FOM: Fw: the 3 circles case / Re: 106:Degenerative Cloning

[Wlodzimierz Holsztynski]

Thank U Harvey for your "sinfully" entertaining
letter. It almost feels unfair that at the same
time it is related to profound matters.

> THEOREM 2. The largest number of times
> that single degenerative cloning can
> occur in any given finite set of k >= 1
> pairwise disjoint circles is exactly one
> less than an exponential stack of 2's of
> height k. This maximum is realized with k
> concentric circles. E.g., 1,  3,  15,  65,535,
> (2^65,536)-1 for 1,2,3,4,5 concentric circles,
> respectively.

Is this just a bound or is it a bound which actually
is achieved?  In the case of  k=3  circles:

        ((()))

I seem to be able to show that the maximal number
of single degenerative clonings is only 8:
 
indeed, it is trivial to show that it never helps
to kill a circle (with no circles inside) hence
it should be delayed until all circles contain no
other circles.  This leaves us with 2 options for
the first d-cloning:
 
   (()) (())

or

   (()())

The second d-cloning is preferable because after the
next d-cloning it achieves a configuration isomorphic
to the first one (but in 2 steps, gaining 1 step).
Hence the next configuration, obtained from the
original with 2 d-clonings is:
 
    (()) (())
 
(we had no other choice but for killing poor small
circles).
 
Now, after 2 more d-clonings (after a total of 4)
we get:
 
    ()()()()

It takes 4 more d-clonings for this to die.
The maximal total number of d-clonings seems
to be  8  rather than  15.

Best regards,

    Wlodzimierz Holsztynski

PS. Are any articles related to this
topic available on the Internet?