the difference between the maps is then determined from the difference between their corresponding output functions. In this present study, we show that the results of this procedure are invariant under a class of transformations of the maps and the domains of the maps. In addition, we study the propagation of errors (observational uncertainties) through this formalism. We show that the uncertainties in the output functions can be controlled provided that the signal to noise ratios in the original astrophysical maps are sufficiently high. The results of this paper thus increase the effectiveness of this formal system for the description, classification, and analysis of astrophysical maps.