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 *