Kavli Affiliate: David N. Spergel
| First 5 Authors: Svitlana Mayboroda, David N Spergel, , ,
| Summary:
Mesoscale structures can often be described as fractional dimensional across
a wide range of scales. We consider a $gamma$ dimensional measure embedded in
an $N$ dimensional space and discuss how to determine its dimension, both in
$N$ dimensions and projected into $D$ dimensions.
It is a highly non-trivial problem to decode the original geometry from lower
dimensional projection of a high-dimensional measure. The projections are
space-feeling, the popular box-counting techniques do not apply, and the
Fourier methods are contaminated by aliasing effects. In the present paper we
demonstrate that under the "Copernican hypothesis” that we are not observing
objects from a special direction, projection in a wavelet basis is remarkably
simple: the wavelet power spectrum of a projected $gamma$ dimensional measure
is $P_j propto 2^{-jgamma}$. This holds regardless of the embedded dimension,
$N$, and the projected dimension, $D$. This approach could have potentially
broad applications in data sciences where a typically sparse matrix encodes
lower dimensional information embedded in an extremely high dimensional field
and often measured in projection to a low dimensional space.
Here, we apply this method to JWST and Chandra observations of the nearby
supernova Cas A. We find that the emissions can be represented by projections
of mesoscale substructures with fractal dimensions varying from $gamma = 1.7$
for the warm CO layer observed by JWST, up to $gamma = 2.5$ for the hot X-ray
emitting gas layer in the supernova remnant. The resulting power law indicates
that the emission is coming from a fractal dimensional mesoscale structure
likely produced by magneto-hydrodynamical instabilities in the expanding
supernova shell.
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