# Exponential growth

From: Greg Kuperberg <greg_at_MATH.UCDAVIS.EDU>
Date: Tue, 7 Nov 2000 11:34:16 -0800

Exponential and linear are examples of mathematical terms whose
lay connotations have strayed somewhat from their rigorous meanings.
Many people say "exponentially" when they really mean "quickly", as in
"journal prices are rising exponentially". If journal prices rose by
0.5% per year (for the sake of argument, after adjusted for inflation),
that would be exponential. But I assume that libraries would prefer
that to seeing journal prices rise by 10 cents per page per year,
even though that is linear. If the exponent varies over time, as it
usually does in the real world, then exponentiation is only a point of
view and not a predictive law. Any trajectory is exponential with a
time-dependent exponent.

Often a system described in exponential language actually follows a
power law. One common reason is that the system expands first in the
locales where it can expand quickly, and then later where it expands
more slowly. For example, HIV/AIDS never spread in the United States
with a constant exponent; I have heard that the curve of total infections
was, at the beginning, closer to a cubic law. A more relevant example
is new submissions per month to the arXiv, whose growth is strikingly
close to linear:

http://xxx.lanl.gov/cgi-bin/show_monthly_submissions

It is also germane to call this a power law, because if new submissions
grow linearly, total submissions grow quadratically. And I suspect
the usual reason, because the first research areas in the arXiv were
turbulent ones such as string theory and quantum computation. More sedate
topics such as enumerative combinatorics and granular materials only came
much later. I don't see why an alternative model, such as distributed
interoperability, would be exempt from the general principle.

Scientifically, then, I can't accept claims that a new standard or a
new project for e-prints will grow exponentially. Mathematically such
claims do not entirely imply the intended hype anyway.
```--
/\  Greg Kuperberg (UC Davis)
/  \
\  / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/  * All the math that's fit to e-print *
```
Received on Mon Jan 24 2000 - 19:17:43 GMT

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