|
From: Osher Doctorow osher@ix.netcom.com, Friday July 14, 2000,
8:47AM
Dear Colleagues:
Dr. S. J. Vickers has found two typographical
errors in my July 12 contribution which might well lead someone to conclude that
an erroneous one-sided operation on an inequality had been made. The
statement "....so the converted form reads: (1 - x + y)y^^2 = -/1 - x + y/y^^2
> k," shoud have a second constant k1 or k2 (it is arbitrary which notation
is used) replacing k. I used the same k by a typographical error and also
because I had skipped an intermediate step and in my haste used the same
constant k from before. The intermediate step was merely to consider what
happens to xy when the conversion of y/x to 1 - x + y is made. Then
xy converts to (1 - x + y)y^^2 = -/1 - x + y/y^^2 in the case when 1 - x +
y < 0, and this is obviously nonpositive, so -/1 - x + y/y^^2 < k2 with k2
= 0 for example. For 1 - x + y > 0, we have xy converting
to /1 - x + y/y^^2 which is nonnegative and therefore /1 - x + y/y^^2 >
= k2 with k2 = 0 again.
Vickers' criticism turned out to be very
fortunate, not only for clarifying the typographical error and avoiding the
wrong conclusion that I operated on only one side of an inequality when
converting xy to the other form, but also in my developing a detailed
argument concerning when the conversion from y/x to 1 - x + y becomes an actual
function. This occurs, for example, when y/x is a reduced
proper or improper fraction in the sense that numerator and
denominator have no common primes, in which case the unique factorization into
primes and consideration of the three cases y/x > 1 and y/x < 1 and y/x =
1 leads to the conclusion that the conversion is a function. Thus, on the
reduced rationals, we have a function. This is not a bad set to work with
mathematically, and certainly provides a nontrivial case where the conversion is
a function and is accurate.
I hope that S. J. Vickers will continue to
contribute to further discussion in this thread because of his important
contributions, provided of course that he continues to emphasize the
correction of errors and ways of further applying the conversions. If
somebody finds any further errors in my future writings, please give me the
benefit of considering the possibility that I made a typographical
error.
Osher Doctorow
|