🚀 go-pugleaf

In article 92Jun10223140@sgi417.msd.lmsc.lockheed.com, slagle@lmsc.lockheed.com (Mark Slagle) writes:
>In article <l3b57gINNloo@jethro.Corp.Sun.COM>, jdr@starflight.Corp.Sun.COM (Jon Roland) writes:
>
>> It arises from the observation that
>>                n
>> lim ( 1 - 1/n )  -> ~.37
>> n>10
>
>> To get an intuitive sense of it, try evaluating the formula for various
>> values of n, which is interpreted as the annual rate of loss of some
>> critical material to the external environment (an entropic sink, from 
>> which recovery is not economically feasible), and the expression following
>> the limit is the amount remaining after n years.
>
>In other words it is a pseudo-scientific, quasi-mathematical
>numbers game that is applicable mostly to fanciful theories of
>apocalypse.  It would also appear to be singularly useful for
>blowing smoke in the faces of the credulous.

It has been useful for a few more things than that. If we let the above
limit be represented by f, for "failure point", then

                   -n
1/f = lim (1 - 1/n)   
      n>..

Compare this to the definition of e, the base of natural logarithms

                   n
  e = lim (1 + 1/n)
      n>..

We get the first by the simple observation that if we reduce the amount
of some initial quantity by the same fraction n each time period for n
such time periods, then for values of n larger than about 10, the amount
remaining converges on a fixed number, which just happens to be about
what you would have left from most resource stocks if you had originally
designed the things out of which they were made with about a 200 percent
safety margin.

We can get almost any other value for the failure point by applying a
constant coefficient, but the point is the relationship between the
values of the periodic loss rate and the number of periods that a given
system can endure. We have that the one varies as the reciprocal of the
other, and thus, we have a good way to estimate either how much we need
to reduce periodic losses in order to have the system last a given time,
or how long the system will last given the periodic loss rate.

I leave it as an exercise for the reader to prove whether e = 1/f, without
looking it up in a textbook.

---

jdr@starflight.corp.sun.com, starflt@uunet.uu.net
Jon Roland
Starflight Corporation, 1755 E Bayshore Rd #9A,
Redwood City, CA 94063-4142, 415/361-8141