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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.

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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