[seqfan] Re: Chewing an old bone

David Wilson davidwwilson at comcast.net
Tue Oct 25 05:31:37 CEST 2011

On 10/24/2011 8:21 PM, David Applegate wrote:
>> I really hope that the qlog plot will remain *supplementary* to the
>> log plot (for large signed sequences), and will not replace it.  Each
>> of these plots can be the more useful one depending on applications.
> Do you have an example of when plotting log(|a(n)|+1) or
> log(a(n)+1) is more informative than plotting qlog(a(n))?
> David Applegate

- For positive n, a qlog(n) plot retains all of the visual information 
of a log(|n|+1) plot.
- qlog(n) is odd and monotonically increasing, hence order and sign 
preserving. In a
qlog(n) plot, positive values lie above the x-axis, negative below. 
Increasing sequences
trend upward, decreasing downward. None of this is true of the 
log(|n|+1) plot.
- For all positive n, qlog(n) is closer to the holy grail, log(n), than 
is log(|n|+1).
- Admittedly, the qlog(n) scale is not a standard plotting scale, but 
then, neither is
the log(|n|+1) scale.  Apart from the OEIS, what other source uses the 
log(|n|+1) scale?
Is it a builtin in any graphing package?

For these reasons, the qlog(n) scale is the equal or superior of the 
log(|n|+1) in every
respect. To answer David Applegate's question, there is no information 
in a log(|n|+1)
plot that is not better represented in a qlog(n) plot. Indeed, this was 
my motivation for
constructing the qlog(n) scale in the first place: to address the 
deficits of the log(|n|+1)
scale in regard to plotting negative values, while retaining its virtues 
with regard to
plotting large ranges (and by sheer luck, qlog(n) happens to be closer 
to log(n)).

The new scale was intended to supplant the old, not to supplement it. If 
we adopt
the qlog(n) scale, the log(|a(n)|+1) scale becomes superfluous and can 
be safely retired.
There would be no technical reason to keep it, and its weakness with 
regard to plotting
negative values would argue to retire it.

In the end I suppose it is NJAS's decision. I hope he reads this mail.

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