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Tue Jul 10 12:34:06 EDT 2012

## Understanding Density

Some things have popped up lately that are all related to the problem
of density: a bunch of discrete things interact in an intractable way,
but when this exact representation is integrated or averaged it can
represent a very nice model. Usually when 1. the granularity is fine
enough and 2. the entropy is high enough, cross terms drop out and
there is a (information-reducing) "morphism" between operations in the
discrete domain and operations in the density domain.
I wonder, how to qualify "enough" for the properties of granularity
and entropy.
Some concrete examples:
- The use of money in the form of discrete transactions in time. To
map this to a density model (money stream) makes it easier to think
about.
- Sigma/Delta modulators: all operations are discrete both in time and
space: fixed sampling frequency and 0/1 bit levels, but the part of
the signal we're interested in is a forgetful time integral = low
pass filter.
- Sound is a phenomenon that can be modeled as fluctuations of
densities of particle positions and velocities.
All density representations seem to be based on some form of smoothing
= convolution of a discrete distribution with a smooth, local kernel
like, e.g. a bell curve.

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